Dr Phill’s Science Made Simple

Newton’s Laws Of Gravitation

Newton’s Laws of Gravitation can be summarized as:

• Time is an independent varaible that passes at the same rate everywhere.
  
• Gravity is an attractive force between masses.
  
• Objects gain negative gravitational potential energy as they fall. This cancels out the kinetic energy gained.

The gravitational force $F$ between objects of masses $M$ and $m$ at a distance $r$ is:

$$F=\frac{GMm}{r^2}$$

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 $c$. They also can’t describe the motion of objects at short distances, such as, the orbit of Mercury around the Sun.

In fact the three statements of the laws, are all false!

Spacetime

Bernhard Riemann extended differential geometry of surfaces to any number of dimensions. This is called Riemannian geometry.

Henri Poincaré formulated a description of four-dimensional spacetime where imaginary time $ict$ is the fourth dimension.

Hermann Minkowski, who was one of Einstein’s teachers, realised that special relativity is easier to understand in four-dimensional spacetime as Poincaré had found but not pursued. This is now known as Minkowski spacetime.

General Relativity

• Albert Einstein worked of a theory of gravity based on curved spacetime 1907-1915.
  
• David Hilbert invited Einstein to Götingen for a week of lectures.
  
• Einstein and Hilbert raced to find the field equations describing gravity in 1915.
  
• Both published results independently in 1915.
  
• Hilbert wrote his paper first but Einstein’s paper was published first.
  
• Hilbert gave Einstein the credit for the theory.

Metric Tensor

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:

$$d{s}^2=c^{2}d{\tau }^2=c^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2$$

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.

$$\begin{array}{rl}x& =r\sin \theta \cos \varphi \\ y& =r\sin \theta \sin \varphi \\ z& =r\cos \theta \end{array}$$

Taking derivatives and substituting values gives:

$$d{s}^2=c^{2}d{\tau }^2=c^{2}d{t}^2-d{r}^2-r^{2}d{\theta }^2-r^{2}si{n}^2\theta d{\varphi }^2$$

Christoffel Symbols

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 ${\mathrm{\Gamma }}_{\mu \nu }^{\lambda }$.

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.

\[g_{\mu\rho,\nu} = \frac{\partial g_{\mu\rho}{\partial x_\nu}\]

$${\mathrm{\Gamma }}_{\mu \nu }^{\lambda }=\frac{1}{2}{g}^{\lambda \rho }(g_{\mu \rho ,\nu }+g_{\nu \rho ,\mu }-g_{\mu \nu ,\rho })$$

Ricci Tensor

Gregorio Ricci-Curbasto discovered the Ricci tensor $R_{ij}$ that 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.

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

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.

$$R=R_i^i=g^{ij}{R}_{ij}$$

Stress Energy Tensor

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.

The Field Equations

The field equations are:

$$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$$

The equations are often written in a simplified form by defining the Einstein tensor $G_{\mu \nu }$ and the Einstein gravitational constant $\kappa$.

$$G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }$$

$$\kappa =\frac{8\pi G}{c^4}\approx 2.0\cdot {10}^{-43}{N}^{-1}$$

The units of $G_{\mu \nu }$ are $m^{-2}$ and $T_{\mu \nu }$ are $N{m}^{-2}$.

Hence:

$$G_{\mu \nu }=\kappa T_{\mu \nu }$$

There is another form of the field equations called the Ricci form. Start by multiplying the field equations by $g^{\gamma \mu }$.

$$g^{\gamma \mu }{R}_{\mu \nu }-\frac{1}{2}R{g}^{\gamma \mu }{g}_{\mu \nu }=\kappa g^{\gamma \mu }{T}_{\mu \nu }$$

The metric $g^{\gamma \mu }$ raises an index in each term.

$$R_{\nu }^{\gamma }-\frac{1}{2}R{\delta }_{\nu }^{\gamma }=\kappa T_{\nu }^{\gamma }$$

Now the Kronecker delta ${\delta }_{\nu }^{\gamma }$ is $0$ if $\gamma \ne \nu$ and $1$ if $\gamma =\nu$. 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 Ricci form of the field equations is:

$$R_{\mu \nu }=\kappa (T_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }T)$$

Interpretation of the Field Equations

The Stress-Energy describes the energy, mass, momentum, and electromagnetic fields in spacetime. This tells spacetime how to curve. The einstein constant is very small so it takes a lot of energy to have a significant effect.

The Einstein tensor describes the shape of spacetime. Even if the stress-energy tensor is zero, as in the vacuum of space, the presence of a nearby massive object will curve spacetime. If spacetime is curved, it has geodesics that affect the motion of objects including light.

Field Equations Exact Solutions

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.

1. The Schwarzschild Outer Metric 1915.
2. The Schwarzschild Interior Metric 1916.
3. The Reissner-Nordström Metric 1916-1921.
4. The Kerr-Newman Metric 1963.