Previous posts have criticised Einstein’s work based upon popular understandings of the theory of Relativity as explained by Wikipedia or an AI engine. In this post we continue to seek clarity, but now from the words of the great man himself as written in one of his own papers.
Such clarity, however, is shockingly absent.
Abstract: This post critiques Einstein’s own introduction to his theory of Relativity and explores the complexities of geometry’s relationship to physical reality. The post argues that geometry is not inherently tied to the real world and emphasizes that, in a theory of physics, geometrical concepts should arise from physical observations. Through the Tamarack mines experiment, it illustrates that observed distances can contradict Euclidean principles, raising questions about defining distance in terms of geometry versus physical realities. The piece advocates for a physical approach to understanding distance and defining straight lines through actual measurements rather than abstract geometrical assumptions, suggesting that Einstein’s theories are flawed as they conflate geometry with physical processes.
Relativity, the special and general theory – Albert Einstein
https://www.marxists.org/reference/archive/einstein/works/1910s/relative/relativity.pdf
Physical Meaning of Geometrical Propositions
The concept “true” does not tally with the assertions of pure geometry, because by the word “true” we are eventually in the habit of designating always the correspondence with a “real” object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves. – Einstein
Einstein wants to respect the purity of geometry and stresses that it really has nothing to do with physical reality and should not be influenced by any events concerning ‘real’ objects.
Fine. Reality is reality and geometry is geometry.
It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry “true.” Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity.
The idea is reiterated. We believe in geometry because it accords with ideas we have formed from our observations of reality. Geometry is, however, not the same as reality.
If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies
Ouch! Geometry may not be tainted by reality, but reality may be regarded as a physical manifestation of geometry! How? How do physical processes act in accordance with geometrical principles?
He will soon be declaring that geometry is a branch of physics!
Geometry which has been supplemented in this way is then to be treated as a branch of physics.
Told you so!
The idea is that concepts such as position, distance and velocity can be described in geometrical terms and that the resulting mathematics is necessarily relevant to physical processes.
The assertion here is that a rigid body will serve to define (physical) distance and that this distance can be transferred to the realm of geometry where calculations can be made, the outcome of which may be used to make predictions concerning events in the Real World.
This may seem intuitively correct but is in fact the single biggest mistake in physics and leads to a theoretical framework which is ill-defined, ambiguous, borderline comprehensible and entirely divorced from physical reality.
The Tamarack mines experiment
In the Tamarack mines experiment, two mine shafts were dug 1.5 km apart and 1.3 km deep. A physical measuring device (a long wire) was used to determine the distance apart of the mine shafts at the top of the shaft and at the bottom.
It was expected that the distance between the shafts would be the lesser in accordance with the rules of Euclidean geometry but to the surprise of everybody, it turned out that the distance between the mines was greater at the bottom of the shafts.
There are a couple of ways of interpreting this, at least:
- The measuring wire has shrunk – in which case we can abandon the notion of a ‘rigid body’
- The gravitational field of the Earth has altered the geometry of space: circumference is no longer proportional to radius!
In the first case, we cannot use a rigid body as a measurement tool as Einstein would wish, and in the second, we cannot transfer geometric calculations from the realm of geometry to the realm of the physical as Einstein would wish.
The foundations of relativity are invalidated by the observed physical evidence and there is therefore no justification for using geometry as a basis for a theory of reality.
On the basis of the physical interpretation of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a ” distance ” (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry ; then, starting from A, we can mark off the distance S time after time until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement of length.
This is confused.
The idea of measuring distance by the repeated marking of a physical rod is of merit and provides a definition of distance which lies squarely within the physical realm and has little to do with geometry. This surely should be the aim of the physicist: to define everything via observed physical processes rather than get involved with abstract geometrical concepts.
Any such measurement of distance, however, is clearly path independent and different paths will measure as having different lengths. Einstein knows this and wants to define the distance between two points as the length of a line as determined by the rules of geometry. He is flicking back and forth between studying physics and geometry, pretending that physical processes are somehow cognisant of the laws of geometry and somehow obliged to follow them.
Where is the merit in this?
If Euclidean geometry lies at the heart of physics then it should be possible to demonstrate this by physical observation rather than simply assuming it and defining physical distance in terms of geometry.
For a theory of physics, Physical Reality should be regarded as the fundamental and geometry as an emergent outcome. Instead, Einstein wants to regard geometry as fundamental and physical reality as a physical manifestation of this ideal.
The Tamarack mines experiment is an indication that this whole idea is misconceived.
If, for instance, a cloud is hovering over Times Square, then we can determine its position relative to the surface of the earth by erecting a pole perpendicularly on the Square, so that it reaches the cloud. The length of the pole measured with the standard measuring-rod, combined with the specification of the position of the foot of the pole, supplies us with a complete place specification. On the basis of this illustration, we are able to see the manner in which a refinement of the conception of position has been developed.
A physical concept of position relative to the surface of the Earth is possible but impractical.
