General relativity

The errors of the general theory of relativity are very different in origin and type from those of special relativity and quantum theory. The errors are simple and basic, and provide no clues as to the physical nature of fundamental reality. They could and should have been exposed a long time ago.

On this page we will look at the origins of general relativity, discuss its mathematical form, the ways in which it does and does not function as a relativity theory, and explain its fatal errors. We will look at the actual mathematics here, and see that it works in a much simpler
way than you might expect.

The general theory of relativity is not tied either to the relativity principle of Einstein or to the particle photon of quantum theory, and this makes it a more purely mathematical theory than the other theories examined.

## Origins

The origins of general relativity lie in a determined effort to continue the mathematical narrative developed with the special theory. It shares a mathematical structure – a metric – with special relativity, but that is superficial and hence misleading: it is entirely unrelated to any other theory.

In special relativity, modern physics believed it has discovered ‘how the world works’ in two ways. It had a principle (relativity) that addressed the failure by 1900 to find the ether, and a form of mathematics (the Lorentz transformation) that accurately modelled changes that were observed when an object was in motion. Neither of these are part of general relativity.

In physics, the gravitational effects on an object, or on light, are treated separately from the effects of motion, and so the general theory of relativity deals with gravitational effects and leaves the effects of motion to the special theory. Consequently, general relativity does not use the Lorentz transformation, and does not rely on the original principle of relativity, that all frames of reference are equivalent. Instead, it has implicitly a weaker version, that all gravitating bodies are mathematically and physically equivalent, a version that had been around since the time of Newton: gravitational effects are calculated and considered first and foremost (and most simply) relative to an individual gravitating body. Gravitational ‘relativity’ has a clearly preferred frame of reference, the non-rotating centre of a gravitating body. It uses a metric, which allows a shift to other frames, but then the mathematics becomes very difficult, so frames of reference are not equivalent.

Prior to November 1915, when Einstein published the outline of the theory, Einstein and Hilbert searched largely separately, but with some cautious communication, for a form of mathematics that would keep the mathematical relationships the same even when there were arbitrary changes or rotations in the space and time coordinate system. This is a mathematical property called covariance, and the search led Einstein, guided by his more mathematically literate friends and associates, such as Marcel Grossman and Hermann Minkowski, to an abstruse form of mathematics developed by Bernhard Riemann.

Riemann had done what mathematicians tend to do. He took a form of mathematics that worked in the real world and generalised it, extended its use beyond what we can actually observe. In school, sixteen year olds use matrix algebra to describe movements, rotations, reflections and distortions to mathematical objects (such as triangles) in two-dimensional space (i.e. on a page). Riemann developed a ‘tensor calculus’ (extended matrix algebra) that would do the same in three, four and higher (mathematical) dimensions.

Discussion of general relativity normally takes place in the language of higher mathematics, and most of this cannot be translated reliably into a simple communicable form. Although this makes it difficult to criticise the detail in a way that is generally intelligible, the important failings of the theory are in areas that we can access and explain.

Riemannian mathematics is the aspect of general relativity that physicists use to contemplate the nature of the universe, but the Schwarzschild metric is the real key to understanding the theory and its somewhat glaring errors.

## Metric

In 1909, Hermann Minkowski, Einstein’s mathematics professor at college, solidified special relativity by showing that it could be presented not as a set of superficial changes to lengths and timekeeping with motion, but instead as a description of the underlying nature
of spacetime in which these observations happened naturalistically.

Mathematically, this meant incorporating the Lorentz transformation into a ‘metric’ that described these effects which could then be considered as the geometry of spacetime. In order for general relativity to be offered as a description of spacetime, it had to be encapsulated into a metric, and in January 1916 Karl Schwarzschild provided this in a letter to Einstein. The benefits of this were seen straightaway; the fatal problems seem to have been entirely ignored.

## Relation to principle

To be clear, there is no principle of relativity in the original sense in general relativity. The theory does not depend upon the idea that the ‘absolute motion of the Earth’ cannot be determined, nor does it offer anything new to support it. It is therefore comfortable with observations of the cosmic microwave background radiation (CMBR) that appear to have identified that motion. It does not conform to the relativistic notion that all frames of reference are equally convenient, and neither does it deal with the inertial frames of reference met earlier, replacing these with 'freefall'.

