Introduction to special relativity
Encyclopedia
In physics
, special relativity
is a fundamental theory
concerning space
and time
, developed by Albert Einstein
in 1905 as a modification of Galilean relativity
. (See "History of special relativity
" for a detailed account and the contributions of Hendrik Lorentz
and Henri Poincaré
.) The theory was able to explain some pressing theoretical
and experimental
issues in the physics of the time involving light
and electrodynamics, such as the failure of the 1887 Michelson–Morley experiment, which aimed to measure differences in the relative speed of light due to the Earth's motion
through the hypothetical luminiferous aether
. The aether was then considered to be the medium of propagation of electromagnetic waves such as light.
Einstein postulated that the speed of light
in free space is the same for all observer
s, regardless of their motion relative to the light source. This postulate stemmed from the assumption that Maxwell's equations
of electromagnetism
(which predict a well-defined speed of light in vacuum) hold in any inertial frame of reference
rather than, as was previously believed, just in the frame of the aether. This prediction contradicted the laws of classical mechanics
, which had been accepted for centuries. Einstein's approach was based on thought experiment
s, calculations, and the principle of relativity
(that is, the notion that all physical laws should appear the same to all inertial observers). Today, scientists are so comfortable with the idea that the speed of light is always the same that the metre
is now defined as "the length of the path travelled by light in vacuum during a time interval of of a second."
This means that the speed of light is by definition 299,792,458 m/s (approximately 1079 million kilometres per hour, or 671 million miles per hour).
The predictions of special relativity are almost identical to those of Galilean relativity for most everyday phenomena, in which speeds are much lower than the speed of light, but it makes different, non-obvious predictions for objects moving at very high speeds. These predictions have been experimentally tested on numerous occasions since the theory's inception and were confirmed by those experiments. The first such prediction described by Einstein is the relativity of simultaneity
; observers who are in motion with respect to each other may disagree on whether two events occurred at the same time or one occurred before the other. The other major predictions of special relativity are time dilation
(a moving clock ticks more slowly than when it is at rest with respect to the observer), length contraction
(a moving rod may be found to be shorter than when it is at rest with respect to the observer), and the equivalence of mass
and energy
(written as E = mc2). Special relativity predicts a non-linear velocity addition formula which prevents speeds greater than that of light
from being observed. In 1908, Hermann Minkowski
reformulated the theory based on different postulates of a more geometrical nature.
This approach considers space and time as being different components of a single entity, the spacetime
, which is "divided" in different ways by observers in relative motion. Likewise, energy and momentum
are the components of the four-momentum
, and the electric
and magnetic field
are the components of the electromagnetic tensor
.
As Galilean relativity is now considered an approximation of special relativity valid for low speeds, special relativity is considered an approximation of the theory of general relativity
valid for weak gravitational field
s. General relativity postulates that physical laws should appear the same to all observers (an accelerating
frame of reference being equivalent to one in which a gravitational field acts), and that gravitation is the effect of the curvature of spacetime caused by energy (including mass).
is simply a selection of what constitutes a stationary object. Once the velocity of a certain object is arbitrarily defined to be zero, the velocity of everything else in the universe can be measured relative to that object.There exists a more technical but mathematically convenient description of reference frames. A reference frame may be considered to be an identification of points in space at different times. That is, it is the identification of space points at different times as being the same point. This concept, particularly useful in making the transition to relativistic spacetime, is described in the language of affine space
by VI Arnold in Mathematical Methods in Classical Mechanics, and in the language of fibre bundles by Roger Penrose
in The Road to Reality.
One oft-used example is the difference in motion of objects on a train as seen from an observer on the train, or on standing on a platform as it passes. A passenger on the train would consider the seat in front of them to be unmoving, as it is moving at the same speed and in the same direction as the rest of the objects around the observer. Someone standing on the platform would see exactly the same objects but interpret them very differently, they would conclude that the seat is moving forward, as is the whole train. Each has their own "frame of reference" that they are using to measure velocities, the rest frame of the platform and the rest frame of the train - or simply the platform frame and the train frame.
So, then, which observer is "right"? Can we select one of these frames to be the "correct" one? Or more generally, is there a frame we can select to be the basis for all measurements, an "absolutely stationary" frame?
Aristotle thought that all objects tend to cease moving and came to rest if there were no forces acting on them. He imagined the Earth lying at the centre of the universe (the geocentric model
), unmoving as other objects moved about it. In this worldview, one could select the surface of the Earth as the absolute frame. However, as the geocentric model was challenged and finally fell in the 1500s, it was realized that the Earth was not stationary at all, but both rotating on its axes as well as orbiting the Sun. In this case the Earth is clearly not the absolute frame. But perhaps there is some other frame one could select, perhaps the Sun's?
Galileo challenged this idea and argued that the concept of an absolute frame, and thus absolute velocity, was unreal; all motion was relative. Galileo gave the common-sense 'formula' for adding velocities: if
In modern terms, we expand the application of this concept from velocity to all physical measurements - according to what we now call the Galilean transformation
, there is no absolute frame of reference. An observer on the train has no measurement that distinguishes whether the train is moving forward at a constant speed, or the platform is moving backwards at that same speed. In fact, the difference is completely arbitrary, who is moving and who is not is something that any observer can select as they see fit. When considering trains moving by platforms it is generally convenient to select the frame of reference of the platform, but such a selection would not be convenient when considering planetary motion.
One can use this formula to explore whether or not any possible measurement would remain the same in different reference frames. For instance, if the passenger on the train threw a ball forward, they would measure one velocity for the ball, and the observer on the platform another. Applying the formula above, both would agree that the velocity of the ball is the same. This means that motion is "invariant
". Laws of classical mechanics
, like Newton's second law of motion
, all obey this principle because they have the same form after applying the transformation. As Newton's law involves the derivative of velocity, any constant velocity added in a Galilean transformation to a different reference frame contributes nothing (the derivative of a constant is zero).
The Galilean transformation, the addition of velocities, only applies to frames that are moving at a constant velocity. Objects tend to retain their current velocity due to a property we call inertia
, so frames that refer to such objects are known as inertial reference frames. Note that the Galilean transformation does not apply to acceleration
s, only velocities, and that classical mechanics is not invariant under acceleration. This mirrors the real world, where acceleration is easily distinguishable from smooth motion in any number of ways. For example, if an observer on a train saw a ball roll backward off their table, they would be able to infer that train was accelerating forward. Under classical mechanics, such changes are the subject of force
s. Addition of a time-varying velocity (corresponding to an accelerated reference frame) will change the formula (see pseudo force).
