Physics in curved spacetimes
Recall the definition of the Ricci tensor and the Ricci scalar,
\[ \begin{align}\begin{aligned} \tensor{R}{_{\beta \nu}} = \tensor{g}{^{\alpha \mu}} \tensor{R}{_{\alpha \beta \mu \nu}} = \tensor{R}{^{\mu}_{\beta \mu \nu}}\\and\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} R = \tensor{g}{^{\beta \nu}} \tensor{R}{_{\beta \nu}}\\Note that the Ricci tensor is symmetric; by differentiating equation ,\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} 2 {R}_{\alpha \beta \mu \nu, \lambda} = \tensor{g}{_{\alpha \beta, \beta \mu \lambda}} - \tensor{g}{_{\alpha \mu, \beta \nu \lambda}}
+\tensor{g}{_{\beta \mu, \alpha \nu \lambda}} - \tensor{g}{_{\beta \nu, \alpha \mu \lambda}}\\Recalling that partial derivatives do commute,\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \label{eq:93}
\tensor{R}{_{\alpha \beta \mu \nu, \lambda}} + \tensor{R}{_{\alpha \beta \lambda \mu, \nu}} + \tensor{R}{_{\alpha \beta \lambda, \mu}} = 0\\Since the Ricci tensor is derived in normal coordinates, where\end{aligned}\end{align} \]
\(\Gamma^{\mu}_{\alpha \beta} = 0\), so partial and covariant
differentiation are equivalent, and so we get the Bianchi identities,
\[ \begin{align}\begin{aligned} \label{eq:93}
\tensor{R}{_{\alpha \beta \mu \nu; \lambda}} + \tensor{R}{_{\alpha \beta \lambda \mu; \nu}} + \tensor{R}{_{\alpha \beta \lambda; \mu}} = 0\\Performing the Ricci contraction from equation on the Bianchi\end{aligned}\end{align} \]
identities,
\[ \begin{align}\begin{aligned} \label{eq:94}
\tensor{R}{\beta \nu; \lambda} - \tensor{R}{_{\beta \lambda; \nu}} + \tensor{R}{^{\mu}_{\beta \nu \lambda; \mu}} = 0\\contracting these in turn,\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \label{eq:95}
\tensor{G}{^{\alpha \beta}_{; \beta}} = 0\\which is the contracted Bianchi identity, with :math:`\ten{G}`, the\end{aligned}\end{align} \]
Einstein tensor, defined
= - R
The equivalence principle
One statement of the equivalence principle is
All local, freely falling, non-rotating laboratories are fully
equivalent for the performance of all physical experiments.
This doesn’t rule out the possibility that the complicated curvature of
spacetime somehow reduces in a locally inertial frame. To preclude this
we can reword the equivalence principle.
Any physical law which can be expressed in tensor notation in
special relativity has exactly the same form in a locally inertial
frame of curved spacetime.
This means that partial differentiation is basically just a special case
of covariant differentiation for a locally inertial frame, that is, in
tensor notation,
\[ \begin{align}\begin{aligned} \label{eq:96}
\tensor{T}{^{\mu \nu}_{;\nu}} = 0\\This also tells us how matter is affected by spacetime; in SR a\end{aligned}\end{align} \]
particle moves along the timelike coordinate of a Minkowski diagram when
at rest, and from the strong equivalence principle this must be the case
with general relativity too; this picks out the curves generated by the
timelike coordinate of a locally inertial frame, that is
Space tells matter how to move: free-falling particles move on
timelike geodesics of the local spacetime.
Einstein’s Equations
Newton’s theory of gravity can be expressed in terms of a gravitational
field, \(\phi\), and the force on a particle of mass \(m\) is
\(f_i = -m \phi_{,i}\), and the source of the field has a mass
density \(\rho\). The field equation connecting the two is
\[\label{eq:90}
\tensor{\phi}{^{,i}_{,i}} = 4 \pi G \rho\]
which is Poisson’s equation; in a vacuum with a mass density
\(\rho\),
\[\label{eq:91}
\tensor{\phi}{^{,i}_{,i}} = 0\]
The acceleration of free-falling particles can be described, from
equation as
\[ \begin{align}\begin{aligned}\dv[2]{\xi^i}{t} = - \tensor{\phi}{^{,i}_{,i}} \xi^j\\comparing this to equation we see both are equations of geodesic\end{aligned}\end{align} \]
- deviation, implying that :math:`tensor{R}{^{mu}_{alpha nu
beta}} U^{alpha} U^{beta}` is analogous to
\(\tensor{\phi}{^{,i}_{,j}}\), where the indices of the Riemann
tensor are swapped using its symmetries. The velocities of the
particles, \(U^{\alpha}\) and \(U^{\beta}\) are arbitrary, so
the \(\ten{\phi}\) of Poisson’s equation is analogous to
\(\tensor{R}{_{\alpha \beta}} =
\tensor{R}{^{\mu}_{\alpha \mu \beta}}\), meaning a good guess at the
relativistic analogue of equation is
\[ \begin{align}\begin{aligned} \label{eq:97}
\tensor{R}{_{\mu \nu}} = 0\\These are Einstein’s vacuum equations for general relativity; if\end{aligned}\end{align} \]
\(\tensor{R}{_{\mu \nu}} = 0\) then
:math:`R = tensor{g}{^{mu nu}} R_{mu
\[ \begin{align}\begin{aligned} \label{eq:98}
G_{\mu \nu} = R_{\mu \nu} - \half R g_{\mu \nu} = 0\\But what if there’s matter? :math:`\rho` is frame-dependent, so the\end{aligned}\end{align} \]
energy-momentum tensor is more likely to be what the field is bound to.
