Einstein's Work After the Special Theory of Relativity (1905)

After the publication of the Special Theory of Relativity Einstein made very important contributions to Quantum mechanics, where he built on Planck's work, but his key objective always lay in extending the Special Theory of Relativity to include accelerating frames.

History leading to the General Theory of Relativity

Aristotle's belief that touching was required for one object to exert a force on another object together with his ideas that a constant velocity requires that a constant force be exerted held back ideas of gravity for nearly 2000 years.
Finally inspirational men such as Copernicus, Kepler and Galileo began to work on various principles that laid the groundwork for Newton to finally produce a complete theory of gravitation. Just like his laws on mechanics, however, by the end of the 19th century his laws of gravitation were in serious trouble and once again it was Einstein to the rescue.

The most evident of these problems was in the case of trying to predict the orbital path of Mercury. In an attempt to try work it out the astronomers following Newton's Universal Law of Gravitation suggested the possibility of another planet inside Mercury's orbit (which was called Vulcan). When it was proved that no such planet existed and there were no moons around Mercury (another 'possibility' to explain the orbit) it was realised that it was probably the theory that was failing. It soon became clear that Newton's theory of gravitation was, just as his laws of mechanics, only a very good approximation, this time breaking down in the presence of very strong gravitational fields.

Einstein's Solution - General Theory of Relativity

General Theory of Relativity was to replace Newton's Universal Law of Gravitation.
What Einstein effectively achieved with general relativity was to extend the Special Theory of Relativity to include accelerating frames (using some of Galileo's principles), and from this developed his own theory of gravitation, which was to be published in 1915.

Einstein's theory of gravity was completely different to Newton's, rather than believing gravity to be a mysterious force that simply attracted objects towards one another he saw gravity as a warping of the shape of space and the flow of time (commonly referred to as spacetime). He described a gravitational field as a curvature of spacetime caused by mass. He also believed that as energy is equivalent to mass (from E = mc²) it would be possible for a gravitational field to interact with the energy of light in the same way that he saw it interacting with mass, with the result being the bending of light in strong gravitational fields. This was a very important realisation, however we are getting ahead of ourselves here, this is a concept we shall return to later.

Third postulate

The first major realisation towards the final theory was achieved in 1907 as he made his 3rd postulate of his complete theory (first 2 detailed under special relativity), and this he called the Principle of Equivalence, what he meant by this was simply that:

Uniform acceleration is equivalent to a gravitational field

To understand this it is important to realise that there are two forms of mass. One is gravitational mass, often referred at as weight, you can feel this mass if you lift an object. The other is called inertial mass and this refers to an object's resistance to acceleration. The important relationship between the two is:

There have been many experiments to verify this to be correct.

What is the significance of this? If we take two objects, one twice the mass of the other, and we can ignore air resistance as we drop the two objects from equal heights they will both accelerate at the same rate and therefore hit the floor at the same time (Galileo's Law of Fall).
This is very strange, you would think the heavier object would fall faster and it does certainly have a greater gravitational force exerted upon it (weight), but the crucial point is that it also has a proportionally higher inertial mass (as inertial mass = gravitational mass) which is a resistance to acceleration, so the two objects actually fall at the same rate.

The conclusion to all this is that an acceleration of a object in a gravitational field is not dependent on its mass. Galileo was the first to realise that this was due to the equality of inertial mass and gravitational mass and that the acceleration of all objects would be at the same rate in any gravitational field.
It was this equality of mass that led Einstein to the principle of equivalence.

What did he mean by this principle of equivalence?

Well if a frame is uniformly accelerated relative to an inertial frame then the frame may be considered at rest by the introduction of a uniform gravitational field relative to it.
This is because, according to Einstein, a gravitational field effect can not be distinguished from uniform acceleration effect.
Let us use a thought experiment to try and demonstrate the principle:

Let us take a man A standing in a box who is unable to see out but an observer B is able to look in. If this box is in Earth's gravitational field anything he drops will accelerate at 9.8m/s². Then we now move this box to deep space with no gravity present (in the real world there is always some some effect from gravity no matter how far away you are from a mass but we shall ignore that in this case) and tie a rope to the top of the box. Next we generate a force pulling the rope and accelerating the box at 9.8m/s², as soon as this force acts A will experience a feeling the same as gravity, they will feel their legs having to apply a force to remain upright. Further to this anything that A drops will accelerate at 9.8m/s² and from inside he will attribute this to gravity, whereas observer B will attribute it to the acceleration of the floor of the box towards the object. Indeed any effect that B attributes to the acceleration, A will attribute to a gravitational force.

