In standard units, each term on the left has units of 1/length2.
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.
Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when Tμν is everywhere zero) define Einstein manifolds.
The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor , since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler (MTW). The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]):
The third sign above is related to the choice of convention for the Ricci tensor:
With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972) is (+ − −), Peebles (1980) and Efstathiou et al. (1990) are (− + +), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) and Peacock (1999) are (− + −).
Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative:
The sign of the cosmological term would change in both these versions if the (+ − − −) metric sign convention is used rather than the MTW (− + + +) metric sign convention adopted here.
where D is the spacetime dimension. Solving for R and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:
In D = 4 dimensions this reduces to
Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace in the expression on the right with the Minkowski metric without significant loss of accuracy).
Einstein then abandoned Λ, remarking to George Gamow "that the introduction of the cosmological term was the biggest blunder of his life".
The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent astronomical observations have shown an accelerating expansion of the universe, and to explain this a positive value of Λ is needed. The cosmological constant is negligible at the scale of a galaxy or smaller.
Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor:
where it is assumed that Λ has SI unit m−2 and κ is defined as above.
The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.
Conservation of energy and momentumEdit
General relativity is consistent with the local conservation of energy and momentum expressed as
The definitions of the Ricci curvature tensor and the scalar curvature then show that
which can be rewritten as
A final contraction with gεδ gives
which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices,
Using the EFE, this immediately gives,
which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.
To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero
and that the metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives
where two factors of dt/dτ have been divided out. This will reduce to its Newtonian counterpart, provided
Our assumptions force α = i and the time (0) derivatives to be zero. So this simplifies to
which is satisfied by letting
Turning to the Einstein equations, we only need the time-time component
the low speed and static field assumptions imply that
From the definition of the Ricci tensor
Our simplifying assumptions make the squares of Γ disappear together with the time derivatives
Combining the above equations together
which reduces to the Newtonian field equation provided
which will occur if
Vacuum field equationsEdit
A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).
in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. However, there are global solutions of the equation that may lack a globally defined potential.
The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.
The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe.
One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum. In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright, self-similar solutions to the Einstein field equations are fixed points of the resulting dynamical system. New solutions have been discovered using these methods by LeBlanc and Kohli and Haslam.
The linearized EFEEdit
The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of gravitational radiation.
Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written
using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as:
Substituting this definition of the inverse of the metric into the equations then multiplying both sides by a suitable power of det(g) to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.
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^With the choice of the Einstein gravitational constant as given here, κ = 8πG/c4, the stress–energy tensor on the right side of the equation must be written with each component in units of energy density (i.e., energy per volume, equivalently pressure). In Einstein's original publication, the choice is κ = 8πG/c2, in which case the stress–energy tensor components have units of mass density.
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