Coordinate conditions
Encyclopedia
In general relativity
, the laws of physics can be expressed in a generally covariant
form. In other words, the real world does not care about our coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions. A coordinate condition selects such coordinate system(s).
do not determine the metric uniquely, even if one knows what the metric tensor equals everywhere at an initial time. This situation is analogous to the failure of the Maxwell Equations to determine the potentials uniquely. In both cases, the ambiguity can be removed by gauge fixing
. Thus, coordinate conditions are a type of gauge condition. No coordinate condition is generally covariant, but many coordinate conditions are Lorentz covariant or rotationally covariant
.
Naively, one might think that coordinate conditions would take the form of equations for the evolution of the four coordinates, and indeed in some cases (e.g. the harmonic coordinate condition) they can be put in that form. However, it is more usual for them to appear as four additional equations (beyond the Einstein field equations) for the evolution of the metric tensor. The Einstein field equations alone do not fully determine the evolution of the metric relative to the coordinate system. It might seem that they would since there are ten equations to determine the ten components of the metric. However, due to the second Bianchi identity of the Riemann curvature tensor
, the divergence of the Einstein tensor
is zero which means that four of the ten equations are redundant, leaving four degrees of freedom which can be associated with the choice of the four coordinates. It should be noted that the same result can be derived from a Kramers-Moyal-van-Kampen expansion of the Master equation
(using the Clebsch-Gordon coefficients for decomposing tensor products).
Here, gamma is a Christoffel symbol
(also known as the "affine connection"), and the "g" with superscripts is the inverse of the metric tensor
. This harmonic condition is frequently used by physicists when working with gravitational wave
s. This condition is also frequently used to derive the post-Newtonian approximation
.
Although the harmonic coordinate condition is not generally covariant, it is Lorentz covariant. This coordinate condition resolves the ambiguity of the metric tensor
by providing four additional differential equations that the metric tensor must satisfy.
and
.
Synchronous coordinates are also known as Gaussian coordinates. They are frequently used in cosmology
.
The synchronous coordinate condition is neither generally covariant nor Lorentz covariant. This coordinate condition resolves the ambiguity of the metric tensor
by providing four algebraic equations that the metric tensor must satisfy.
An example of an under-determinative condition is the algebraic statement that the determinant of the metric tensor is negative one, which still leaves considerable gauge freedom. This condition would have to be supplemented by other conditions in order to remove the ambiguity in the metric tensor.
An example of an over-determinative condition is the algebraic statement that the difference between the metric tensor and the Minkowski tensor is simply a null four-vector times itself, which is known as a Kerr-Schild form of the metric. This Kerr-Schild condition goes well beyond removing coordinate ambiguity, and thus also prescribes a type of physical space-time structure. The determinant of the metric tensor in a Kerr-Schild metric is negative one, which by itself is an under-determinative coordinate condition.
When choosing coordinate conditions, it is important to beware of illusions or artifacts that can be created by that choice. For example, the Schwarzschild metric may include an apparent singularity at a surface that is separate from the point-source, but that singularity is merely an artifact of the choice of coordinate conditions, rather than arising from actual physical reality.
, then one gets a theory which is in some sense consistent with both special and general relativity. Among the simplest examples of such coordinate conditions are these:
where one can fix the constant k to be any convenient value.
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...
, the laws of physics can be expressed in a generally covariant
General covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
form. In other words, the real world does not care about our coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions. A coordinate condition selects such coordinate system(s).
Indeterminacy in general relativity
The Einstein field equationsEinstein 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...
do not determine the metric uniquely, even if one knows what the metric tensor equals everywhere at an initial time. This situation is analogous to the failure of the Maxwell Equations to determine the potentials uniquely. In both cases, the ambiguity can be removed by gauge fixing
Gauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
. Thus, coordinate conditions are a type of gauge condition. No coordinate condition is generally covariant, but many coordinate conditions are Lorentz covariant or rotationally covariant
Rotational invariance
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument...
.
Naively, one might think that coordinate conditions would take the form of equations for the evolution of the four coordinates, and indeed in some cases (e.g. the harmonic coordinate condition) they can be put in that form. However, it is more usual for them to appear as four additional equations (beyond the Einstein field equations) for the evolution of the metric tensor. The Einstein field equations alone do not fully determine the evolution of the metric relative to the coordinate system. It might seem that they would since there are ten equations to determine the ten components of the metric. However, due to the second Bianchi identity of the Riemann curvature tensor
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
, the divergence of 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...
is zero which means that four of the ten equations are redundant, leaving four degrees of freedom which can be associated with the choice of the four coordinates. It should be noted that the same result can be derived from a Kramers-Moyal-van-Kampen expansion of the Master equation
Master equation
In physics and chemistry and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically...
(using the Clebsch-Gordon coefficients for decomposing tensor products).
Harmonic coordinates
A particularly useful coordinate condition is the harmonic condition (also known as the "de Donder gauge"):Here, gamma is a 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...
(also known as the "affine connection"), and the "g" with superscripts is the inverse of 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...
. This harmonic condition is frequently used by physicists when working with gravitational wave
Gravitational wave
In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...
s. This condition is also frequently used to derive the post-Newtonian approximation
Parameterized post-Newtonian formalism
Post-Newtonian formalism is a calculational tool that expresses Einstein's equations of gravity in terms of the lowest-order deviations from Newton's theory. This allows approximations to Einstein's equations to be made in the case of weak fields...
.
Although the harmonic coordinate condition is not generally covariant, it is Lorentz covariant. This coordinate condition resolves the ambiguity of the metric tensor
by providing four additional differential equations that the metric tensor must satisfy.Synchronous coordinates
Another particularly useful coordinate condition is the synchronous condition:and
.Synchronous coordinates are also known as Gaussian coordinates. They are frequently used in cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
.
The synchronous coordinate condition is neither generally covariant nor Lorentz covariant. This coordinate condition resolves the ambiguity of 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...
by providing four algebraic equations that the metric tensor must satisfy.Other coordinates
Many other coordinate conditions have been employed by physicists, though none as pervasively as those described above. Almost all coordinate conditions used by physicists, including the harmonic and synchronous coordinate conditions, would be satisfied by a metric tensor that equals the Minkowski tensor everywhere. Unlike the harmonic and synchronous coordinate conditions, some commonly used coordinate conditions may be either under-determinative or over-determinative.An example of an under-determinative condition is the algebraic statement that the determinant of the metric tensor is negative one, which still leaves considerable gauge freedom. This condition would have to be supplemented by other conditions in order to remove the ambiguity in the metric tensor.
An example of an over-determinative condition is the algebraic statement that the difference between the metric tensor and the Minkowski tensor is simply a null four-vector times itself, which is known as a Kerr-Schild form of the metric. This Kerr-Schild condition goes well beyond removing coordinate ambiguity, and thus also prescribes a type of physical space-time structure. The determinant of the metric tensor in a Kerr-Schild metric is negative one, which by itself is an under-determinative coordinate condition.
When choosing coordinate conditions, it is important to beware of illusions or artifacts that can be created by that choice. For example, the Schwarzschild metric may include an apparent singularity at a surface that is separate from the point-source, but that singularity is merely an artifact of the choice of coordinate conditions, rather than arising from actual physical reality.
Lorentz covariant coordinate conditions
If one combines a coordinate condition which is Lorentz covariant, such as the harmonic coordinate condition mentioned above, with the Einstein field equationsEinstein 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...
, then one gets a theory which is in some sense consistent with both special and general relativity. Among the simplest examples of such coordinate conditions are these:
where one can fix the constant k to be any convenient value.