Both the Aristotelian and Galilean views of motion contain an important assumption. Motion is defined as the change of position over time, but both of these quantities, position and time, are not defined within the system. It is assumed, explicitly in the Greek worldview, that space and time lie outside physical existence, they are theoretical constructs and are absolute even if the objects within them are measured relative to each other. The Galilean transform can only be applied because both observers are assumed to be able to measure the same time and space, regardless of their frame. So in spite of there being no absolute motion, it is assumed there is some, perhaps unknowable, absolute space and time.
to light
, and calculus
made the direct calculation of these effects tractable. Over time, new mathematical techniques, notably the Lagrangian
, greatly simplified the application of these physical laws to more complex problems.
As electricity and magnetism were better explored, it became clear that the two concepts were related. Over time, this work culminated in Maxwell's equations
, a set of four equations that could be used to calculate the entirety of electromagnetism. One of the most interesting results of the application of these equations was that it was possible to construct a self-sustaining wave of electrical and magnetic fields that could propagate through space. When reduced, the math demonstrated that the speed of propagation was dependant on two universal constants, and their ratio was the speed of light
. Light was an electromagnetic wave.
Under the classic model, wave
s are displacements within a medium
. In the case of light, the waves were thought to be displacements of a special medium known as the luminiferous aether
, which extend through all space. This being the case, light travels in its own frame of reference, the frame of the aether. According to the Galilean transform, we should be able to measure the difference in velocities between the aether's frame and any other - a universal frame at last.
Designing an experiment to actually carry out this measurement proved very difficult, however, as the speeds and timing involved made accurate measurement difficult. The measurement problem was eventually solved with the Michelson–Morley experiment. To everyone's surprise, no relative motion was seen. Either the aether was travelling at the same velocity as the Earth, difficult to imagine given the Earth's complex motion, or there was no aether. Follow-up experiments tested various possibilities, and by the turn of the century it was becoming increasing difficult to escape the conclusion that the aether did not exist.
These experiments all showed that light simply did not follow the Galilean transformation. And yet it was clear that physical objects emitted light, which led to unsolved problems. If one were to carry out the experiment on the train by "throwing light" instead of balls, if light does not follow the Galilean transformation then the observers should not agree on the results. Yet it was apparent that the universe disagreed; physical systems known to be at great speeds, like distant stars, had physics that were as similar to our own as measurements allowed. Some sort of transformation had to be acting on light, or better, a single transformation for both light and matter.
The development of a suitable transformation to replace the Galilean transformation is the basis of special relativity.
. To gain a sense of what spacetime is like, we must first look at the Euclidean space
of classical Newtonian physics. This approach to explaining the theory of special relativity begins with the concept of "length
".
In everyday experience, it seems that the length of objects remains the same no matter how they are rotated or moved from place to place; as a result the simple length of an object doesn't appear to change or is invariant. However, as is shown in the illustrations below, what is actually being suggested is that length seems to be invariant in a three-dimensional coordinate system.
The length of a line in a two-dimensional Cartesian coordinate system
is given by Pythagoras' theorem
:
One of the basic theorems of vector algebra is that the length of a vector does not change when it is rotated. However, a closer inspection tells us that this is only true if we consider rotations confined to the plane. If we introduce rotation in the third dimension, then we can tilt the line out of the plane. In this case the projection of the line on the plane will get shorter. Does this mean the line's length changes? - obviously not. The world is three-dimensional and in a 3D Cartesian coordinate system the length is given by the three-dimensional version of Pythagoras's theorem:
This is invariant under all rotations. The apparent violation of invariance of length only happened because we were 'missing' a dimension. It seems that, provided all the directions in which an object can be tilted or arranged are represented within a coordinate system, the length of an object does not change under rotations. With time and space considered to be outside the realm of physics itself, under classical mechanic a 3-dimensional coordinate system is enough to describe the world.
Note that invariance of length is not ordinarily considered a principle or law, not even a theorem. It is simply a statement about the fundamental nature of space itself. Space as we ordinarily conceive it is called a three-dimensional Euclidean space
, because its geometrical structure is described by the principles of Euclidean geometry
. The formula for distance between two points is a fundamental property of a Euclidean space, it is called the Euclidean metric tensor (or simply the Euclidean metric). In general, distance formulas are called metric tensors.
Note that rotations are fundamentally related to the concept of length. In fact, one may define length or distance to be that which stays the same (is invariant) under rotations, or define rotations to be that which keep the length invariant. Given any one, it is possible to find the other. If we know the distance formula, we can find out the formula for transforming coordinates in a rotation. If, on the other hand, we have the formula for rotations then we can find out the distance formula.
built on mathematical approaches used in non-euclidean geometry and on the mathematical work of Lorentz and Poincaré. Minkowski showed in 1908 that Einstein's new theory could also be explained by replacing the concept of a separate space and time with a four-dimensional continuum called spacetime. This was a groundbreaking concept, and Roger Penrose
has said that relativity was not truly complete until Minkowski reformulated Einstein's work.
The concept of a four-dimensional space is hard to visualise. It may help at the beginning to think simply in terms of coordinates. In three-dimensional space, one needs three real numbers to refer to a point. In the Minkowski space
, one needs four real numbers (three space coordinates and one time coordinate) to refer to a point at a particular instant of time. This point, specified by the four coordinates, is called an event. The distance between two different events is called the spacetime interval.
A path through the four-dimensional spacetime (usually known as Minkowski space) is called a world line
. Since it specifies both position and time, a particle having a known world line has a completely determined trajectory and velocity. This is just like graphing the displacement of a particle moving in a straight line against the time elapsed. The curve contains the complete motional information of the particle.
In the same way as the measurement of distance in 3D space needed all three coordinates, we must include time as well as the three space coordinates when calculating the distance in Minkowski space (henceforth called M). In a sense, the spacetime interval provides a combined estimate of how far apart two events occur in space as well as the time that elapses between their occurrence.
But there is a problem; time is related to the space coordinates, but they are not equivalent. Pythagoras's theorem treats all coordinates on an equal footing (see Euclidean space
for more details). We can exchange two space coordinates without changing the length, but we can not simply exchange a space coordinate with time — they are fundamentally different. It is an entirely different thing for two events to be separated in space and to be separated in time. Minkowski proposed that the formula for distance needed a change. He found that the correct formula was actually quite simple, differing only by a sign from Pythagoras's theorem:
where c is a constant and t is the time coordinate.Originally Minkowski tried to make his formula look like Pythagoras's theorem by introducing the concept of imaginary
time and writing −1 as i2. But Wilson, Gilbert, Borel and others proposed that this was unnecessary and introduced real time with the assumption that, when comparing coordinate systems, the change of spatial displacements with displacements in time can be negative. This assumption is expressed in differential geometry using a metric tensor
that has a negative coefficient. The different signature of the Minkowski metric means that the Minkowski space has hyperbolic rather than Euclidean geometry.
