Einstein's theory of General Relativity

1. General relativity

1.1 The curvature of spacetime

In his theory of special relativity, German-Swiss-American physicist Albert Einstein had shown that time, length, and distance can all be measured to have different values depending on their relative speed and direction. However, special relativity only applies to objects moving at a constant velocity. If an object accelerates, then the theory breaks down.

Einstein’s theory of general relativity applies special relativity to the concept of acceleration[1a]. The fact that acceleration due to a gravitational field is the same as any other type of acceleration meant that the results of general relativity also describes the gravitational force, and hence unite theories of space and time with theories of light and matter.

Gravity can be equated with acceleration because the effects of a gravitational field can be produced by accelerating. In a closed room, a person cannot tell if objects are falling because they are experiencing the gravitational force of the Earth, or because they are in deep space, accelerating at the same rate. This allowed Einstein to consider the effect of acceleration on spacetime and apply his findings to gravitation.

Newton had already shown that the equivalence between gravitational acceleration and ordinary acceleration. Einstein realised that a gravitational field would cause objects to accelerate if Minkowski’s spacetime were curved. This acceleration is the same as when an object rolls down a curved surface, like a hill. The amount of acceleration is higher for a higher mass, because a higher mass causes a steeper, longer curve.

This means that objects accelerate because of the curvature of spacetime, and we call this effect gravity.

Diagram showing that you cannot tell the difference between the force of acceleration and the force of gravity.

Image credit: Pbroks13/Markus Poessel/CC-SA.

Einstein showed how the curvature of spacetime is related to the distribution of matter using equations developed by German mathematician Bernhard Riemann in the 1850s[2]. Einstein showed that:

Gμν = Rμν -
Rgμν =

Here, Gμν is the Einstein tensor, which describes the curvature of spacetime. Rμν is the Ricci tensor, which describes the relationship between Euclidean and non-Euclidean geometry. gμν is the metric tensor for Minkowski space, and describes all the intrinsic properties of the spacetime manifold, including time periods, distances, and volumes, as well as the curvature. Finally, Tμν is the energy-momentum tensor, which describes the distribution of matter. This is the source of the gravitational field, just as mass is the source of the gravitational field in Newton’s equations.

A tensor is like a vector - which contains two properties. Velocity, for example, is a vector as it represents speed in a given direction. A tensor contains more than two properties, which may be written in a matrix. A ‘matrix’ refers to numbers or symbols that are arranged in rows and columns.

General relativity shows that observers in any frame will agree on how spacetime is curved by objects, and hence their gravitational field, whether they are moving relative the object or not. This means that the curvature of space, and hence the force of gravity is invariant.

Light moves through curved spacetime, taking the shortest possible path, however the shortest path across a curved surface is not a straight line. The shortest path, which may be curved, is known as a geodesic. This means that the path of light is bent by heavy objects, like the Sun.

1.2 The cosmological principle

In order to apply his theories to the universe as a whole, Einstein applied the cosmological principle. This assumes that the universe is homogeneous and isotropic when averaged over very large scales.

Homogeneity assumes that our observations are representational of the whole universe, and isotropy means that the universe is the same in whichever direction we look.

These ideas were formulised by British astronomer Edward Milne in the 1930s[3][4], and verified by NASA's WMAP (Wilkinson Microwave Anisotropy Probe) satellite, which launched in 2001.

2. Confirmation of general relativity

British physicist Arthur Eddington confirmed general relativity after the 1919 solar eclipse. Eddington knew that if gravity curves spacetime, then light would travel in a curved path as it approaches a massive object like the Sun. This means that if the glare of the Sun were blocked out, like it is during an eclipse, then he would be able to see stars that should be behind it. Einstein's theory predicted that the gravitational force of the Sun can cause starlight to deflect by up to 1.75 arc seconds (0.0005°), and Eddington confirmed this[5].

Photograph of the 1919 eclipse with the position of stars marked.

Eddington's photograph of the 1919 eclipse. Image credit: F. W. Dyson, A. S. Eddington, and C. Davidson/Public domain.

3. Consequences of general relativity

3.1 The orbit of Mercury

The curvature of spacetime forces planets to orbit in open ellipses that are rotating. This effect is more noticeable the closer a planet is to the Sun, and so this explained why these effects were first observed in Mercury's orbit.

3.2 Gravitational time dilation

By 1907, Einstein had already predicted that gravity would create an effect similar to the Doppler effect, which causes light to be red shifted if it is moving away from an observer and blue shifted if it is moving towards them.

Einstein predicted that from the point of view of a stationary observer, light will appear to be red shifted as it moves away from a strong gravitational force, and blue shifted as it moves towards one. This change corresponds to the fact that time appears to run slower for observers in a stronger gravitational field[6].

Gravitational time dilation was first observed in 1971, when American physicists Joseph Hafele and Richard Keating flew four atomic clocks twice around the world on a commercial plane, and then compared their time to clocks left on Earth[7][8].

All satellite navigation systems, including Global Positioning Systems (GPS), have to take the effects of gravitational time dilation into account in order to be accurate.

3.3 Black holes

Black holes occur when an object becomes so massive that even light, which has no rest mass, cannot escape its gravitational well.

It is now known that supermassive black holes reside at the centre of most galaxies, including our own. These are millions of times the mass of the Sun.

3.4 Time travel

If mass curves space, then it will also curve time, and in 1949, Austrian mathematician Kurt Gödel suggested that time travel is possible because 'closed timelike curves' can occur[9]. These are regions where time is curved, so that an observer would find themselves in an earlier time.

Time travel would violate our notion of causality, and so there is still much debate over whether it is possible.

3.5 Gravitational lensing

The curvature of spacetime means that it's also possible for the same image to be projected more than once, as its image is deflected in different directions. This effect is known as gravitational lensing, and it can affect the shape of an event's light cone.

