Einstein field equations

Cosmological constant

In the Einstein field equations $${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu },}$$ the term containing the cosmological constant ${\displaystyle \mathrm{\Lambda }}$ was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a universe that is not expanding or contracting. This effort was unsuccessful because:

• any desired steady state solution described by this equation is unstable, and
• observations by Edwin Hubble showed that our universe is expanding.

Einstein then abandoned ${\displaystyle \mathrm{\Lambda }}$, 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 ${\displaystyle \mathrm{\Lambda }}$ is needed. The effect of 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: $${\displaystyle T_{\mu \nu }^{(\mathrm{v}\mathrm{a}\mathrm{c})}=-\frac{\mathrm{\Lambda }}{\kappa }{g}_{\mu \nu }.}$$

This tensor describes a vacuum state with an energy density ${\displaystyle {\rho }_{\text{vac}}}$ and isotropic pressure ${\displaystyle p_{\text{vac}}}$ that are fixed constants and given by $${\displaystyle {\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}=-p_{\mathrm{v}\mathrm{a}\mathrm{c}}=\frac{\mathrm{\Lambda }}{\kappa },}$$ where it is assumed that ${\displaystyle \mathrm{\Lambda }}$ has SI unit m⁻² and ${\displaystyle \kappa }$ 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.

Features

Conservation of energy and momentum

General relativity is consistent with the local conservation of energy and momentum expressed as $${\displaystyle {\mathrm{\nabla }}_{\beta }{T}^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.}$$

Derivation of local energy–momentum conservation

Contracting the differential Bianchi identity $${\displaystyle R_{\alpha \beta [\gamma \delta ;\epsilon ]}=0}$$ with ${\displaystyle g^{\alpha \beta }}$ gives, using the fact that the metric tensor is covariantly constant, i.e. ${\displaystyle {g^{\alpha \beta }}_{;\gamma }=0}$, $${\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\epsilon }+{R^{\gamma }}_{\beta \epsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \epsilon ;\gamma }=0}$$

The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten: $${\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\epsilon }-{R^{\gamma }}_{\beta \gamma \epsilon ;\delta }+{R^{\gamma }}_{\beta \delta \epsilon ;\gamma }=0}$$ which is equivalent to $${\displaystyle R_{\beta \delta ;\epsilon }-R_{\beta \epsilon ;\delta }+{R^{\gamma }}_{\beta \delta \epsilon ;\gamma }=0}$$ using the definition of the Ricci tensor.

Next, contract again with the metric $${\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\epsilon }-R_{\beta \epsilon ;\delta }+{R^{\gamma }}_{\beta \delta \epsilon ;\gamma }\right)=0}$$ to get $${\displaystyle {R^{\delta }}_{\delta ;\epsilon }-{R^{\delta }}_{\epsilon ;\delta }+{R^{\gamma \delta }}_{\delta \epsilon ;\gamma }=0.}$$

The definitions of the Ricci curvature tensor and the scalar curvature then show that $${\displaystyle R_{;\epsilon }-2{R^{\gamma }}_{\epsilon ;\gamma }=0,}$$ which can be rewritten as $${\displaystyle {\left({R^{\gamma }}_{\epsilon }-\frac{1}{2}{g^{\gamma }}_{\epsilon }R\right)}_{;\gamma }=0.}$$

A final contraction with g^εδ gives $${\displaystyle {\left(R^{\gamma \delta }-\frac{1}{2}{g}^{\gamma \delta }R\right)}_{;\gamma }=0,}$$ which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices, $${\displaystyle {G^{\alpha \beta }}_{;\beta }=0.}$$

Using the EFE, this immediately gives, $${\displaystyle {\mathrm{\nabla }}_{\beta }{T}^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0}$$

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.

Nonlinearity

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is the Schrödinger equation of quantum mechanics, which is linear in the wavefunction.

Correspondence principle

The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the low-velocity approximation. The constant ${\displaystyle G}$ appearing in the EFE is determined by making these two approximations.