Multiplication by c, which has the dimensions
, converts the time to units of length and this constant has the same value as the speed of light
. So the spacetime interval between two distinct events is given by
There are two major points to be noted. Firstly, time is being measured in the same units as length by multiplying it by a constant conversion factor. Secondly, and more importantly, the time-coordinate has a different sign than the space coordinates. This means that in the four-dimensional spacetime, one coordinate is different from the others and influences the distance differently. This new 'distance' may be zero or even negative. This new distance formula, called the metric
of the spacetime, is at the heart of relativity. This distance formula is called the metric tensor
of M. This minus sign means that a lot of our intuition about distances can not be directly carried over into spacetime intervals. For example, the spacetime interval between two events separated both in time and space may be zero (see below). From now on, the terms distance formula and metric tensor will be used interchangeably, as will be the terms Minkowski metric and spacetime interval.
In Minkowski spacetime the spacetime interval is the invariant length, the ordinary 3D length is not required to be invariant. The spacetime interval must stay the same under rotations, but ordinary lengths can change. Just like before, we were missing a dimension. Note that everything thus far is merely definitions. We define a four-dimensional mathematical construct which has a special formula for distance, where distance means that which stays the same under rotations (alternatively, one may define a rotation to be that which keeps the distance unchanged).
Now comes the physical part. Rotations in Minkowski space have a different interpretation than ordinary rotations. These rotations correspond to transformations of reference frames. Passing from one reference frame to another corresponds to rotating the Minkowski space. An intuitive justification for this is given below, but mathematically this is a dynamical postulate just like assuming that physical laws must stay the same under Galilean transformations (which seems so intuitive that we don't usually recognise it to be a postulate).
Since by definition rotations must keep the distance same, passing to a different reference frame must keep the spacetime interval between two events unchanged. This requirement can be used to derive an explicit mathematical form for the transformation that must be applied to the laws of physics (compare with the application of Galilean transformations to classical laws) when shifting reference frames. These transformations are called the Lorentz transformations. Just like the Galilean transformations are the mathematical statement of the principle of Galilean relativity in classical mechanics, the Lorentz transformations are the mathematical form of Einstein's principle of relativity. Laws of physics must stay the same under Lorentz transformations. Maxwell's equations
and Dirac's equation satisfy this property, and hence they are relativistically correct laws (but classically incorrect, since they don't transform correctly under Galilean transformations).
With the statement of the Minkowski metric, the common name for the distance formula given above, the theoretical foundation of special relativity is complete. The entire basis for special relativity can be summed up by the geometric statement "changes of reference frame correspond to rotations in the 4D Minkowski spacetime, which is defined to have the distance formula given above". The unique dynamical predictions of SR stem from this geometrical property of spacetime. Special relativity may be said to be the physics of Minkowski spacetime. In this case of spacetime, there are six independent rotations to be considered. Three of them are the standard rotations on a plane in two directions of space. The other three are rotations in a plane of both space and time: These rotations correspond to a change of velocity
, and the Minkowski diagram
s devised by him describe such rotations.
As has been mentioned before, one can replace distance formulas with rotation formulas. Instead of starting with the invariance of the Minkowski metric as the fundamental property of spacetime, one may state (as was done in classical physics with Galilean relativity) the mathematical form of the Lorentz transformations and require that physical laws be invariant under these transformations. This makes no reference to the geometry of spacetime, but will produce the same result. This was in fact the traditional approach to SR, used originally by Einstein himself. However, this approach is often considered to offer less insight and be more cumbersome than the more natural Minkowski formalism.
Suppose there are two people, Bill and John, on separate planets that are moving away from each other. Bill and John are on separate planets so they both think that they are stationary. John draws a graph of Bill's motion through space and time and this is shown in the illustration below:
John sees that Bill is moving through space as well as time but Bill thinks he is moving through time alone. Bill would draw the same conclusion about John's motion. In fact, these two views, which would be classically considered a difference in reference frames, are related simply by a coordinate transformation in M. Bill's view of his own world line and John's view of Bill's world line are related to each other simply by a rotation of coordinates. One can be transformed into the other by a rotation of the time axis. Minkowski geometry handles transformations of reference frames in a very natural way.
Changes in reference frame, represented by velocity transformations in classical mechanics, are represented by rotations in Minkowski space. These rotations are called Lorentz transformation
s. They are different from the Galilean transformations because of the unique form of the Minkowski metric. The Lorentz transformations are the relativistic equivalent of Galilean transformations. Laws of physics, in order to be relativistically correct, must stay the same under Lorentz transformations. The physical statement that they must be the same in all inertial reference frames remains unchanged, but the mathematical transformation between different reference frames changes. Newton's laws of motion are invariant under Galilean rather than Lorentz transformations, so they are immediately recognisable as non-relativistic laws and must be discarded in relativistic physics. The Schrödinger equation
is also non-relativistic.
Maxwell's equations are trickier. They are written using vectors and at first glance appear to transform correctly under Galilean transformations. But on closer inspection, several questions are apparent that can not be satisfactorily resolved within classical mechanics (see History of special relativity
). They are indeed invariant under Lorentz transformations and are relativistic, even though they were formulated before the discovery of special relativity. Classical electrodynamics can be said to be the first relativistic theory in physics. To make the relativistic character of equations apparent, they are written using 4-component vector-like quantities called 4-vectors. 4-vectors transform correctly under Lorentz transformations, so equations written using 4-vectors are inherently relativistic. This is called the manifestly covariant form of equations. 4-Vectors form a very important part of the formalism of special relativity.
Proposition 1:
Proof:
Proposition 2:
Proof:
The paths of light rays have a zero spacetime interval, and hence all observers will obtain the same value for the speed of light. Therefore, when assuming that the universe has four dimensions that are related by Minkowski's formula, the speed of light appears as a constant, and does not need to be assumed (postulated) to be constant as in Einstein's original approach to special relativity.