As a further example let us take a ray of light entering the box. B will observe the light entering and striking the opposite wall slightly lower than the entry level. This he will realise will be due to the box being accelerated upwards. Meanwhile A will see the light enter and hit the wall slightly lower that it entered and will ascribe this to the gravitational field as it would interact with the energy of light to bend it, in this case downwards.

Einstein may have considered it along the following lines:
If we take two frames, A and B, and B has a uniform acceleration relative to A then all objects at rest relative to A must have a uniform acceleration relative to B. As we know in a gravitational field all objects fall at the same rate of acceleration, so Einstein asserted that the effect of B's uniform acceleration is the same as B being at rest and a uniform gravitational field being present.

The importance of being able to consider B as being at rest with a gravitational field present rather than as an accelerating frame is simple (as long as you have read the previous page on The Special theory of Relativity, if not and you would like to do so click here) B could then be considered as an inertial frame and so the laws of mechanics can be applied and studied.

Fourth Postulate

For completeness it is worth pointing out that he also made a fourth postulate, or rather he restated his first postulate to say:

'The laws of nature are the same in every frame'

Einstein continues to develop his General Theory of Relativity

1911 was an important year for Einstein, by this point he had completed formulating his postulates and he also realised an important fact that light from a massive body would redshift (a Doppler shift in the visible spectrum to red) as it loses energy in trying to escape the gravitational field.
Further to this, due to the equivalence of mass and energy, a gravitational field would be able to interact with the energy of light to cause a bending of light in very strong gravitational fields.

A crucial problem, however, remained; he was unable to make any gravitational field interchangeable with a uniformly accelerating frame as he was unable to substitute such a frame with a gravitational field that possessed radial symmetry (such as the Earth).

In 1912 he began to solve this problem. His first step in this new phase of his work was to realise the foundations of geometry have physical significance and, working together with Grossman, he began using tensor calculus in his work, which he found assisted him in his calculations in 4 dimensions, which allowed him to solve this difficult problem.

Einstein Expresses His Theory - Geometry

Riemann Geometry (a non-Euclidean form of geometry, the principle of which was originally formed by Gauss) was a crucial part that Einstein incorporated into his theory. This together with his use of tensor calculus allowed him to formulate and importantly find a way to express his theory.

The profound insight of the General Theory of Relativity was that Einstein was able to describe gravitation not as a force but was able to reduce it to a statement of the geometry of spacetime. In other words mass has the effect of warping, or bending/curving, spacetime and any objects in that spacetime have their path deflected as if some force had acted.
This is a perhaps a difficult idea to visualise, the following example should help, though it must be remembered that this only demonstrates the idea in 3 dimensions, it actually works in many more dimensions.

The best analogy I have read of visualising the General Theory of Relativity is to imagine a stretched rubber sheet attached to 4 high poles, this represents spacetime. Then if a large heavy ball is placed on the sheet it causes the sheet to 'warp' or bend as the ball sinks downwards.
If a much smaller ball is then placed on the sheet it will roll 'down' the sheet towards the larger ball.
This represents a large mass, such as a star, warping spacetime and a smaller mass, such as a planet, being attracted to it by gravitational force.

The important point to note is that this attraction takes place, not by some mysterious force as Newton had believed, but by actually travelling through and following the warped spacetime's geometry.

This analogy has been 'flattened' to 3 dimensions to visualise it, of course spacetime is believed to contain many more spatial dimensions than this, with ideas ranging from 4 to infinity.

Spacetime itself can be a very difficult concept to grasp, Hans von Baeyer gave an explanation of his visualisation of spacetime as an invisible stream. The stream is flowing ever onwards, bending in response to objects in its path, carrying everything in the universe along its twists and turns, just as a stream of water bending in response to heavy objects and carrying light objects along with it.