Derivation of Newton's law of gravity

Newtonian gravitation can be written as the theory of a scalar field, ${\displaystyle \mathrm{\Phi }}$, which is the gravitational potential in joules per kilogram of the gravitational field ${\displaystyle g=-\mathrm{\nabla }\mathrm{\Phi }}$, see Gauss's law for gravity $${\displaystyle {\mathrm{\nabla }}^2\mathrm{\Phi }\left(\overrightarrow{x},t\right)=4\pi G\rho \left(\overrightarrow{x},t\right)}$$ where ${\displaystyle \rho }$ is the mass density. The orbit of a free-falling particle satisfies $${\displaystyle \ddot{\overrightarrow{x}}(t)=\overrightarrow{g}=-\mathrm{\nabla }\mathrm{\Phi }\left(\overrightarrow{x}(t),t\right).}$$

In tensor notation, these become $${\displaystyle \begin{array}{cc}{\mathrm{\Phi }}_{,ii}& =4\pi G\rho \\ \frac{d^2{x}^i}{d{t}^2}& =-{\mathrm{\Phi }}_{,i}.\end{array}}$$

In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form $${\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-\frac{1}{2}T{g}_{\mu \nu }\right)}$$ for some constant, ${\displaystyle K}$, and the geodesic equation $${\displaystyle \frac{d^2{x}^{\alpha }}{d{\tau }^2}=-{\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }\frac{d{x}^{\beta }}{d\tau }\frac{d{x}^{\gamma }}{d\tau }.}$$

To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero $${\displaystyle \frac{d{x}^{\beta }}{d\tau }\approx \left(\frac{dt}{d\tau },0,0,0\right)}$$ and thus $${\displaystyle \frac{d}{dt}\left(\frac{dt}{d\tau }\right)\approx 0}$$ 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 $${\displaystyle \frac{d^2{x}^i}{d{t}^2}\approx -{\mathrm{\Gamma }}_{00}^i}$$ where two factors of $\frac{\text{d}t}{\text{d}\tau }$ have been divided out. This will reduce to its Newtonian counterpart, provided $${\displaystyle {\mathrm{\Phi }}_{,i}\approx {\mathrm{\Gamma }}_{00}^i=\frac{1}{2}{g}^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right).}$$

Our assumptions force ${\displaystyle \alpha =i}$ and the time (0) derivatives to be zero. So this simplifies to $${\displaystyle 2{\mathrm{\Phi }}_{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}}$$ which is satisfied by letting $${\displaystyle g_{00}\approx -c^2-2\mathrm{\Phi }.}$$

Turning to the Einstein equations, we only need the time-time component $${\displaystyle R_{00}=K\left(T_{00}-\frac{1}{2}T{g}_{00}\right)}$$ the low speed and static field assumptions imply that $${\displaystyle T_{\mu \nu }\approx \mathrm{diag}\left(T_{00},0,0,0\right)\approx \mathrm{diag}\left(\rho c^4,0,0,0\right).}$$

So $${\displaystyle T=g^{\alpha \beta }{T}_{\alpha \beta }\approx g^{00}{T}_{00}\approx -\frac{1}{c^2}\rho c^4=-\rho c^2}$$ and thus $${\displaystyle K\left(T_{00}-\frac{1}{2}T{g}_{00}\right)\approx K\left(\rho c^4-\frac{1}{2}\left(-\rho c^2\right)\left(-c^2\right)\right)=\frac{1}{2}K\rho c^4.}$$

From the definition of the Ricci tensor $${\displaystyle R_{00}={\mathrm{\Gamma }}_{00,\rho }^{\rho }-{\mathrm{\Gamma }}_{\rho 0,0}^{\rho }+{\mathrm{\Gamma }}_{\rho \lambda }^{\rho }{\mathrm{\Gamma }}_{00}^{\lambda }-{\mathrm{\Gamma }}_{0\lambda }^{\rho }{\mathrm{\Gamma }}_{\rho 0}^{\lambda }.}$$