John observes the length of Bill's spacetime interval as:
whereas Bill doesn't think he has traveled in space, so writes:
The spacetime interval, s2, is invariant. It has the same value for all observers, no matter who measures it or how they are moving in a straight line. This means that Bill's spacetime interval equals John's observation of Bill's spacetime interval so:
and
hence
So, if John sees a clock that is at rest in Bill's frame record one second, John will find that his own clock measures between these same ticks an interval t, called coordinate time, which is greater than one second. It is said that clocks in motion slow down, relative to those on observers at rest. This is known as "relativistic time dilation
of a moving clock". The time that is measured in the rest frame of the clock (in Bill's frame) is called the proper time
of the clock.
In special relativity, therefore, changes in reference frame affect time also. Time is no longer absolute. There is no universally correct clock; time runs at different rates for different observers.
Similarly it can be shown that John will also observe measuring rods at rest on Bill's planet to be shorter in the direction of motion than his own measuring rods.It should also be made clear that the length contraction result only applies to rods aligned in the direction of motion. At right angles to the direction of motion, there is no contraction. This is a prediction known as "relativistic length contraction
of a moving rod". If the length of a rod at rest on Bill's planet is
, then we call this quantity the proper length
of the rod. The length
of that same rod as measured on John's planet, is called coordinate length, and given by
.
These two equations can be combined to obtain the general form of the Lorentz transformation in one spatial dimension:
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
is a fundamental theory
Theory
The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...
concerning space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
and time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, developed by Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
in 1905 as a modification of Galilean relativity
Galilean invariance
Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames...
. (See "History of special relativity
History of special relativity
The history of special relativity consists of many theoretical results and empirical findings obtained by Albert Michelson, Hendrik Lorentz, Henri Poincaré and others...
" for a detailed account and the contributions of Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
and Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
.) The theory was able to explain some pressing theoretical
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
and experimental
Experimental physics
Within the field of physics, experimental physics is the category of disciplines and sub-disciplines concerned with the observation of physical phenomena in order to gather data about the universe...
issues in the physics of the time involving light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
and electrodynamics, such as the failure of the 1887 Michelson–Morley experiment, which aimed to measure differences in the relative speed of light due to the Earth's motion
Earth's orbit
In astronomy, the Earth's orbit is the motion of the Earth around the Sun, at an average distance of about 150 million kilometers, every 365.256363 mean solar days .A solar day is on average 24 hours; it takes 365.256363 of these to orbit the sun once in the sense of returning...
through the hypothetical luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....
. The aether was then considered to be the medium of propagation of electromagnetic waves such as light.
Einstein postulated that the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in free space is the same for all observer
Observer (special relativity)
The term observer in special relativity refers most commonly to an inertial reference frame. Less often it may refer to an arbitrary non-inertial reference frame; in particular, a Rindler frame is sometimes called an "accelerating observer". In such cases an inertial reference frame may be called...
s, regardless of their motion relative to the light source. This postulate stemmed from the assumption that Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
of electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
(which predict a well-defined speed of light in vacuum) hold in any inertial frame of reference
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...
rather than, as was previously believed, just in the frame of the aether. This prediction contradicted the laws of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, which had been accepted for centuries. Einstein's approach was based on thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
s, calculations, and the principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
(that is, the notion that all physical laws should appear the same to all inertial observers). Today, scientists are so comfortable with the idea that the speed of light is always the same that the metre
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...
is now defined as "the length of the path travelled by light in vacuum during a time interval of of a second."
This means that the speed of light is by definition 299,792,458 m/s (approximately 1079 million kilometres per hour, or 671 million miles per hour).
The predictions of special relativity are almost identical to those of Galilean relativity for most everyday phenomena, in which speeds are much lower than the speed of light, but it makes different, non-obvious predictions for objects moving at very high speeds. These predictions have been experimentally tested on numerous occasions since the theory's inception and were confirmed by those experiments. The first such prediction described by Einstein is the relativity of simultaneity
Relativity of simultaneity
In physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur...
; observers who are in motion with respect to each other may disagree on whether two events occurred at the same time or one occurred before the other. The other major predictions of special relativity are time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
(a moving clock ticks more slowly than when it is at rest with respect to the observer), length contraction
Length contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...
(a moving rod may be found to be shorter than when it is at rest with respect to the observer), and the equivalence of mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
and energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
(written as E = mc2). Special relativity predicts a non-linear velocity addition formula which prevents speeds greater than that of light
Faster-than-light
Faster-than-light communications and travel refer to the propagation of information or matter faster than the speed of light....
from being observed. In 1908, Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
reformulated the theory based on different postulates of a more geometrical nature.
This approach considers space and time as being different components of a single entity, the spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, which is "divided" in different ways by observers in relative motion. Likewise, energy and momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
are the components of the four-momentum
Four-momentum
In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...
, and the electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
are the components of the electromagnetic tensor
Electromagnetic tensor
The electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism...
.
As Galilean relativity is now considered an approximation of special relativity valid for low speeds, special relativity is considered an approximation of the theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
valid for weak gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
s. General relativity postulates that physical laws should appear the same to all observers (an accelerating
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
frame of reference being equivalent to one in which a gravitational field acts), and that gravitation is the effect of the curvature of spacetime caused by energy (including mass).
Reference frames and Galilean relativity: a classical prelude
A reference frameFrame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
is simply a selection of what constitutes a stationary object. Once the velocity of a certain object is arbitrarily defined to be zero, the velocity of everything else in the universe can be measured relative to that object.There exists a more technical but mathematically convenient description of reference frames. A reference frame may be considered to be an identification of points in space at different times. That is, it is the identification of space points at different times as being the same point. This concept, particularly useful in making the transition to relativistic spacetime, is described in the language of affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
by VI Arnold in Mathematical Methods in Classical Mechanics, and in the language of fibre bundles by Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
in The Road to Reality.
One oft-used example is the difference in motion of objects on a train as seen from an observer on the train, or on standing on a platform as it passes. A passenger on the train would consider the seat in front of them to be unmoving, as it is moving at the same speed and in the same direction as the rest of the objects around the observer. Someone standing on the platform would see exactly the same objects but interpret them very differently, they would conclude that the seat is moving forward, as is the whole train. Each has their own "frame of reference" that they are using to measure velocities, the rest frame of the platform and the rest frame of the train - or simply the platform frame and the train frame.
So, then, which observer is "right"? Can we select one of these frames to be the "correct" one? Or more generally, is there a frame we can select to be the basis for all measurements, an "absolutely stationary" frame?
