Einstein's constant
Encyclopedia
Einstein's constant or Einstein's gravitational constant, denoted κ (kappa
), is the coupling constant
appearing in the Einstein field equation
which can be written:
where Gαγ is the Einstein tensor
and Tαγ is the stress-energy tensor
.
This equation relates to the curvature
of space
and time
, telling that stress-energy
is what causes the disturbance of spacetime
, thus gravitation
. Einstein
used Newton's law of universal gravitation
in his field equations, and the constant of κ is found to have a value of:
N.B.: Writing Einstein's constant depends on how the stress-energy tensor is defined, so the Einstein field equations are always invariant (see details in the section "About the two possible writings" further).
Λ is equal to zero is taken, with a steady state hypothesis
. Then we use the Newtonian approximation with hypothesis of a weak field and low velocities with respect to the speed of light.
The Newton law will arise and its corollary Poisson's equation
.
In this approximation, Poisson's equation appears as the approached form of the field equation (or the field equation appears as a generalization of Poisson's equation). The identification gives the expression of Einstein's constant related to quantities G and c.
to describe the geometry of space in the presence of an energy field. Einstein proposed this equation in 1917, written as:
(const) is what will become Einstein's constant. We will take the cosmological constant Λ equal to zero (one of the requirements of the properties of the gravitational equations is that they reduce to the free-space field equations when the density of energy in space Tαγ is zero, therefore that the cosmological constant λ appearing in this equation is zero) so the field equation becomes:
where Rαγ s the Ricci tensor, gαγ is the metric tensor, R the scalar curvature and κ is Einstein's constant we will calculate in the next section.
This equation can be written in another form, contracting indexes:
Thus:
where T is the scalar Tαα which we shall refer to as the Laue scalar.
Using this result we can write the field equation as:
|
>
and 
respectively indicate 
and 
. Thus, 
means 
We will consider a field of matter with low proper density ρ, moving at low velocity v. The stress-energy tensor can be written:
If the terms of order
and 
are neglected, it becomes:
We assume the flow to be stationary and therefore expect the metric to be time-independent. We use the coordinates of special relativity ct, x, y, z that we write as x0, x1, x2, and x3. The first coordinate is time, and the three others are the space coordinates.
Applying a perturbation method
, we will consider a metric appearing through a two-terms summation. The first is the Lorentz metric, ημν which is that of the Minkowski space
, locally flat. Formulating we get:
The second term corresponds to the small perturbation (due to the presence of a gravitating body) and is also time-independent:
Thus we write the metric:
Clarifying the length element:
If we neglect terms of order
, the Laue scalar 
is:
And the right side of the field equations is to first order in all the small quantities
, 
and 
is written:
Neglecting second-order terms in
gives the following approximate form for the contracted Riemann tensor:
Thus the approximate field equations may be expressed as:
At first let us consider the case μ = ν = 0. As the metric is time-independent, the first term of the equation above is zero. What remains is:
The Christoffel symbol
of the first kind is defined by:
Since the Lorentz metric is constant in space and time, this simplifies to:
Moreover
is time-independent, so [00,0] is zero. Neglecting second-order terms in the perturbation term 
, we get:
which is zero for β = 0 (which then corresponds to the derivative with respect to time). Substituting inside (*) we obtain the following approximate field equation for
:
or, by virtue of time independence:
This notation is just a writing convention. The equation can be written:
which can be identified to Poisson's equation if we write:
Therefore we have established that the classical theory (Poisson's equation) is the limiting case (weak field, low velocities with respect to the speed of light) of a relativistic theory where the metric is time-independent.
To be complete, gravity has to be demonstrated as a metric phenomenon. In the following, without detailing all calculation the simplistic description of the complete calculation is given. Again, at first we start from a perturbed Lorentz metric:
made explicit:
We suppose the velocity v is low with respect to the speed of light c, with a small parameter
.
We have:
We can write:
Limiting to the first degree in β and ε we get:
Then we write, as a classical calculation, the differential equation
system giving the geodesic
s. Christoffel symbols are calculated. The geodesic equation becomes:
The approximate form of the Christoffel symbol is:
Introducing this result into the geodesic equation (**) we get:
This is a vector equation. Since the metric is time-independent, only space variables are concerned. Therefore the second member of the equation is a gradient.