Our simplifying assumptions make the squares of ${\displaystyle \mathrm{\Gamma }}$ disappear together with the time derivatives $${\displaystyle R_{00}\approx {\mathrm{\Gamma }}_{00,i}^i.}$$

Combining the above equations together $${\displaystyle {\mathrm{\Phi }}_{,ii}\approx {\mathrm{\Gamma }}_{00,i}^i\approx R_{00}=K\left(T_{00}-\frac{1}{2}T{g}_{00}\right)\approx \frac{1}{2}K\rho c^4}$$ which reduces to the Newtonian field equation provided $${\displaystyle \frac{1}{2}K\rho c^4=4\pi G\rho ,}$$ which will occur if $${\displaystyle K=\frac{8\pi G}{c^4}.}$$

Vacuum field equations

If the energy–momentum tensor ${\displaystyle T_{\mu \nu }}$ is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting ${\displaystyle T_{\mu \nu }=0}$ in the trace-reversed field equations, the vacuum field equations, also known as 'Einstein vacuum equations' (EVE), can be written as $${\displaystyle R_{\mu \nu }=0.}$$

In the case of nonzero cosmological constant, the equations are $${\displaystyle R_{\mu \nu }=\frac{\mathrm{\Lambda }}{\frac{D}{2}-1}{g}_{\mu \nu }.}$$

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

Manifolds with a vanishing Ricci tensor, ${\displaystyle R_{\mu \nu }=0}$, are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.

Einstein–Maxwell equations

If the energy–momentum tensor ${\displaystyle T_{\mu \nu }}$ is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor $${\displaystyle T^{\alpha \beta }=-\frac{1}{{\mu }_0}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+\frac{1}{4}{g}^{\alpha \beta }{F}_{\psi \tau }{F}^{\psi \tau }\right)}$$ is used, then the Einstein field equations are called the Einstein–Maxwell equations (with cosmological constant ${\displaystyle \mathrm{\Lambda }}$, taken to be zero in conventional relativity theory): $${\displaystyle G^{\alpha \beta }+\mathrm{\Lambda }{g}^{\alpha \beta }=\frac{\kappa }{{\mu }_0}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+\frac{1}{4}{g}^{\alpha \beta }{F}_{\psi \tau }{F}^{\psi \tau }\right).}$$

Additionally, the covariant Maxwell equations are also applicable in free space: $${\displaystyle \begin{array}{cc}{F^{\alpha \beta }}_{;\beta }& =0\\ F_{[\alpha \beta ;\gamma ]}& =\frac{1}{3}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)=\frac{1}{3}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0,\end{array}}$$ where the semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first equation asserts that the 4-divergence of the 2-form ${\displaystyle F}$ is zero, and the second that its exterior derivative is zero. From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential ${\displaystyle A_{\alpha }}$ such that $${\displaystyle F_{\alpha \beta }=A_{\alpha ;\beta }-A_{\beta ;\alpha }=A_{\alpha ,\beta }-A_{\beta ,\alpha }}$$ 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.

Solutions

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.

Linearized EFE

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.

Polynomial form

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 $${\displaystyle det(g)=\frac{1}{24}{\epsilon }^{\alpha \beta \gamma \delta }{\epsilon }^{\kappa \lambda \mu \nu }{g}_{\alpha \kappa }{g}_{\beta \lambda }{g}_{\gamma \mu }{g}_{\delta \nu }}$$ using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as: $${\displaystyle g^{\alpha \kappa }=\frac{\frac{1}{6}{\epsilon }^{\alpha \beta \gamma \delta }{\epsilon }^{\kappa \lambda \mu \nu }{g}_{\beta \lambda }{g}_{\gamma \mu }{g}_{\delta \nu }}{det(g)}.}$$

Substituting this expression of the inverse of the metric into the equations then multiplying both sides by a suitable power of ${\displaystyle det(g)}$ to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The Einstein–Hilbert action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.