Aristotle thought that all objects tend to cease moving and came to rest if there were no forces acting on them. He imagined the Earth lying at the centre of the universe (the geocentric model
Geocentric model
In astronomy, the geocentric model , is the superseded theory that the Earth is the center of the universe, and that all other objects orbit around it. This geocentric model served as the predominant cosmological system in many ancient civilizations such as ancient Greece...
), unmoving as other objects moved about it. In this worldview, one could select the surface of the Earth as the absolute frame. However, as the geocentric model was challenged and finally fell in the 1500s, it was realized that the Earth was not stationary at all, but both rotating on its axes as well as orbiting the Sun. In this case the Earth is clearly not the absolute frame. But perhaps there is some other frame one could select, perhaps the Sun's?
Galileo challenged this idea and argued that the concept of an absolute frame, and thus absolute velocity, was unreal; all motion was relative. Galileo gave the common-sense 'formula' for adding velocities: if
- particle P is moving at velocity v with respect to reference frame A and
- reference frame A is moving at velocity u with respect to reference frame B, then
- the velocity of P with respect to B is given by v + u.
In modern terms, we expand the application of this concept from velocity to all physical measurements - according to what we now call the Galilean transformation
Galilean transformation
The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation point of view...
, there is no absolute frame of reference. An observer on the train has no measurement that distinguishes whether the train is moving forward at a constant speed, or the platform is moving backwards at that same speed. In fact, the difference is completely arbitrary, who is moving and who is not is something that any observer can select as they see fit. When considering trains moving by platforms it is generally convenient to select the frame of reference of the platform, but such a selection would not be convenient when considering planetary motion.
One can use this formula to explore whether or not any possible measurement would remain the same in different reference frames. For instance, if the passenger on the train threw a ball forward, they would measure one velocity for the ball, and the observer on the platform another. Applying the formula above, both would agree that the velocity of the ball is the same. This means that motion is "invariant
Invariant (physics)
In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...
". Laws of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, like Newton's second law of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
, all obey this principle because they have the same form after applying the transformation. As Newton's law involves the derivative of velocity, any constant velocity added in a Galilean transformation to a different reference frame contributes nothing (the derivative of a constant is zero).
The Galilean transformation, the addition of velocities, only applies to frames that are moving at a constant velocity. Objects tend to retain their current velocity due to a property we call inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
, so frames that refer to such objects are known as inertial reference frames. Note that the Galilean transformation does not apply to acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
s, only velocities, and that classical mechanics is not invariant under acceleration. This mirrors the real world, where acceleration is easily distinguishable from smooth motion in any number of ways. For example, if an observer on a train saw a ball roll backward off their table, they would be able to infer that train was accelerating forward. Under classical mechanics, such changes are the subject of force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s. Addition of a time-varying velocity (corresponding to an accelerated reference frame) will change the formula (see pseudo force).
Both the Aristotelian and Galilean views of motion contain an important assumption. Motion is defined as the change of position over time, but both of these quantities, position and time, are not defined within the system. It is assumed, explicitly in the Greek worldview, that space and time lie outside physical existence, they are theoretical constructs and are absolute even if the objects within them are measured relative to each other. The Galilean transform can only be applied because both observers are assumed to be able to measure the same time and space, regardless of their frame. So in spite of there being no absolute motion, it is assumed there is some, perhaps unknowable, absolute space and time.
Classical physics and electromagnetism
Through the era between Newton and the turn of the 20th century, the development of classical physics had made great strides. Newton's application of the inverse square law to gravity was the key to unlocking a wide variety of physical events, from heatHeat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
to light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
, and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
made the direct calculation of these effects tractable. Over time, new mathematical techniques, notably the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
, greatly simplified the application of these physical laws to more complex problems.
As electricity and magnetism were better explored, it became clear that the two concepts were related. Over time, this work culminated in Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
, a set of four equations that could be used to calculate the entirety of electromagnetism. One of the most interesting results of the application of these equations was that it was possible to construct a self-sustaining wave of electrical and magnetic fields that could propagate through space. When reduced, the math demonstrated that the speed of propagation was dependant on two universal constants, and their ratio was the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. Light was an electromagnetic wave.
Under the classic model, wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
s are displacements within a medium
Transmission medium
A transmission medium is a material substance that can propagate energy waves...
. In the case of light, the waves were thought to be displacements of a special medium known as the luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....
, which extend through all space. This being the case, light travels in its own frame of reference, the frame of the aether. According to the Galilean transform, we should be able to measure the difference in velocities between the aether's frame and any other - a universal frame at last.
Designing an experiment to actually carry out this measurement proved very difficult, however, as the speeds and timing involved made accurate measurement difficult. The measurement problem was eventually solved with the Michelson–Morley experiment. To everyone's surprise, no relative motion was seen. Either the aether was travelling at the same velocity as the Earth, difficult to imagine given the Earth's complex motion, or there was no aether. Follow-up experiments tested various possibilities, and by the turn of the century it was becoming increasing difficult to escape the conclusion that the aether did not exist.
These experiments all showed that light simply did not follow the Galilean transformation. And yet it was clear that physical objects emitted light, which led to unsolved problems. If one were to carry out the experiment on the train by "throwing light" instead of balls, if light does not follow the Galilean transformation then the observers should not agree on the results. Yet it was apparent that the universe disagreed; physical systems known to be at great speeds, like distant stars, had physics that were as similar to our own as measurements allowed. Some sort of transformation had to be acting on light, or better, a single transformation for both light and matter.
The development of a suitable transformation to replace the Galilean transformation is the basis of special relativity.
Invariance of length: the Euclidean picture
In special relativity, space and time are joined into a unified four-dimensional continuum called spacetimeSpacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. To gain a sense of what spacetime is like, we must first look at the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
of classical Newtonian physics. This approach to explaining the theory of special relativity begins with the concept of "length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
".
In everyday experience, it seems that the length of objects remains the same no matter how they are rotated or moved from place to place; as a result the simple length of an object doesn't appear to change or is invariant. However, as is shown in the illustrations below, what is actually being suggested is that length seems to be invariant in a three-dimensional coordinate system.
The length of a line in a two-dimensional Cartesian coordinate system
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
is given by Pythagoras' theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
:
One of the basic theorems of vector algebra is that the length of a vector does not change when it is rotated. However, a closer inspection tells us that this is only true if we consider rotations confined to the plane. If we introduce rotation in the third dimension, then we can tilt the line out of the plane. In this case the projection of the line on the plane will get shorter. Does this mean the line's length changes? - obviously not. The world is three-dimensional and in a 3D Cartesian coordinate system the length is given by the three-dimensional version of Pythagoras's theorem:
This is invariant under all rotations. The apparent violation of invariance of length only happened because we were 'missing' a dimension. It seems that, provided all the directions in which an object can be tilted or arranged are represented within a coordinate system, the length of an object does not change under rotations. With time and space considered to be outside the realm of physics itself, under classical mechanic a 3-dimensional coordinate system is enough to describe the world.