Instead it recognises a central truth of gravity: that it stems in some fashion from large astronomical bodies. These are therefore ‘preferred frames’ in general relativity, as they have always been in gravitational theory. Newton’s inverse square law of gravity measures distances from the centres of the stars and planets, and general relativity is just the same. Newton’s gravity is proportional to mass, and so is Einstein’s. General relativity’s key metric, the Schwarzschild metric, relates changes in lengths, timekeeping and velocity to this preferred frame, and becomes completely unwieldy if applied in any other frame.

It does use a form of (advanced, Riemannian) mathematics that maintains the mathematical relationships and physical laws when we move from one frame of reference to another. This is scientifically neat and mathematically convenient.

## How the metric works

The Schwarzschild metric is very simple. It lists the adjustments to timekeeping and lengths at any point in a gravitational field. Each of the four adjustments is also very simple, as you can see here. These adjustments can be written into a square array (a matrix) so as to be convenient to use in matrix calculations, or they can be applied directly to timekeeping and lengths.

Each adjustment is simply dependent on gravitational field strength, and so it depends directly on the mass of the gravitating body (such as the Earth or the Sun), and on the distance from the centre of that body.

Relativity is famously a four dimensional theory, meaning that it attempts to treat the dimension of time in the same way as the three known dimensions of space. Critics of this approach are cited here.

The Schwarzschild metric requires four adjustments, one for each of its ‘dimensions’. The three length dimensions it uses are not the three we learn in school (up-down, right-left, forward-back), visualised easily as three connected edges of a cube. Instead it recognises that gravity is (pretty much) spherically symmetrical, and that the distance from the centre of the Earth (or Sun) is one of the key distances required. This is known as the radial distance.

The other two directions or dimensions are measured at right angles to that, and are known as ‘tangential’. (For example: north-south and east-west) A curiosity of the metric is that the tangential adjustments to lengths are different from the radial ones.

## The ‘key observations’

There are a small number of ‘key observations’ that are used to validate general relativity, but this is not nearly enough.

1. We know that atomic clocks above the surface of the Earth, (whether in planes or satellites), tick faster than those on the ground. This is seen in the classic clocks in planes experiment of Hafele and Keating, and in satellites used for GPS, where precise timekeeping is important in calculating positions correctly. The difference is tiny but measurable.

2. We know that light emitted is also different in frequency depending upon the strength of gravitational field in which it is emitted. This is seen in the light emitted from the surfaces of stars and planets, and was measured in 1960 by Pound and Rebka at Harvard University, as discussed and calculated here. This is sometimes described by relativists as light changing in frequency as it leaves a star or planet, but this switch in language appears to do no more than obscure the basic facts.

Observations 1 and 2 are modelled (described precisely by mathematical formulae) by the ‘time’ adjustment in the Schwarzschild metric, and validate only the time adjustment in the metric.

3 & 4. Although relativists normally prefer to describe light as travelling at a constant velocity (constant speed in a straight line), they are happy to discuss this effect as ‘bending and delay’. We know from eclipse observations from 1919 and earlier that light from distant stars changes direction or bends as it passes the Sun. In 1964, Irwin Shapiro recognised that this would also involve a delay, and calculated it. This calculation has been confirmed, for example by bouncing radar signals off Mars.

The assumptions of special relativity seem to have inhibited this enquiry, although the insight that light travels at different speeds at different points in a gravitational field is inherent in the 1911 Einstein paper where he famously calculates the bending as half the actual value, and observes that:

‘The principle of the constancy of the velocity of light holds good according to this theory in a different form from that which usually underlies the ordinary theory of relativity’^{i}.

This is also known as gravitational refraction, and it is a fairly straightforward matter to use this change in velocity to calculate the observed bending and delay. The calculation for bending assumes that light is a wave that travels at different speeds across the wave front near the Sun. The calculation of delay similarly assumes that light travels more slowly in a stronger gravitational field.

Since speed is distance divided by time, we use the adjustment to distances (lengths) in the metric divided by the adjustment to timekeeping to calculate this effect. Unexpectedly, all of the observations we have been able to make are predicted precisely by radial distance divided by time (radial velocity), ignoring the different tangential adjustments in the metric. Only a minority of theoretical physicists are aware of this feature, although it is important in judging whether the metric has been fully validated by observation.