Coding the position-vector by the letter X and the gradient by the vector ∇ we can write:
This is no more than Newton's law of universal gravitation in classical theory, derivating from the gravitational potential φ if we make the identification:
Conversely, if we set a gravitational potential φ, the movement of a particle will follow a space-time geodesic if the first term of the metric tensor is like:
That step is important. Newton's law appears as a particular aspect of the general relativity with the double approximation:
With the calculation above, we have made the following statements:
And this rewards the value of the constant κ, called "Einstein's constant" (which is not the cosmological constant Λ nor the speed of light c):
We can then write the Einstein field equation:
and 
, that the Laue scalar could be written:
which gives the corresponding Einstein's constant:
But another valid choice for writing the form of the stress-energy tensor is:
Neglecting the same term orders, the corresponding Laue scalar is:
which owns an additional term c2, so the corresponding Einstein's constant in the field equations is then:
This is just a question of similar choices, since for each chosen writing the Einstein field equations are the same.
However since Einstein's constant had been evaluated by a calculation based on a time-independent metric, this by no mean requires that G and c must be unvarying constants themselves, but that the only absolute constant is their ratio:
Kappa
Kappa is the 10th letter of the Greek alphabet, used to represent the voiceless velar stop, or "k", sound in Ancient and Modern Greek. In the system of Greek numerals it has a value of 20. It was derived from the Phoenician letter Kaph...
), is the coupling constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...
appearing in the Einstein field equation
Einstein field equations
The 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...
which can be written:
where Gαγ is the Einstein tensor
Einstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
and Tαγ is the stress-energy tensor
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
.
This equation relates to the curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of 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 in physics
Time in physics is defined by its measurement: time is what a clock reads. It is a scalar quantity and, like length, mass, and charge, is usually described as a fundamental quantity. Time can be combined mathematically with other physical quantities to derive other concepts such as motion, kinetic...
, telling that stress-energy
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
is what causes the disturbance of 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...
, thus gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
. 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...
used Newton's law of universal gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...
in his field equations, and the constant of κ is found to have a value of:
N.B.: Writing Einstein's constant depends on how the stress-energy tensor is defined, so the Einstein field equations are always invariant (see details in the section "About the two possible writings" further).
Calculation
In the following, the value of Einstein's constant will be calculated. To do so, at the beginning a field equation where the cosmological constantCosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
Λ is equal to zero is taken, with a steady state hypothesis
Steady State theory
In cosmology, the Steady State theory is a model developed in 1948 by Fred Hoyle, Thomas Gold, Hermann Bondi and others as an alternative to the Big Bang theory...
. Then we use the Newtonian approximation with hypothesis of a weak field and low velocities with respect to the speed of light.
The Newton law will arise and its corollary Poisson's equation
Poisson's equation
In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics...
.
In this approximation, Poisson's equation appears as the approached form of the field equation (or the field equation appears as a generalization of Poisson's equation). The identification gives the expression of Einstein's constant related to quantities G and c.
The Einstein field equations in non-empty space
We have to obtain a suitable tensorTensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
to describe the geometry of space in the presence of an energy field. Einstein proposed this equation in 1917, written as:
(const) is what will become Einstein's constant. We will take the cosmological constant Λ equal to zero (one of the requirements of the properties of the gravitational equations is that they reduce to the free-space field equations when the density of energy in space Tαγ is zero, therefore that the cosmological constant λ appearing in this equation is zero) so the field equation becomes:
where Rαγ s the Ricci tensor, gαγ is the metric tensor, R the scalar curvature and κ is Einstein's constant we will calculate in the next section.
This equation can be written in another form, contracting indexes:
Thus:
where T is the scalar Tαα which we shall refer to as the Laue scalar.
Using this result we can write the field equation as:
>
The classical limit of the gravitational equations
We will show that the field equations are a generalization of Poisson's classical field equation. The reduction to the classical limit, besides being a validity check on the field equations, gives as a by-product the value of the constant κ. and respectively indicate and . Thus, means We will consider a field of matter with low proper density ρ, moving at low velocity v. The stress-energy tensor can be written:
If the terms of order
and are neglected, it becomes:We assume the flow to be stationary and therefore expect the metric to be time-independent. We use the coordinates of special relativity ct, x, y, z that we write as x0, x1, x2, and x3. The first coordinate is time, and the three others are the space coordinates.