Note that invariance of length is not ordinarily considered a principle or law, not even a theorem. It is simply a statement about the fundamental nature of space itself. Space as we ordinarily conceive it is called a three-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, because its geometrical structure is described by the principles of Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. The formula for distance between two points is a fundamental property of a Euclidean space, it is called the Euclidean metric tensor (or simply the Euclidean metric). In general, distance formulas are called metric tensors.
Note that rotations are fundamentally related to the concept of length. In fact, one may define length or distance to be that which stays the same (is invariant) under rotations, or define rotations to be that which keep the length invariant. Given any one, it is possible to find the other. If we know the distance formula, we can find out the formula for transforming coordinates in a rotation. If, on the other hand, we have the formula for rotations then we can find out the distance formula.
The Minkowski formulation: introduction of spacetime
After Einstein derived special relativity formally from the (at first sight counter-intuitive) assumption that the speed of light is the same to all observers, Hermann MinkowskiHermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
built on mathematical approaches used in non-euclidean geometry and on the mathematical work of Lorentz and Poincaré. Minkowski showed in 1908 that Einstein's new theory could also be explained by replacing the concept of a separate space and time with a four-dimensional continuum called spacetime. This was a groundbreaking concept, and Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
has said that relativity was not truly complete until Minkowski reformulated Einstein's work.
The concept of a four-dimensional space is hard to visualise. It may help at the beginning to think simply in terms of coordinates. In three-dimensional space, one needs three real numbers to refer to a point. In the Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
, one needs four real numbers (three space coordinates and one time coordinate) to refer to a point at a particular instant of time. This point, specified by the four coordinates, is called an event. The distance between two different events is called the spacetime interval.
A path through the four-dimensional spacetime (usually known as Minkowski space) is called a world line
World line
In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
. Since it specifies both position and time, a particle having a known world line has a completely determined trajectory and velocity. This is just like graphing the displacement of a particle moving in a straight line against the time elapsed. The curve contains the complete motional information of the particle.
In the same way as the measurement of distance in 3D space needed all three coordinates, we must include time as well as the three space coordinates when calculating the distance in Minkowski space (henceforth called M). In a sense, the spacetime interval provides a combined estimate of how far apart two events occur in space as well as the time that elapses between their occurrence.
But there is a problem; time is related to the space coordinates, but they are not equivalent. Pythagoras's theorem treats all coordinates on an equal footing (see Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
for more details). We can exchange two space coordinates without changing the length, but we can not simply exchange a space coordinate with time — they are fundamentally different. It is an entirely different thing for two events to be separated in space and to be separated in time. Minkowski proposed that the formula for distance needed a change. He found that the correct formula was actually quite simple, differing only by a sign from Pythagoras's theorem:
where c is a constant and t is the time coordinate.Originally Minkowski tried to make his formula look like Pythagoras's theorem by introducing the concept of imaginary
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
time and writing −1 as i2. But Wilson, Gilbert, Borel and others proposed that this was unnecessary and introduced real time with the assumption that, when comparing coordinate systems, the change of spatial displacements with displacements in time can be negative. This assumption is expressed in differential geometry using a metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
that has a negative coefficient. The different signature of the Minkowski metric means that the Minkowski space has hyperbolic rather than Euclidean geometry.
Multiplication by c, which has the dimensions
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
, converts the time to units of length and this constant has the same value as the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. So the spacetime interval between two distinct events is given by
There are two major points to be noted. Firstly, time is being measured in the same units as length by multiplying it by a constant conversion factor. Secondly, and more importantly, the time-coordinate has a different sign than the space coordinates. This means that in the four-dimensional spacetime, one coordinate is different from the others and influences the distance differently. This new 'distance' may be zero or even negative. This new distance formula, called the metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
of the spacetime, is at the heart of relativity. This distance formula is called the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
of M. This minus sign means that a lot of our intuition about distances can not be directly carried over into spacetime intervals. For example, the spacetime interval between two events separated both in time and space may be zero (see below). From now on, the terms distance formula and metric tensor will be used interchangeably, as will be the terms Minkowski metric and spacetime interval.
In Minkowski spacetime the spacetime interval is the invariant length, the ordinary 3D length is not required to be invariant. The spacetime interval must stay the same under rotations, but ordinary lengths can change. Just like before, we were missing a dimension. Note that everything thus far is merely definitions. We define a four-dimensional mathematical construct which has a special formula for distance, where distance means that which stays the same under rotations (alternatively, one may define a rotation to be that which keeps the distance unchanged).
Now comes the physical part. Rotations in Minkowski space have a different interpretation than ordinary rotations. These rotations correspond to transformations of reference frames. Passing from one reference frame to another corresponds to rotating the Minkowski space. An intuitive justification for this is given below, but mathematically this is a dynamical postulate just like assuming that physical laws must stay the same under Galilean transformations (which seems so intuitive that we don't usually recognise it to be a postulate).
Since by definition rotations must keep the distance same, passing to a different reference frame must keep the spacetime interval between two events unchanged. This requirement can be used to derive an explicit mathematical form for the transformation that must be applied to the laws of physics (compare with the application of Galilean transformations to classical laws) when shifting reference frames. These transformations are called the Lorentz transformations. Just like the Galilean transformations are the mathematical statement of the principle of Galilean relativity in classical mechanics, the Lorentz transformations are the mathematical form of Einstein's principle of relativity. Laws of physics must stay the same under Lorentz transformations. Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
and Dirac's equation satisfy this property, and hence they are relativistically correct laws (but classically incorrect, since they don't transform correctly under Galilean transformations).
With the statement of the Minkowski metric, the common name for the distance formula given above, the theoretical foundation of special relativity is complete. The entire basis for special relativity can be summed up by the geometric statement "changes of reference frame correspond to rotations in the 4D Minkowski spacetime, which is defined to have the distance formula given above". The unique dynamical predictions of SR stem from this geometrical property of spacetime. Special relativity may be said to be the physics of Minkowski spacetime. In this case of spacetime, there are six independent rotations to be considered. Three of them are the standard rotations on a plane in two directions of space. The other three are rotations in a plane of both space and time: These rotations correspond to a change of velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
, and the Minkowski diagram
Minkowski diagram
The Minkowski diagram was developed in 1908 by Hermann Minkowski and provides an illustration of the properties of space and time in the special theory of relativity. It allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical...
s devised by him describe such rotations.
