Theory of relativity/General relativity/Einstein equations

Einstein's Field Equations

$G^{\mu \nu }-\lambda g^{\mu \nu }=k{T}^{\mu \nu }$

$k=\frac{8\pi G}{c^4}$ where this latter G is the gravitational constant of the universe, and $\lambda$ is the cosmological constant, sometimes referred to as the universe's dark energy, and is often taken to be zero.

Einstein's field equations couple the stress energy momentum tensor description of the matter contained by the spacetime to a constraint on the spacetime curvature through nonlinear coupled second order differential equations of 4 dimensional spacetime.

Bianchi Identities

Let us start with the Riemann spacetime curvature tensor in terms of the Christoffel symbols

$R_{\mu \rho \nu }^{\lambda }={\mathrm{\Gamma }}_{\mu \nu }^{\lambda }{,}_{\rho }-{\mathrm{\Gamma }}_{\mu \rho }^{\lambda }{,}_{\nu }+{\mathrm{\Gamma }}_{\sigma \rho }^{\lambda }{\mathrm{\Gamma }}_{\mu \nu }^{\sigma }-{\mathrm{\Gamma }}_{\sigma \nu }^{\lambda }{\mathrm{\Gamma }}_{\mu \rho }^{\sigma }$

First consider going to a local free fall frame so that the Christoffel symbols vanish even though their first derivatives do not where there is spacetime curvature

$R_{\mu \rho \nu }^{\lambda }={\mathrm{\Gamma }}_{\mu \nu }^{\lambda }{,}_{\rho }-{\mathrm{\Gamma }}_{\mu \rho }^{\lambda }{,}_{\nu }$

Taking a partial derivative

$R_{\mu \rho \nu }^{\lambda }{,}_{\sigma }={\mathrm{\Gamma }}_{\mu \nu }^{\lambda }{,}_{\rho \sigma }-{\mathrm{\Gamma }}_{\mu \rho }^{\lambda }{,}_{\nu \sigma }$

Permute the indeces and do the following sum

$R_{\mu \rho \nu }^{\lambda }{,}_{\sigma }+R_{\mu \nu \sigma }^{\lambda }{,}_{\rho }+R_{\mu \sigma \rho }^{\lambda }{,}_{\nu }={\mathrm{\Gamma }}_{\mu \nu }^{\lambda }{,}_{\rho \sigma }-{\mathrm{\Gamma }}_{\mu \rho }^{\lambda }{,}_{\nu \sigma }+{\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }{,}_{\nu \rho }-{\mathrm{\Gamma }}_{\mu \nu }^{\lambda }{,}_{\sigma \rho }+{\mathrm{\Gamma }}_{\mu \rho }^{\lambda }{,}_{\sigma \nu }-{\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }{,}_{\rho \nu }$

You can switch order of partial derivatives

$R_{\mu \rho \nu }^{\lambda }{,}_{\sigma }+R_{\mu \nu \sigma }^{\lambda }{,}_{\rho }+R_{\mu \sigma \rho }^{\lambda }{,}_{\nu }={\mathrm{\Gamma }}_{\mu \nu }^{\lambda }{,}_{\rho \sigma }-{\mathrm{\Gamma }}_{\mu \rho }^{\lambda }{,}_{\sigma \nu }+{\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }{,}_{\nu \rho }-{\mathrm{\Gamma }}_{\mu \nu }^{\lambda }{,}_{\rho \sigma }+{\mathrm{\Gamma }}_{\mu \rho }^{\lambda }{,}_{\sigma \nu }-{\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }{,}_{\nu \rho }$

And note now that all the Christoffel symbol terms add to zero

$R_{\mu \rho \nu }^{\lambda }{,}_{\sigma }+R_{\mu \nu \sigma }^{\lambda }{,}_{\rho }+R_{\mu \sigma \rho }^{\lambda }{,}_{\nu }=0$

Going back to an arbitrary frame from free fall corresponds to merely replacing the partial derivative comma with the covariant derivative semicolin which derivative opperation involves the use of Christoffel symbols yeilding the Bianchi identities

$R_{\mu \rho \nu }^{\lambda }{;}_{\sigma }+R_{\mu \nu \sigma }^{\lambda }{;}_{\rho }+R_{\mu \sigma \rho }^{\lambda }{;}_{\nu }=0$

