Newton’s Laws of Gravitation can be summarized as:
The gravitational force \(F\) between objects of masses \(M\) and \(m\) at a distance \(r\) is:
\[\begin{equation} F = \frac{GMm}{r^2} \end{equation}\]
Where \(G\) is the gravitational constant.
Newton’s laws are very accurate unless the masses are very large or objects are moving at a significant percentage of the speed of light \(c\).
In fact the three statements of the laws, are all false.
General Relativity is based on the principle of spacetime, where time is the fourth dimension. The simplest form of spacetime is Minkowski spacetime which is flat.
General Relativity is based on curved spacetime. Bernhard Riemann, Gregorio Ricci-Curbasto and others formulated a description of curved spacetime on which General Relativity is based.
A metric tensor is a function which enables the calculation of the distance between two points in a given space of any number of dimensions. The metric is usually given the notation \(g_{ij}\).
The metric tensor is symmetric \(g_{ij}=g_{ji}\). The inverse of the metric tensor is \(g^{ij}\). The metric tensor can also be used to raise and lower indexes of other tensors.
In flat Minkowski space the line element is given by:
\[ds^2 = c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2\]
Where \(\tau\), the proper time, is the time measured by a clock on a body in motion or under gravity and \(t\) is the coordinate time as measured by a clock an infinite distance away from any masses.
Now transform Cartesian coordinates into spherical polar coordinates.
\[\begin{align*} x &= r \sin \theta \cos \phi\\ y &= r \sin \theta \sin \phi\\ z &= r \cos \theta \end{align*}\]
Taking derivatives and substituting values gives:
\[ds^2 = c^2d\tau^2 = c^2dt^2 - dr^2 - r^2d\theta^2 - r^2 sin^2\theta d\phi^2\]
In order to do calculations on spaces it is necessary to define the derivatives with respect to the coordinates. This process was made easier by the German mathematician and physicist Elwin Bruno Christoffel. He introduced the Christoffel symbols which are defined in terms of the metric \(g_{ij}\). They are used to study the geometry of the metric.
The Christoffel symbols are tensor like objects \(\Gamma^\lambda_{\mu\nu}\).
The Einstein summation principle applies where every term with a repeated index implies a summation over all posible values of the index. The comma notation is also used.
\[g_{\mu\rho,\nu} = \frac{\partial g_{\mu\rho}{\partial x_\nu}\]
\[\begin{equation} \Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\rho}(g_{\mu\rho,\nu} + g_{\nu\rho,\mu} - g_{\mu\nu,\rho}) \end{equation}\]
The Ricci tensor \(R_{ij}\) represents how the volume of a small wedge of a geodesic ball in a curved space differs from the volume of a ball in Euclidean space. In particular, the Ricci tensor measures how a volume between geodesics changes due to curvature. In general relativity, the Ricci tensor represents volume changes due to gravitational tides.
\[\begin{equation} R_{\mu\nu} = \Gamma^\lambda{\mu\lambda,\nu} - \Gamma^\lambda{\mu\nu,\lambda} + \Gamma^\lambda{\mu\eta}\Gamma^\eta_{\nu\lambda} - \Gamma^\lambda{\lambda\eta}\Gamma^\eta_{\mu\nu} \end{equation}\]
The scalar curvature \(R\) is the difference in volume between a small geodesic ball in Riemannian space and the corresponding ball in Euclidean space. It is the trace of the Ricci tensor. It is also referred to as the Ricci scalar.
\[R=R^i_i=g^{ij}R_{ij}\]
The stress-energy tensor \(T_{\alpha\beta}\) describes energy, momentum, and flux, which is a flow of something. Electromagnetic fields behave like mass in some respects and curve space time.
Energy is usually regarded as a scalar quantity and momentum is a vector quantity. If the momentum vector is extended to four dimensions then energy can be placed in the time dimension. It acts as a form of pressure in the time direction. This four vector forms the first row and column of the stress-energy tensor.
The diagonal terms describe pressure.
The remaining terms describe shear stress.
Einstein’s field equations are:
\[R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}\]
The equations are often written in a simplified from by defining the Eisnstein tensor \(G_{\mu\nu}\) and the Einstein gravitational constant \(\kappa\).
\[G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}\]
\[\kappa = \frac{8\pi G}{c^4}T_{\mu\nu}\]
Hence:
\[G_{\mu\nu} = \kappa T_{\mu\nu}\]
There is another form of the field equations. Start by multiplying the firled equations by \(g_^{\gamma\mu}\).
\[g_^{\gamma\mu}R_{\mu\nu} - \frac{1}{2}Rg_^{\gamma\mu}g_{\mu\nu} = \kappa g_^{\gamma\mu}T_{\mu\nu}\]
The metric \(g_^{\gamma\mu}\) raises an index in each term.
\[R^\gamma_\nu - \frac{1}{2}R\delta^\gamma_\nu = \kappa T^\gamma_\nu\]
No the Kronecker delta \(\delta^\gamma_\nu\) is \(0\) if \(\gamma \neq \nu\) an \(1\) if \(\gamma \ne \nu\). Contract the indices \(\gamma = \nu\), \(\delta^\mu_\mu = 4\).
\[R^\gamma_\gamma - 2R = \kappa T^\gamma_\gamma\]
\[R = -\kappa T\]
Note that if \(T = 0\) then \(R = 0\).
The alternative form of the field equations is:
\[R_{\mu\nu} = \kappa\bigg(T_{\mu\nu} -\frac{1}{2}g_{\mu\nu}T\bigg)\]
The field equations consist of ten nonlinear partial differential equations in four variables. They are very difficult to solve exactly. There are however, a number of exact solutions.