As has been mentioned before, one can replace distance formulas with rotation formulas. Instead of starting with the invariance of the Minkowski metric as the fundamental property of spacetime, one may state (as was done in classical physics with Galilean relativity) the mathematical form of the Lorentz transformations and require that physical laws be invariant under these transformations. This makes no reference to the geometry of spacetime, but will produce the same result. This was in fact the traditional approach to SR, used originally by Einstein himself. However, this approach is often considered to offer less insight and be more cumbersome than the more natural Minkowski formalism.
Reference frames and Lorentz transformations: relativity revisited
We have already discussed that in classical mechanics coordinate frame changes correspond to Galilean transformations of the coordinates. Is this adequate in the relativistic Minkowski picture?Suppose there are two people, Bill and John, on separate planets that are moving away from each other. Bill and John are on separate planets so they both think that they are stationary. John draws a graph of Bill's motion through space and time and this is shown in the illustration below:
John sees that Bill is moving through space as well as time but Bill thinks he is moving through time alone. Bill would draw the same conclusion about John's motion. In fact, these two views, which would be classically considered a difference in reference frames, are related simply by a coordinate transformation in M. Bill's view of his own world line and John's view of Bill's world line are related to each other simply by a rotation of coordinates. One can be transformed into the other by a rotation of the time axis. Minkowski geometry handles transformations of reference frames in a very natural way.
Changes in reference frame, represented by velocity transformations in classical mechanics, are represented by rotations in Minkowski space. These rotations are called Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
s. They are different from the Galilean transformations because of the unique form of the Minkowski metric. The Lorentz transformations are the relativistic equivalent of Galilean transformations. Laws of physics, in order to be relativistically correct, must stay the same under Lorentz transformations. The physical statement that they must be the same in all inertial reference frames remains unchanged, but the mathematical transformation between different reference frames changes. Newton's laws of motion are invariant under Galilean rather than Lorentz transformations, so they are immediately recognisable as non-relativistic laws and must be discarded in relativistic physics. The Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
is also non-relativistic.
Maxwell's equations are trickier. They are written using vectors and at first glance appear to transform correctly under Galilean transformations. But on closer inspection, several questions are apparent that can not be satisfactorily resolved within classical mechanics (see History of special relativity
History of special relativity
The history of special relativity consists of many theoretical results and empirical findings obtained by Albert Michelson, Hendrik Lorentz, Henri Poincaré and others...
). They are indeed invariant under Lorentz transformations and are relativistic, even though they were formulated before the discovery of special relativity. Classical electrodynamics can be said to be the first relativistic theory in physics. To make the relativistic character of equations apparent, they are written using 4-component vector-like quantities called 4-vectors. 4-vectors transform correctly under Lorentz transformations, so equations written using 4-vectors are inherently relativistic. This is called the manifestly covariant form of equations. 4-Vectors form a very important part of the formalism of special relativity.
Einstein's postulate: the constancy of the speed of light
Einstein's postulate that the speed of light is a constant comes out as a natural consequence of the Minkowski formulation.Proposition 1:
- When an object is travelling at c in a certain reference frameFrame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
, the spacetime interval is zero.
Proof:
- The spacetime interval between the origin-event (0,0,0,0) and an event (x, y, z, t) is
-
- The distance travelled by an object moving at velocity v for t seconds is:
-
- giving
-
- Since the velocity v equals c we have
-
- Hence the spacetime interval between the events of departure and arrival is given by
-
Proposition 2:
- An object travelling at c in one reference frame is travelling at c in all reference frames.
Proof:
- Let the object move with velocity v when observed from a different reference frame. A change in reference frame corresponds to a rotation in M. Since the spacetime interval must be conserved under rotation, the spacetime interval must be the same in all reference frames. In proposition 1 we showed it to be zero in one reference frame, hence it must be zero in all other reference frames. We get that
-
- which implies
-
The paths of light rays have a zero spacetime interval, and hence all observers will obtain the same value for the speed of light. Therefore, when assuming that the universe has four dimensions that are related by Minkowski's formula, the speed of light appears as a constant, and does not need to be assumed (postulated) to be constant as in Einstein's original approach to special relativity.
Clock delays and rod contractions: more on Lorentz transformations
Another consequence of the invariance of the spacetime interval is that clocks will appear to go slower on objects that are moving relative to you. This is very similar to how the 2D projection of a line rotated into the third-dimension appears to get shorter. Length is not conserved simply because we are ignoring one of the dimensions. Let us return to the example of John and Bill.John observes the length of Bill's spacetime interval as:
whereas Bill doesn't think he has traveled in space, so writes:
The spacetime interval, s2, is invariant. It has the same value for all observers, no matter who measures it or how they are moving in a straight line. This means that Bill's spacetime interval equals John's observation of Bill's spacetime interval so:
and
hence
-
.
So, if John sees a clock that is at rest in Bill's frame record one second, John will find that his own clock measures between these same ticks an interval t, called coordinate time, which is greater than one second. It is said that clocks in motion slow down, relative to those on observers at rest. This is known as "relativistic time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
of a moving clock". The time that is measured in the rest frame of the clock (in Bill's frame) is called the proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
of the clock.
In special relativity, therefore, changes in reference frame affect time also. Time is no longer absolute. There is no universally correct clock; time runs at different rates for different observers.
Similarly it can be shown that John will also observe measuring rods at rest on Bill's planet to be shorter in the direction of motion than his own measuring rods.It should also be made clear that the length contraction result only applies to rods aligned in the direction of motion. At right angles to the direction of motion, there is no contraction. This is a prediction known as "relativistic length contraction
Length contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...
of a moving rod". If the length of a rod at rest on Bill's planet is
, then we call this quantity the proper lengthProper length
In relativistic physics, proper length is an invariant measure of the distance between two spacelike-separated events, or of the length of a spacelike path within a spacetime....
of the rod. The length
of that same rod as measured on John's planet, is called coordinate length, and given by.These two equations can be combined to obtain the general form of the Lorentz transformation in one spatial dimension:
-
or equivalently:-
where the Lorentz factorLorentz factorThe Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
is given by
The above formulas for clock delays and length contractions are special cases of the general transformation.
