
List of relativistic equations
Encyclopedia
Lorentz Transformation


The Lorentz transformation:




Time dilation:

Length contraction:

Velocity Subtraction:



Four-Vectors and Matrices
The matrix version of the Lorentz transformation:
The metric tensor:

The space time interval:

Four velocity:

Four-momentum:

Energy in Relativity
Energy and the four-momentum's time term:
Kinetic energy:


Other useful energy-momentum relations:


Four-force:

Doppler Shift
General Doppler shift:
Doppler shift for emitter and observer moving right towards each other (or directly away):

Doppler shift for emitter and observer moving in a direction perpendicular to the line connecting them:

Derivation of the equations of Special Relativity
To derive the equations of special relativity, one must start with two postulates:- The laws of physics are invariant under transformations between inertial frames. Basically, the laws of physics will be the same whether you are testing them in a frame 'at rest', or a frame moving with a constant velocity relative to the 'rest' frame.
- The speed of light in a vacuum is measured to be the same by all observers in inertial frames. So, if I shine a beam of light, and measure how fast it's going, and then start moving really fast, and then measure the speed of the light beam again, both measurements will produce the same result.
From these two postulates, all of special relativity follows.
Time Dilation
Now, with the above postulates in mind, imagine you are in a train, a car, a bus, or any other vehicle, moving with a velocity



However, to the observer on the ground, the situation is very different. Because the train is moving by the observer on the ground, the light beam appears to move diagonally instead of straight up and down. To visualize this, picture the light being emitted at one point, then having the vehicle move until the light hits the mirror at the top of the vehicle, and then having the train move still more until the light beam returns to the bottom of the vehicle. The light beam will have appeared to have moved diagonally upward with the train, and then diagonally downward. This path will help form two-right sided triangles, with the height as one of the sides, and the two straight parts of the path being the respective hypotenuses:

Rearranging to get




Taking out a factor of



This is the formula for time dilation. In particular, the following notations are used very often in special relativity:


This makes the formula for time dilation:

There are a few things worth pointing out here before moving on. First,



Length Contraction and the Lorentz Transformation
Consider a long train, moving with velocity


However, the observer on the ground, making the same measurement, comes to a different conclusion. This observer finds that time


This is the formula for length contraction. As there existed a proper time for time dilation, there exists a proper length for length contraction, which in this case is

If you now plug in this result into the Galilean transformation, substituting




Or:

And going from the primed frame to the unprimed frame:

Going from the primed frame to the unprimed frame was accomplished by making



Finally, to figure out how









Plugging in the value for





Finally, dividing through by


Or more commonly:

And the converse can again be gotten by changing the sign of





Velocity in Relativity
The Lorentz transformations also apply to differentials, so:



Now velocity is




Now, putting in





Also, the velocities in the directions perpendicular to the frame changes are affected, as shown above. This is due to time dilation, as encapsulated in the






The Metric and Four-Vectors
It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace











And, finally, in matrix form:

The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors. In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows:

In the above,





As with four vectors, there is a concept of the dot product, or the inner product. The form of this is:




In the above,


On a final note about the matrix and four-vector formulation of special relativity, the sign of the metric and the placement of the






Velocity and Momentum
The relativistic four-velocity, that is the four-vector representing velocity in relativity, is defined as follows:
In the above,

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...
, called the world-line, followed by the object velocity the above represents. As stated above in the derivation of time dilation, the proper time is the time between two events in a frame of reference where they take place at the same location. With the example of time dilation in mind, one can use the formula for it,



The first three terms, excepting the factor of


The four-momentum of an object is relatively straightforward:

This definition is identical in form to the relation to the classical momentum and the classical velocity. The mass,


In the above, the factor of



This formulation makes the new relation between the spatial velocity and the spatial momentum look practically identical. However, this can be misleading, as it is not appropriate in special relativity in all circumstances. For instance, kinetic energy and force in special relativity can not be written exactly like their classical analogues by only replacing the mass with the relativistic mass. Moreover, under Lorentz transformations, this relativistic mass is not invariant, while the regular mass is. That is why many people find it easier to just stick with the regular mass, and discard the relativistic mass.
Energy
There is one last feature of the four-momentum worth discussing, and that is the role of the time-based term, which will be called

As it turn out, this is directly related to the energy in special relativity as follows:

This is the reason that the momentum four-vector is sometimes called the momentum-energy four-vector. However, it is not clear that this definition corresponds to the definition of the energy of a free particle classically, which is just the particle's kinetic energy. However, if you use the binomial series expansion of



The second term above is the classical kinetic energy. The first term is something new entirely, and reduces to Einstein's famous equation


Taking the above expansion into account, noting that an object with zero velocity has energy equal to


A few other useful energy and momentum relations
If one moves to the rest frame of an object, and then takes the dot-product of their four-momentums, one gets:



As the four-momentum is Lorentz invariant, the dot product of the four-momentum with itself is invariant under Lorentz transformations, so the above relation is true for the four-momentum in any frame of reference. In fact, assuming that there are spatial momentum terms, you can produce another relation:


In the above,

Force
A relativistic force, one that is invariant under the Lorentz transformation, can be formulated in a similar fashion to how the four-velocity was formulated. In the case of the four-velocity, this was arrived at by taking the derivative of four-position with respect to proper time, in an analogous fashion to the classical definition of valeocity as the derivative of position by time. Similarly, classical force was defined as the time derivative of momentum with time. So you can, in a similar fashion you can take the derivative of four-momentum with respect to proper time:
You can simplify the last term by noting that




It should be noted that in the above, it is assumed that the mass is constant over time. Plugging this into the first result:

This is the relativistic four-force, which is invariant under Lorentz transformation.
Doppler Shift
If an object emits a beam of light or radiation, the frequency, wavelength, and energy of that light or radiation will look different to a moving observer than to one at rest with respect to the emitter. If one assumes that the observer is moving with respect to the emitter along the x-axis, then the standard Lorentz transformation of the four-momentum, which includes energy, becomes:

Now, if







This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis. In that case






This is the equation for doppler shift in the case where the velocity between the emitter and observer is along the x-axis. The second special case is that where the relative velocity is perpendicular to the x-axis, and thus



This is actually completely analogous to time dilation, as frequency is one over time. So, doppler shift for emitters and observers moving perpendicular to the line connecting them is completely due to the effects of time dilation.