The Relativistic Universe

Relativity

Special relativity indicates that humans live in a four-dimensional space-time where the 'distance' $s$ between points in space-time can be regarded as:

$s^2 = -c^2 \Delta t^2 +\Delta x^2 + \Delta y^2 +\Delta z^2 = X^T \eta\ X$

The last equation is a matrix relation in which $^T$ denotes transpose (for vectors $A^T B = A \cdot B$ ), $X$ is the vector $(c \Delta t, \Delta x, \Delta y, \Delta z)$, and $\eta$ is the matrix:

$\eta = \left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) $

A matrix that goes in between two vectors to give a length is called a metric. In mathematics, a metric or distance function is a function which defines a distance between elements of a set. In this case, the set is the space-time and the elements are points in that space-time. A space-time with the $\eta$ metric is called Minkowski space and $\eta$ is the Minkowski metric.

Four-dimensional Minkowski space-time is only one of many different possible space-times (geometries) which differ in their metric matrix. An arbitrary metric matrix can be denoted as $g$, raising questions as to what space-times with different metrics represent. In 1916, Einstein found the importance of these space-times in his theory of general relativity.

General Relativity

General relativity, or the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or space-time. In particular, the curvature of space-time is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

$g \leftrightarrow \text{curvature} \leftrightarrow \text{Energy and Momentum}$

People can use the metric to calculate curvature and then use the Einstein field equations to relate the curvature to the energy and momentum of the space-time. Going in the reverse order, energy and momentum affect the curvature and the space-time. Thus, energy and momentum curves space-time. Minkowski space is the special space devoid of matter, and as a result, it is completely flat. The precise definition of curvature requires knowledge of advanced mathematics, but an intuitive way to understand it is that the definition of a straight line changes in curved spacetime.

A pictorial example is shown in the figure below, where anything that is near the Earth has its motion altered due to the curved space-time. This altering of motion affects satellites, the moon, and even humans. If the space around the Earth were flat, humans would simply float away if they jumped upwards. Since the Earth alters the space-time, humans are pulled toward the Earth. In essence, gravity is a geometrical effect.

Curvature of space-time

The massive Earth is altering the curvature of space-time.