Stress Energy Momentum Tensor

The stress-energy tensor for short of elements $T^{\mu \nu }$ contains information about the stress, pressure, energy density, and momentum density of the matter in the spacetime. For rectilinear inertial frame coordinates

$T^{\mu \nu }{,}_{\mu }=0$

is a statment of energy conservation. The closest thing to such a statment for general relativity where globally rectilinear inertial frames don't exist when there is Riemannian spacetime curvature present is

$T^{\mu \nu }{;}_{\mu }=0$

which is a statment of energy conservation for local free fall frames for which the Christoffel symbols vanish reducing it to the expression just above.

For such a rectilinear inertial frame the elements of the stress energy tensor have the following interpretations

$T^{00}$ is the coordinate frame energy density.

$T^{ii}$ is a flow of momentum per area in the $x^i$ direction or the pressure on a plane whose normal is in the $x^i$ direction.

$T^{ij}$ is the $x^i$ component of momentum per area in the $x^j$ direction or describes a shearing from stresses.

$T^{0i}$ is the volume density of the ith component of momentum flow.

Coupling of Spacetime Curvature to Stress-Energy

Start with the Bianchi identities

$R_{\mu \rho \nu }^{\lambda }{;}_{\sigma }+R_{\mu \nu \sigma }^{\lambda }{;}_{\rho }+R_{\mu \sigma \rho }^{\lambda }{;}_{\nu }=0$

Contract with the metric tensor over $\mu$ and $\nu$, then contract again over $\lambda$ and $\rho$ and use the antisymetry of the Riemann tensor in the last two indices to arrive at

$R{;}_{\sigma }-R_{\sigma }^{\rho }{;}_{\rho }-R_{\sigma }^{\nu }{;}_{\nu }=0$

Renaming the repeated index in common

$R{;}_{\sigma }-R_{\sigma }^{\mu }{;}_{\mu }-R_{\sigma }^{\mu }{;}_{\mu }=0$

combining like terms and dividing by -2

$R_{\sigma }^{\mu }{;}_{\mu }-\frac{1}{2}R{;}_{\sigma }=0$

A chronecker delta may be inserted to yield the equivelent expression

$R_{\sigma }^{\mu }{;}_{\mu }-\frac{1}{2}{\delta }_{\sigma }^{\mu }R{;}_{\mu }=0$

Raise an index with the contravariant metric tensor $g^{\sigma \nu }$ and opperate the chronecker delta on the metric tensor

$R^{\mu \nu }{;}_{\mu }-\frac{1}{2}{g}^{\mu \nu }R{;}_{\mu }=0$

Since the covariant divergence of the metric tensor is zero, the following is equivalent

$(R^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }R){;}_{\mu }=0$

Here we define the Einstein tensor $G^{\mu \nu }$

$G^{\mu \nu }=R^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }R$

So we find that the covariant divergence of the Einstein tensor is zero

$G^{\mu \nu }{;}_{\mu }=0$

As we want to retain a statement of energy conservation for general relativity which was

$T^{\mu \nu }{;}_{\mu }=0$

the simplest way to ensure this energy conservation statement in coupling the Einstein curvature tensor to the stress-energy is to write one proportional to the other

$G^{\mu \nu }=k{T}^{\mu \nu }$

In a low speed weak field limit, for this to correspond to Newtonian gravitation we find that the coupling constant must be $k=\frac{8\pi G}{c^4}$ where this G is the gravitational constant of the universe. Since the covariant divergence of the metric tensor is also zero, a more general possibility exists that the field equations should be

$G^{\mu \nu }-\lambda g^{\mu \nu }=k{T}^{\mu \nu }$

as this would also correspond to Newtonian gravitation in the low speed weak field limit in the case of a small enough cosmological constant $\lambda$. Einstein originally included this in his field equations, but his motivation for doing so was an effort to seek static state universe solutions for it was not yet known that the universe was expanding. When the expansion was discovered, because of his motive, he thought including it was his "greatest blunder", but we have some evidence now that this cosmological constant, sometimes referred to as dark energy is not zero, so he was likely right to include it afterall. A positive cosmological constant yields accelerating universe solutions, and the evidence found is that the observed universe's expansion does look to have such an acceleration term.

References

• Relativity