## General Relativity

Globular Cluster IC 4499. Image Credits: Hubble Space Telescope, NASA

Often quoted as a theory of extraordinary beauty, with Nobel Prize winner Paul Dirac calling it “probably the greatest scientific discovery ever made,” Einstein’s general theory of relativity revolutionized our understanding of the universe.

## Introduction

Newton’s law of gravitation describes gravitation as the force of attraction between two objects. General relativity extends this theory and attempts to describe how gravity behaves by generalising the special theory of relativity (which describes non-accelerating frames of reference). While Newton’s theory attributes gravity to mass, Einstein’s theory reveals that it is a more general quantity known as the Energy-Momentum tensor.

Special relativity was first introduced in 1905 in the third of a set of four outstanding papers which he published in 1905. It made amendments to Newtonian mechanics, which were necessitated once it was clear that Maxwell’s Equations, the governing equations of electrodynamics, were incompatible with the classical theory. The theory abandoned the notion that one dimensional time and space three-dimensional space were independent and instead introduced a four-dimensional model known as spacetime. But still, special relativity did not consider the effects of gravitation on spacetime, which is what general relativity would build upon.

Remarkably, Einstein developed his theory starting in 1907, making the simple observation that the mass responsible for “generating” gravity is equivalent to the mass experiencing a pseudo-force in an accelerated frame of reference. This insight led Einstein to deduce that spacetime is not flat but rather curved, necessitating the use of differential geometry to comprehend it more fully.

At the heart of general relativity lies the principle that gravity is not simply a force acting between massive objects, but rather a result of the curvature of the fabric of spacetime itself. According to Einstein’s theory, massive objects with energy and mass, such as planets or stars, warp the structure of spacetime around them, creating what we perceive as gravitational attraction.

This is most clearly seen in the Einstein field equation: –

## Gμν = Rμν − 0.5Rgμν = kTμν

where:

T is the Energy-Momentum tensor; R is the Ricci Tensor; G is the Einstein Tensor; K is a constant; and g is the metric (which is relevant since it measures distances in curved space).

Thus, the right hand-side, which describes matter is equivalent to the left hand-side which describes the geometry of spacetime i.e. matter teaches space how to curve and space teaches matter how to move.

## Geometry of spacetime

To visualise this concept, imagine spacetime as a grid-like structure. Massive objects distort the grid and pull grid lines toward it as shown in the picture. As opposed to the 3D of space, there’s also time which unfortunately can’t be imagined in this way as it adds another overall dimension. A simpler analogy is to think of this in 2D and look at light ‘bending’ along these gridlines.

## Gravitational lensing

It was hypothesised that the warping of spacetime predicted by Einstein’s theory would cause gravitational lensing, where light would bend around massive bodies like the Sun. Theoretically, this suggests that the light from stars located behind the Sun (from Earth’s perspective) should be visible.[6] However, the challenge lies in distinguishing the light from these stars amid the overpowering brightness of the Sun. The answer to this lies within a total solar eclipse. During such an eclipse, the Moon passes directly between the Earth and the Sun, momentarily obscuring the Sun’s luminous disk. This rare event enables scientists to search for the light from stars that are otherwise hidden behind the Sun. [?]

## Verification of general relativity

Einstein’s prediction specified that during a total solar eclipse, the apparent positions of stars located near the Sun would be shifted due to the Sun’s gravitational influence. These measurable shifts, if accurately observed, would provide a direct test of general relativity.

Thus, having made his first experimentally testable prediction, in 1911, Einstein issued a challenge to astronomers worldwide, urging them to observe total solar eclipses and search for the apparent positions of these stars.[8]This initiated a global quest for eclipses to further test the theory, with more information available here [linked]!

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Earman, J. and Glymour, C. (1980) ‘Relativity and eclipses: The British Eclipse Expeditions of 1919 and their predecessors’, Historical Studies in the Physical Sciences, 11(1), pp. 49–85. doi:10.2307/27757471.

LANDAU, L.D. and LIFSHITZ, E.M. (1975a) ‘Particle in a gravitational field’, The Classical Theory of Fields, p 228. doi:10.1016/b978-0-08-025072-4.50017-4.

Misner, C.W., Thorne, K.S. and Wheeler, J.A. (1973) ‘Tests of Geodesic Motion: Gravitational Redshift Experiments’, in Gravitation. San Francisco: W. H. Freeman.

Rindler, W. (2001) in Relativity: Special, general, and cosmological. Oxford: Oxford University Press, pp. 24–26.

Schmidhuber, J. (2006) Einstein and the ‘Greatest Scientific Discovery Ever’ (according to Paul Dirac). Available at: <https://people.idsia.ch/~juergen/einstein.html> [Accessed: 25 June 2023].

Schutz, B.F. (2018) A first course in general relativity. Cambridge etc.: Cambridge University Press.

The fabric of space-time? (no date) Physics Stack Exchange. Available at: <https://physics.stackexchange.com/questions/309369/the-fabric-of-space-time> [Accessed: 25 June 2023].

Wheeler, J.A. and Ford, K.W. (2000) Geons, black holes, and quantum foam: A life in physics. New York: Norton.

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