Alternatively, these equations for time dilation and length contraction (here obtained from the invariance of the spacetime interval), can be obtained directly from the Lorentz transformation by setting X = 0 for time dilation, meaning that the clock is at rest in Bill's frame, or by setting t = 0 for length contraction, meaning that John must measure the distances to the end points of the moving rod at the same time.
A consequence of the Lorentz transformations is the modified velocity-addition formula:
Simultaneity and clock desynchronisation
The last consequence of Minkowski's spacetime is that clocks will appear to be out of phase with each other along the length of a moving object. This means that if one observer sets up a line of clocks that are all synchronised so they all read the same time, then another observer who is moving along the line at high speed will see the clocks all reading different times. This means that observers who are moving relative to each other see different events as simultaneous. This effect is known as "Relativistic Phase" or the "Relativity of Simultaneity". Relativistic phase is often overlooked by students of special relativity, but if it is understood, then phenomena such as the twin paradoxTwin paradoxIn physics, the twin paradox is a thought experiment in special relativity, in which a twin makes a journey into space in a high-speed rocket and returns home to find he has aged less than his identical twin who stayed on Earth...
are easier to understand.
Observers have a set of simultaneous events around them that they regard as composing the present instant. The relativity of simultaneity results in observers who are moving relative to each other having different sets of events in their present instant.
The net effect of the four-dimensional universe is that observers who are in motion relative to you seem to have time coordinates that lean over in the direction of motion, and consider things to be simultaneous that are not simultaneous for you. Spatial lengths in the direction of travel are shortened, because they tip upwards and downwards, relative to the time axis in the direction of travel, akin to a skew or shear of three-dimensional space.
Great care is needed when interpreting spacetime diagrams. Diagrams present data in two dimensions, and cannot show faithfully how, for instance, a zero length spacetime interval appears.
General relativity: a peek forward
Unlike Newton's laws of motion, relativity is not based upon dynamical postulates. It does not assume anything about motion or forces. Rather, it deals with the fundamental nature of spacetime. It is concerned with describing the geometry of the backdrop on which all dynamical phenomena take place. In a sense therefore, it is a meta-theory, a theory that lays out a structure that all other theories must follow. In truth, Special relativity is only a special case. It assumes that spacetime is flat. That is, it assumes that the structure of Minkowski space and the Minkowski metric tensor is constant throughout. In General relativityGeneral relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, Einstein showed that this is not true. The structure of spacetime is modified by the presence of matter. Specifically, the distance formula given above is no longer generally valid except in space free from mass. However, just like a curved surface can be considered flat in the infinitesimal limit of calculus, a curved spacetime can be considered flat at a small scale. This means that the Minkowski metric written in the differential form is generally valid.
One says that the Minkowski metric is valid locally, but it fails to give a measure of distance over extended distances. It is not valid globally. In fact, in general relativity the global metric itself becomes dependent on the mass distribution and varies through space. The central problem of general relativity is to solve the famous Einstein field equationsEinstein field equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
for a given mass distribution and find the distance formula that applies in that particular case. Minkowski's spacetime formulation was the conceptual stepping stone to general relativity. His fundamentally new outlook allowed not only the development of general relativity, but also to some extent quantum field theories.
Mass-energy equivalence: sunlight and atom bombs
Einstein showed that mass is simply another form of energy. The energy equivalent of rest mass m is
. This equivalence implies that mass should be interconvertible with other forms of energy. This is the basic principle behind atom bombs and production of energy in nuclear reactors and stars (like the Sun).
Applications
There is a common perception that relativistic physics is not needed for practical purposes or in everyday life. This is not true. Without relativistic effects, goldGoldGold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...
would look silvery, rather than yellow. Many technologies are critically dependent on relativistic physics:- Cathode ray tubeCathode ray tubeThe cathode ray tube is a vacuum tube containing an electron gun and a fluorescent screen used to view images. It has a means to accelerate and deflect the electron beam onto the fluorescent screen to create the images. The image may represent electrical waveforms , pictures , radar targets and...
s , - Particle acceleratorParticle acceleratorA particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field accelerators.In...
s, - Global Positioning SystemGlobal Positioning SystemThe Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...
(GPS) - although this really requires the full theory of general relativityGeneral relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
The postulates of special relativity
Einstein developed special relativity on the basis of two postulates:
- First postulateAxiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
- Special principle of relativityPrinciple of relativityIn physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
- The laws of physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
are the same in all inertial frames of referenceInertial frame of referenceIn physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...
. In other words, there are no privileged inertial frames of reference. - Second postulate - Invariance of c - The speed of light in a vacuumVacuumIn everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
is independent of the motion of the lightLightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
source.
Special Relativity can be derived from these postulates, as was done by Einstein in 1905. Einstein's postulates are still applicable in the modern theory but the origin of the postulates is more explicit. It was shown above how the existence of a universally constant velocity (the speed of light) is a consequence of modeling the universe as a particular four dimensional space having certain specific properties. The principle of relativity is a result of Minkowski structure being preserved under Lorentz transformations, which are postulated to be the physical transformations of inertial reference frames.
See also
- Introduction to general relativityIntroduction to general relativityGeneral relativity is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915. According to general relativity, the observed gravitational attraction between masses results from their warping of space and time....
- Special relativitySpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
- History of special relativityHistory of special relativityThe history of special relativity consists of many theoretical results and empirical findings obtained by Albert Michelson, Hendrik Lorentz, Henri Poincaré and others...
- Speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
- InvarianceInvariant (physics)In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...
- SpacetimeSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
- Light clock
- Andromeda paradox
- SymmetrySymmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
- Symmetry in physicsSymmetry in physicsIn physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...
Special relativity for a general audience (no math knowledge required)
- Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
- Einstein Online Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.
Special relativity explained (using simple or more advanced math)
- Wikibooks: Special Relativity
- Albert Einstein. Relativity: The Special and General Theory. New York: Henry Holt 1920. BARTLEBY.COM, 2000
- Usenet Physics FAQ
- Sean Carroll's online Lecture Notes on General Relativity
- Hyperphysics Time Dilation
- Hyperphysics Length Contraction
- Greg Egan's Foundations
- Special Relativity Simulation
- A Primer on Special Relativity - MathPages
- Caltech Relativity Tutorial A basic introduction to concepts of Special and General Relativity, requiring only a knowledge of basic geometry.
- Special Relativity in film clips and animations from the University of New South Wales.
- Relativity Calculator - Learn Special Relativity Mathematics Mathematics of special relativity presented in as simple and comprehensive manner possible within philosophical and historical contexts.
- Special relativity made stupid ☺.
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