Applying a perturbation method
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
, we will consider a metric appearing through a two-terms summation. The first is the Lorentz metric, ημν which is that of 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...
, locally flat. Formulating we get:
The second term corresponds to the small perturbation (due to the presence of a gravitating body) and is also time-independent:
Thus we write the metric:
Clarifying the length element:
If we neglect terms of order
, the Laue scalar is:And the right side of the field equations is to first order in all the small quantities
, and is written:Neglecting second-order terms in
gives the following approximate form for the contracted Riemann tensor:Thus the approximate field equations may be expressed as:
At first let us consider the case μ = ν = 0. As the metric is time-independent, the first term of the equation above is zero. What remains is:
The Christoffel symbol
Christoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
of the first kind is defined by:
Since the Lorentz metric is constant in space and time, this simplifies to:
Moreover
is time-independent, so [00,0] is zero. Neglecting second-order terms in the perturbation term , we get:which is zero for β = 0 (which then corresponds to the derivative with respect to time). Substituting inside (*) we obtain the following approximate field equation for
:or, by virtue of time independence:
This notation is just a writing convention. The equation can be written:
which can be identified to Poisson's equation if we write:
Therefore we have established that the classical theory (Poisson's equation) is the limiting case (weak field, low velocities with respect to the speed of light) of a relativistic theory where the metric is time-independent.
To be complete, gravity has to be demonstrated as a metric phenomenon. In the following, without detailing all calculation the simplistic description of the complete calculation is given. Again, at first we start from a perturbed Lorentz metric:
made explicit:
We suppose the velocity v is low with respect to the speed of light c, with a small parameter
.We have:
We can write:
Limiting to the first degree in β and ε we get:
Then we write, as a classical calculation, the differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
system giving the geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s. Christoffel symbols are calculated. The geodesic equation becomes:
The approximate form of the Christoffel symbol is:
Introducing this result into the geodesic equation (**) we get:
This is a vector equation. Since the metric is time-independent, only space variables are concerned. Therefore the second member of the equation is a gradient.
Coding the position-vector by the letter X and the gradient by the vector ∇ we can write:
This is no more than Newton's law of universal gravitation in classical theory, derivating from the gravitational potential φ if we make the identification:
Conversely, if we set a gravitational potential φ, the movement of a particle will follow a space-time geodesic if the first term of the metric tensor is like:
That step is important. Newton's law appears as a particular aspect of the general relativity with the double approximation:
- weak gravitational field
- low velocity with respect to the speed of light
With the calculation above, we have made the following statements:
- A metric g, solution of the Einstein field equation (with a cosmological constant Λ equal to zero).
- This metric would be a weak perturbation in relation to a Lorentz metric η (relativistic and flat space).
- The perturbation term would not depend of time. Since the Lorentz metric does not depend of time neither, that metric g is time-independent too.
- The expansion into a series gives a linearization of the Einstein field equations.
- This linearized form is found to identify to Poisson's equation using the fact that a field is a curvature, linking the perturbation term to the metric and to the gravitational potential thanks to the relation:
And this rewards the value of the constant κ, called "Einstein's constant" (which is not the cosmological constant Λ nor the speed of light c):
We can then write the Einstein field equation:
About the two possible writings
We have seen, neglecting the terms of order and , that the Laue scalar could be written:which gives the corresponding Einstein's constant:
But another valid choice for writing the form of the stress-energy tensor is:
Neglecting the same term orders, the corresponding Laue scalar is:
which owns an additional term c2, so the corresponding Einstein's constant in the field equations is then:
This is just a question of similar choices, since for each chosen writing the Einstein field equations are the same.
About constants
The Einstein field equation has zero divergence. The zero divergence of the stress-energy tensor is the geometrical expression of the conservation law. So it appears constants in the Einstein equation cannot vary, otherwise this postulate would be violated.However since Einstein's constant had been evaluated by a calculation based on a time-independent metric, this by no mean requires that G and c must be unvarying constants themselves, but that the only absolute constant is their ratio: