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X. Curvature
In the
Newton theory of gravitation, mass is the source of the gravitational force.
This suggests for GRT that mass, or rather energy since it is a
conserved quantity, is affecting geometry. However, in relativity theory,
energy is only conserved in combination with momentum. Therefore Einstein
surmised that the energy-momentum tensor ${\mathbf{T}} =
{\text{T(}}{{\mathbf{g}}_\mu })$ , with components ${T^{\mu \nu }}$, is the
frame-independent geometric object that must act as the source of gravity.
It is a
theorem of relativistic field theories that the energy-momentum tensor is symmetric:
${T^{\mu \nu }} = {T^{\nu \mu }}$; this makes index raising/lowering
unambiguous. Moreover, to uphold the law of conservation of energy-momentum,
this tensor must be covariantly conserved
\[{D_{_\mu }}{T^{\mu \nu }} =
0{\qquad}{D_\mu }\operatorname{T} {\text{(}}{{\mathbf{g}}^\mu }) = 0\]
In a
famous article published in 1915, Albert Einstein derived his gravitational
field equation by setting the Einstein tensor proportional to the
energy-momentum tensor:
\[{G_{\mu \nu }} = \kappa {T_{\mu
\nu }}{\qquad} \operatorname{G} ({{\mathbf{g}}_\mu }) = \kappa
\operatorname{T} {\text{(}}{{\mathbf{g}}_\mu })\]
The left-hand side describes the geometry of space-time; the right-hand side
the matter content of the universe. This simple looking equation actually
stands for a complicated system of second order non-linear differential
equations that relate the local spacetime curvature, i.e. the metric
tensor, to the local density and flow of energy and momentum, quantified by
the energy-momentum tensor. Hence, one usually speaks of Einstein equations
(plural).
The
Einstein gravitational constant $\kappa$ is a proportionality factor
that is determined by a comparison with the Newton theory of gravitation in
the appropriate limit. This yields $\kappa = 8\pi $, or in physical units
$\kappa = 8\pi G/{c^4}$, with Newtons gravitational constant and the speed of
light, made explicit.