We speak of the height of the cloud even when the pole which reaches the cloud has not been erected. By means of optical observations of the cloud from different positions on the ground, and taking into account the properties of the propagation of light, we determine the length of the pole we should have required in order to reach the cloud.
Here we go. Distance, once described by comparison with a measuring rod, is now defined by geometry and the properties of the propagation of light.
We can now measure distances all over the place without having to there. We can measure the distance to the stars by means of geometrical calculations and declare them to be accurate even though we never visited them and have no practical means by which to verify our claims.
A Euclidean geometry is now assumed to lie at the root of physics and Einstein will now propose various theories of relativity based upon this assumption. However, these are theories of geometry, not theories of physical processes.
If experimental evidence seems to accord with Einstein’s theories, it is because the results are interpreted in the realm of geometry and Einstein has effectively declared geometry to be a good substitute for physical reality. Observations may sometimes give consistent distances but since distance is now defined by geometrical calculations instead of physical measurements we can ask: “Why should we care?” and “What have we actually learnt about physics?”
What is ‘fundamental’?
In the above, Einstein is attempting to position geometry as the very basis of physical theory whilst at the same time trying to define geometry in terms of the “properties of the propagation of light” (i.e. a physical process).
This is a clear circularity. He needs to decide whether:
- Geometry is fundamental and light propagates according to geometrical regularity
- ..or..
- Physical processes are fundamental and geometry is deduced from observations of such
Note that the “properties of the propagation of light” are not known in their entirety anyhow and that the word ‘properties‘ very likely means ‘geometrical properties‘.
The Galileian System of Co-ordinates
As is well known, the fundamental law of the mechanics of Galilei-Newton, which is known as the law of inertia, can be stated thus: A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line.
This is hopeless!
A ‘body removed sufficiently far..’ – How far is sufficiently far? How do we know that such a sufficiency exists? How do we test this experimentally?
There is said to be a gravitational field everywhere in the cosmos and therefore there is no ‘sufficiently far‘!
Since all motion is deemed to be relative to something else then a ‘state of rest’ must also be deemed relative, but relative to what? Neither ‘state of rest‘ or ‘uniform motion in a straight line‘ have been defined in this context and hence the ‘fundamental law of mechanics‘ is meaningless for both theoretical and practical purposes.
If K is a Galilean co-ordinate system. then every other coordinate system K’ is a Galilean one, when, in relation to K, it is in a condition of uniform motion of translation. Relative to K1 the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K.
The Laws of Newton are said to hold in a coordinate system!
No! The Laws of Physics reside within the physical word and hold good (or otherwise) irrespective of any geometrical representation.
A reasonable attitude here is that the laws of physics unfold everywhere in the cosmos according to local forces only and independently of any reference to any coordinate system. you cannot change the Laws of Physics by running away from them very quickly!
So what is ‘distance’?
Physical distance is that which is counted out by the repeated laying down of a measuring rod.
Geometric distance is that which is defined by geometry.
These two are not the same, as indicated by the Tamarack mines experiment. One interpretation is that the length of a rod (wire) varies according to the local gravitational field and another is that the geometry of space varies in accordance with gravitational field strength. The two are equivalent.
Physical theory should reflect physical reality
A reasonable attitude towards developing a physical theory of reality is to first make experimental observations and then make deductions from them.
We first take a physical object to use as a measuring tool, make some measurements and interpret the resulting data as physical distance and hence physical geometry. If the length of the measuring tool were to ‘vary’ at every point in the universe then so be it; we will still use this as a definition of local length and hence we are now potentially defining a different geometry (curvature) at each point in the universe. This is the correct way to proceed.
What is a straight line?
Geometrically straight lines are just about defined in Euclidean geometry but we now need a physical definition.
We can try:
- A straight line in physical space is that which is taken by a beam of light
- A straight line in physical space is that which is taken by a freely moving mass in space
Importantly:
- Neither of these has anything to do with geometry
- They may both give different results
- Both depend upon actual physical processes
- Both are measurable
- These now serve as fundamental definitions rather than laws
This approach makes no assumptions about the ‘underlying’ geometry of the universe but gives us every chance of determining such a structure by making physical measurements and drawing conclusions.
We no longer need laws concerning bodies travelling in straight lines as we have now defined a straight line in terms of moving bodies. The theory has a sound basis in terms of physical observations.
Einstein’s approach is now seen to be highly artificial and ultimately flawed, based as it is upon unjustified assumptions about the physical world and a spurious argument attempting to connect it to abstract geometry.
Summary
Einstein’s clear aim here is to present an argument to the effect that physical reality is best described in terms of events occurring within a Euclidean coordinate system.
Subsequent arguments from Einstein confirm this in his attempts to describe gravity (a physical process) in purely geometric terms, that is to say as ‘curved space’.
As shown above this idea is flawed from the start and therefore any downstream theory is also doomed to failure.
The idea of Euclidean space as a fundamental coordinate system may seem to be intuitively attractive, but turns out to be a completely arbitrary choice, unrelated to physical reality.