While \(R^{\mu \nu}= - \kappa T^{\mu \nu}\) seems plausible, but
equation implies that :math:`tensor{R}{^{mu nu}_{;
nu}}=0`, which, via the Bianchi identity implies
- :math:`tensor{R}{_{;
nu}} = 0`, and
\((g_{\alpha \beta} T^{\alpha \beta})_{; \nu} = 0\), so this would
imply that all matter has constant density, which is not the case. What
about
\[ \begin{align}\begin{aligned} \label{eq:99}
\tensor{G}{^{\mu \nu}} = - \kappa \tensor{T}{^{\mu \nu}}\\numerous experiments have shown this to describe physical reality.\end{aligned}\end{align} \]
These are the Einstein field equations; ten second-order non-linear
differential equations, which reduce to six independent equations when
the Bianchi identities are used.
One variation of the field equations which is now being taken very
seriously is
\[ \begin{align}\begin{aligned} \label{eq:100}
\tensor{G}{^{\mu \nu}} + \Lambda \tensor{g}{^{\mu \nu}}
= - \kappa \tensor{T}{^{\mu \nu}}\\which we can do because :math:`\tensor{g}{_{\alpha \beta; \mu}}=0`.\end{aligned}\end{align} \]
This contains an extra term, \(\Lambda\), the cosmological constant.
The Newtonian limit
In the weak-field approximation we can take the spacetime around a small
object to be nearly Minkowskian, with
\[ \begin{align}\begin{aligned} \label{eq:101}
\tensor{g}{_{\alpha \beta}} = \tensor{\eta}{_{\alpha \beta}} + \tensor{h}{_{\alpha \beta}}\\Thus :math:`g_{\alpha \beta}` is the result of a perturbation on flat\end{aligned}\end{align} \]
spacetime, and \(\ten{h}\), which encodes the perturbation is a
tensor in Minkowskian space, and using this form in Einstein’s equations
gives a mathematically tractable problem. In the Newtonian limit we have
\(\abs{\phi} \ll 1\), and speeds \(\abs{\vec{v}} \ll 1\), so
this implies \(\abs{T^{00}} \gg \abs{T^{0i}} \gg \abs{T^{ij}}\). We
then identify \(T^{00} = \rho + \mathcal{O}(\rho v^2)\). Matching
the resulting form of Einstein’s equation with Newton’s equation we fix
the constant \(\kappa\), so, in geometrical units,
\[ \begin{align}\begin{aligned} \label{eq:102}
G^{\mu \nu} = 8 \pi T^{\mu \nu}\\The solution in this approximation is then\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \label{eq:103}
h^{00} = h^{11} = h^{22} = h^{33} = -2 \phi\\So the Newtonian metric is then\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \label{eq:104}
\ten{g} \to \diag( -(1+ 2\phi), 1- 2\phi, 1-2\phi, 1 - 2 \phi)\\and its interval is then\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \label{eq:105}
\dd{s^2} = -(1+2 \phi) \dd{t}^2 + (1- 2 \phi) (\dd{x}^2 + \dd{y}^2 + \dd{z}^2 )\\The geodesic equation is :math:`\nabla_U U = 0`; this geodesic has an\end{aligned}\end{align} \]
affine parameter \(\tau\), but rescaling, \(\tau \to \tau/m\),
we can express it in terms of momentum, \(p = m U\), so
\[ \begin{align}\begin{aligned} \label{eq:106}
\nabla_p p = 0\\In component form this looks like\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \label{eq:107}
p^{\alpha} \tensor{p}{^{\mu}_{,\alpha}} + \Gamma^{\mu}_{\alpha \beta} p^{\alpha} p^{\beta} = 0\\Restricting the motion to non-relativistic particles,\end{aligned}\end{align} \]
\(\abs{p^0} \gg \abs{p^i}\), and so
\[ \begin{align}\begin{aligned} \label{eq:108}
m \dv{\tau} p^{\mu} + \Gamma_{00}^{\mu} (p^0)^2 = 0\\The 0-0 symbols in this metric and approximation are\end{aligned}\end{align} \]
\[\begin{split}\begin{aligned}
\label{eq:110}
\Gamma^0_{00} &= \phi_{,0} + \mathcal{O}(\phi^2) \\
\label{eq:111}
\Gamma_{00}^i &= -\half \tensor{(-2 \phi)}{_{,j}} \delta^{ij}
\end{aligned}\end{split}\]
Thus,
\[\begin{split}\begin{aligned}
\label{eq:113}
\dv{p^0}{\tau} &= -m \pdv{\phi}{\tau} \\
\label{eq:114}
\dv{p^i}{\tau} &= -m \phi^{,i}
\end{aligned}\end{split}\]
Which is Newton’s law of gravitation.