General Theory of Relativity and the Future of the Universe

It is interesting to note General Relativity's predictions for the future of the universe. Since the time of Edwin Hubble we have known that the galaxies are moving apart and thus that the universe is expanding. (If you have not checked out TheSpaceSite.com page on Edwin Hubble and you would like some more information on him and his theories of the expanding universe then click here). From general relativity we know that space is curved, and this leads us to three possibilities for the future of the universe. The possibilities are determined by the total gravitational strength of the universe (in other words the total amount of mass in the universe).

An Open Universe

The universe has a negative curvature (i.e. insufficient mass to counteract and stop the universe's current expansion), so the it will continue to expand over an infinite time scale.

Flat/Euclidean Universe

The universe has no curvature (i.e. the total mass is exactly what is required to stop the current expansion), so the universe will stop expanding but this will only occur over an infinite time scale, so the effect will be that the rate of expansion will continually slow to eventually reach an equilibrium over an infinite time period.

A Closed Universe

The universe has a positive curvature (i.e. a greater total mass than is required to overcome the universe's current expansion), so the expansion will eventually stop and the contraction of the universe will begin, where the galaxies start moving towards each other.

It is unknown as to which of this 3 possibilities actually represent the true future of our universe as we are unable to calculate the total mass of the universe.
It was believed until fairly recently that we could only see about 10% of our universe and the other 90% was made up of dark matter that we could not see, which suggested a closed universe.
Relatively recently, however, Stephen Hawking among others has suggested that we know about 5% of the universe, another 25% to 30% is made up of dark matter and the other 65% to 70% is comprised of energy, hence this could be suggesting that the universe is actually an open universe, and the rate of expansion could be increasing.

The Cosmological Constant

At the time that Einstein published the general theory of relativity the theory of the big bang start to the universe was not widely accepted. This being the case Einstein found that general relativity could not satisfy cosmological modelling without adding a 'cosmological' constant.
Much later Stephen Hawking has virtually proved the big band theory, using General Theory of Relativity. Once this theory of the origin of the universe was accepted it was realised that this constant was not actually needed.
The discovery of dark matter making up a large part of the universe then changed everything again and suggested that the constant was needed after all.
The latest development, following Hawking's suggestion of as much as 70% of the universe is made up of an unaccounted energy, suggests once again that the cosmological constant is not required.

This is an interesting representation of how the theories of science can develop and evolve over time, and it makes me wonder: How will this evolution change our now widely accepted theories in the future?

General Relativity and Time

The General Theory of Relativity predicts that gravitation slows down time, just as the time dilation that occurs under special relativity. Here it is not the speed that effects time but the strength of the gravitational field. Once again this effect is not noticeable in everyday life and does not become truly significant unless we are in the presence of a massive gravitational field, such as that developed by a blackhole.
It is thought that time, as we know it, no longer exists at the bottom of the blackhole.

General Theory of Relativity vs Newton's Universal Gravitation

The General Theory of Relativity and Newton's Law of Universal Gravitation essentially make the same predictions within a weak gravitational field. Just as the modern view of Newtonian mechanics is that it makes extremely good approximations at sub-relativistic speeds, so the law of gravitation makes very good predictions in weak gravitational fields.
Once we begin to study very strong gravitational fields, however, the Universal Law of Gravitation begins to fail.
An excellent example of this are blackholes, where we find that these phenomena can only really be described by general relativity, but we shall return to this later.

There are three major predictions where the theories diverge.

1. The Mercury orbit - Here we are interested in a phenomenon known as 'precession of perihelion', which refers to Mercury's curious orbit where the perihelion actually moves over time. General relativity is able to make its accurate prediction without having to resort to making postulates such as moons or another planet inside the orbit of mercury, that universal gravitation had to rely upon.
2. Bending light - General relativity predicts, contrary to universal gravitation, the direction of light should change in gravitational fields (to be visible requires a very strong gravitational field)
3. Redshift of light - General relativity predicts, again contrary to Newton, that light from a strong gravitational field will redshift (a Doppler shift in the visible spectrum to red) as energy is expended escaping the field.