Swiss astrophysicist Fritz Zwicky first considered the effects of gravitational lensing in 1937[10]. These effects were first observed by astronomers Dennis Walsh, Robert Carswell, and Ray Weymann, at the Kitt Peak National Observatory in the United States, in 1979[11].

Animation of a black hole passing in front of a galaxy. The image of the galaxy is distorted as the black hole passes over it.

Simulation of gravitational lensing caused by a black hole moving past a background galaxy. Image credit: Urbane Legend/Alain r/CC-SA.

Photograph of galaxies, some of their shapes are distorted by gravitational lensing.

Gravitational lensing. Image credit: NASA/ESA/Caltech/Public domain.

3.6 Gravitational waves

Gravitational waves are another consequence of general relativity. This is because a massive moving object creates a wave as the curvature of spacetime changes at the speed of light.

Scientists in the LIGO (Laser Interferometer Gravitational-Wave Observatory) Scientific Collaboration and the Virgo Collaboration discovered evidence for gravitational waves from a pair of merging black holes in 2016[12].

3.7 Absolute space

Einstein suggested that general relativity disproves English natural philosopher Isaac Newton's idea that space and time exist independently of matter and energy, as a type of substance, stating that general relativity "takes away the last remnants of physical objectivity from space and time"[1b].

This is because metric fields cannot exist independently of matter and energy, and people who believe the manifold can exist as a type of substance face the hole argument[13].

General relativity shows that spacetime coordinate systems are arbitrary. This means that metric fields can be spread over the manifold in as many ways as there are coordinate systems. Any system will give the same solution to relativistic equations, and so there's no way to know which represents the manifold’s 'correct' shape.

The hole argument shows that if you accept that coordinate systems do represent the true nature of spacetime, then there's no reason why there couldn't be a 'hole', where spacetime suddenly follows another coordinate system.

There's no way to determine when a hole will appear, since both systems are observationally identical, and so proponents of absolute spacetime must accept that general relativity possesses the same objective uncertainty as quantum mechanics. If we remove the idea that spacetime is absolute, then we do not face this problem.

This does not necessarily address Newton's argument that absolute space can be derived from the fact that acceleration is absolute; we can always determine if we are accelerating. We cannot feel the effects of the movement of the Earth, for example, but we can prove that we are moving, and so some continue to argue that spacetime must be absolute.

3.8 A non-static universe

For Einstein, the most startling consequence of general relativity was that it shows that the shape of the universe depends on its energy density. This means that if all the mass in the universe is not sufficiently distributed, then everything will eventually fall in on itself, and the universe will end in a 'big crunch'.

Einstein devised the cosmological constant in order to counterbalance this force and create a static universe.

In 1922, Russian mathematician Alexander Friedmann showed that it's also possible that the universe is expanding[14]. This could occur if it begun at an extremely high density and temperature. If this were the case, and if mass is distributed too sparsely, then matter will continue to move apart forever.

American astronomer Edwin Hubble proved that the universe is expanding in 1929[15]. It's now thought that the universe will expand forever due to dark energy, which was discovered in 1998[16][17].

4. References

  1. (a, b) Einstein, A., 1916, 'The foundation of the generalised theory of relativity', Annalen der Physik, 354, pp.769-822, reprinted in 'The principle of relativity; original papers', 1920, The University of Calcutta.

  2. Einstein, A., 1922, 'Geometry and Experience', Methuen & Co. Ltd.

  3. Milne, E. A., 1934, 'World-models and the world-picture', The Observatory, 57, pp.24-27.

  4. Gale, G., 'Cosmology: Methodological Debates in the 1930s and 1940s', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  5. Dyson, F. W., Eddington, A. S. and Davidson, C., 1920, 'A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919', Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 220, pp.291-333.

  6. Einstein, A. and Schwartz, H. M. (trans), 1977 (1907), 'On the relativity principle and the conclusions drawn from it'. Part I in American Journal of Physics, 45, pp.512–517. Part II in American Journal of Physics, 45, pp.811–817. Part III in American Journal of Physics, 45, pp.899–902.

  7. Hafele, J. C. and Keating, R. E., 1972, 'Around-the-World Atomic Clocks: Predicted Relativistic Time Gains', Science, 177, pp.166–168.

  8. Hafele, J. C. and Keating, R. E., 1972, 'Around-the-World Atomic Clocks: Observed Relativistic Time Gains', Science, 177, pp.168–170.

  9. Gödel, K., 1949, 'An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation', Reviews of Modern Physics, 21, pp.447.

  10. Zwicky, F., 1937, 'Nebulae as gravitational lenses', Physical Review, 51, pp.290.

  11. Walsh, D., Carswell, R. F., and Weymann, R. J., 1979, '0957+ 561 A, B- Twin quasistellar objects or gravitational lens?', Nature, 279, pp.381-384.

  12. Abbott, B. P., et al, 2016, 'Observation of Gravitational Waves from a Binary Black Hole Merger', Physical Review Letters, 116.

  13. Norton, J. D., 'The Hole Argument', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  14. Friedmann, A., 1922, 'On the Curvature of Space', Zeitschrift für Physik, 10, pp.377-386.

  15. Hubble, E., 1929, 'A relation between distance and radial velocity among extra-galactic nebulae', Proceedings of the National Academy of Sciences, 15, pp.168-173.

  16. Riess, A. G., et al, 1998, 'Observational evidence from supernovae for an accelerating universe and a cosmological constant', The Astronomical Journal, 116, pp.1009-1038.

  17. Perlmutter, S., et al., 1999, 'Measurements of Omega and Lambda from 42 high-redshift supernovae', The Astrophysical Journal, 517, pp.565-586.

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