Observations 3 and 4 are therefore modelled by combining the radial length adjustment with the time adjustment. Since we have checked the time adjustment above, these validate the radial adjustment (only) in the metric.

5. The final ‘key observation’ is what is known as the advance of perihelion of Mercury. This refers to a very gradual change in the angle of the axis of its elliptical orbit, and specifically to the component of that change that physics has been unable to explain by other influences. This is the only observation that requires more than two of the four possible adjustments in the metric.

## Problems

It is the Schwarzschild metric that has been checked by a small number of ‘key observations’. Not all of the metric has been properly validated, and none of the higher (Riemannian) mathematics. This creates problems that are simple and easy to see, and appear to be immediately fatal to the theory.

Problem 1: The use of Riemannian mathematics, and hence Einstein and Hilbert’s famous equation of general relativity, do not appear to have any relationship to anything we can reliably measure. They allow physicists to play with grand ideas about the Universe and form a considerable defence against prying eyes, but do not impact on any of the successful predictions.

Problem 2: There are not enough ‘key observations’ to check four elements in the metric. Even when we make the not unreasonable assumption that the two tangential directions are equivalent, and therefore must have the same adjustment, there are three (known and measured) effects, and three adjustments available. In mathematical modelling we require more observations than assumptions before we can have confidence in a model. Simply on this basis, the general theory has not been properly checked.

Problem 3: There are actually more assumptions than three, because we also assume in gravitational relativity the general Riemannian mathematics into which we insert the metric, and make a further assumption that it is appropriate to address the motions of both light and matter (planets) in a single metric.

Problem 4: I have been unable to find a derivation of the advance of perihelion in which I have confidence. The six I have looked at are all different, and lengthy. The three I have examined in detail all have a dubious step tucked away inside.

Problem 5: If we stop for a moment to ask whether a single metric covering the motions of both light and matter is advisable or even possible, we find that it is not. Light is known to slow down as it approaches the Sun, whereas matter accelerates. Equally important is that the speed of light is determined directly by the local gravitational field strength, whereas the speed of matter is determined indirectly, with gravity affecting instead acceleration.

To be clear, it is certainly possible to have an effective metric for gravitation. We could specify the motion of matter (planets and suns) as natural, and use that for the metric, but then the motion of light would have to be specified separately, outside the metric. We could instead apply the metric to light, but then planetary motion would be extra (and would appear bizarre).

General relative does neither; it pretends that both forms of motion are natural, and hence can be described by the same mathematics, which is manifestly false. It pulls off this trick by subterfuge, sleight of hand, and only gets away with it because our brains have been softened up by other theories.

This is an impossible combination to model by a single metric, and it is not entirely clear how this could have been overlooked for so long. Problems 1 to 4 may be considered ‘picky’, but problem 5 is immediately fatal. With hindsight, we can see that physics is careful to apply the metric only to a very limited number of situations, so there has long been some degree of awareness that all is not right.

## Conclusions

The general theory of relativity is neither general nor relativistic. It utilises the same preferred frames of reference as Newtonian theory, and deals specifically with the effects of gravitation.

The theory is based squarely in mathematics. The overarching mathematics is based on the requirements of covariance, but has no other validating observation. The Schwarzschild metric is itself inadequately supported by observational evidence, and is based on an impossible premise, that the motions of light and matter under gravity can be described by a single mathematics. This assumption is seen to be nonsense as soon as it is clearly stated.

Only two elements of the metric are adequately supported by observation: in a stronger gravitational field, two things happen, clocks tick more slowly and light travels more slowly. We can even discern from both observation and the mathematics that, for the low-strength gravitational fields we have been able to examine, the percentage adjustment to light speed is precisely double that for clocks.

## The principle of equivalence

This is the famous principle that the effects of gravity and of acceleration are indistinguishable. Curiously, this is also clearly false.

It is a simple matter to show that the bending of light is double what it would be if the principle were true, and that we can therefore tell if the force we are experiencing is gravitational or due to acceleration by measuring the observed bending of light. In 1911, Einstein famously made the false assumption that they were the same (referenced below), and so the concomitant failure of this principle has been known and recognised for a full century.

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i. A Einstein, On the Effect of Gravitation on the Propagation of Light, Annln Phys 35, 898 (1911)