Confirmation of the General Theory of Relativity

It was in 1919 that the General Theory of Relativity was actually confirmed. This was achieved by observation, during a solar eclipse, that the Sun's gravitational strength was able to bend light, contrary to Newton's theory. This was to bring Einstein world wide recognition, and indeed fame. The London Times 7 Nov 1919 ran the headline:

Revolution in Science - New Theory of the Universe - Newtonian Ideas Overthrown

For many years after the publication of the general theory of relativity there was an increasing rate of experiments that were designed to prove general relativity's predictions.
Eventually all of general relativity's predictions, including those that are contrary to Newton's Universal Law of Gravitation were proved correct, thus confirming beyond any doubt that general relativity supersedes the Universal Law of Gravitation.

All this work that was done on general relativity became encapsulated in a series of mathematical equations, that became known as the Einstein Field Equations (or just Einstein Equations).
These equations created the foundations for all work in the field of relativity and most of the work in cosmology. They are still used today and it was by solving these equations that Stephen Hawking has been able to virtually prove the big bang theory.

The Einstein Field Equations

The Einstein Field Equations are extremely complex (due largely to their non-linear nature) and to solve the equations often takes many pages of mathematics. For very complex anomalies such as blackholes, advanced computational techniques and specialised software are usually required to solve the equations.
The equations can be used to describe all possible spacetime scenarios, by describing the properties of a gravitational field, i.e. the degree and the effect of the curvature. They can also produce some bizarre predictions, such as blackholes and gravitational waves, which at the time were both dismissed as mathematical aberrations, of course today it is widely accepted that such things do indeed exist.

Note this is the symbolic form for the equations, it can not be used for calculations.

The first part of this equation describes how the spacetime is warped, in other words the degree of curvature, and the second part describes the location and motion of matter, in other words the effect of the curvature.
This equation accurately demonstrates the principle:

'matter 'tells' spacetime how to curve, curved spacetime 'tells' matter how to move'

The first mathematical solutions found to Einstein's equations came in 1916 and were produced by Karl Schwarzschild for curvature due to massive gravitational field. At the time this work was done the ideas were purely theoretical but now the modern work done on neutron stars, pulsars and blackholes rely entirely on his solutions to Einstein's Equations.

It should be noted that while the mathematics is very complex, the use of the theory's predictions can be extremely useful. Astronomers and Cosmologists can search the universe for objects that exhibit the theory's predicted behaviours, without having to wade through the mathematics.

After General Theory of Relativity

Following the confirmation of his theory, Einstein was awarded the Nobel Prize for Physics in 1922.

Einstein was very much a pacifist and publicly criticised Germany for their involvement in the First World War and once Hitler was in power he moved to the US. Here he renounced his pacifist stance in the face of the growing danger of the Nazi Germany threat. In 1939 he wrote a letter to the then President of the US, Franklin D. Roosevelt, explaining the possibility of an atomic bomb, warning that Germany could be developing one. This led directly to the US development of the atomic bomb, although Einstein had nothing to do with the development, indeed he campaigned feverishly against it. After the war he worked hard publicly for disarmament.

Einstein spent the latter portion of his life attempting to create a unified theory, to explain all the fundamental forces of nature that he believed in, namely electromagnetism and gravity as manifestations of a single fundamental force (he was uninterested in the possibilities of the other two modern fundamental forces - strong and weak nuclear forces). Unfortunately he was not to succeed in this attempt before his death and was known to remark that he believed he had taken on a task too great to be finished in his lifetime.
Many people today are still seeking to unify the laws of physics, but little progress has been made, Stephen Hawking has remarked that a unified theory encompassing everything would probably be too difficult to find if indeed it exists at all. He himself is interested in trying to unify Quantum Mechanics (which encompasses electromagnetism and nuclear forces (although not without its problems!)) and General Relativity for one theory from the subatomic level to the cosmos.

Einstein was almost unique in the way he regarded his own greatest achievements as mere stepping stones. Indeed Einstein himself noted that he believed that relativity was almost certainly part of a wider theory that was yet to be worked out.
He contributed to the majority of the great problems of physics, and his work covered a wide range of ideas, for instance one of his less famous achievements was that it was actually he who invented the mechanism by which today's lasers operate.

Einstein is generally regarded as the greatest mathematical physicist of all time.

Go on to the Approach of Modern Rocketry
Go back to the Contents Page

Username	Posts
Antares	71
Ender	66
Pulsar	33
MysteryStevenson	26
dunkst3r	11