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The Trouble with Tensors
by Miles Mathis
[This paper addreses tensors in general. To see a full critique of tensors as used in Einstein's derivation of General Relativity, you may read my multipart analysis of his original papers, where I show the actual fudges in the linebyline proof of his field equations.]
I would like to begin this paper with a quote from Wikipedia. Some will try to cast doubt on my source of quoting by claiming that Wikipedia is an informal site that can be written and edited by anyone. Actually, the science and math pages are edited and written by the universities and governmental organizations, and are heavily policed. These pages are extensively sourced and linked, and many topics (including the ones below) are nearly book length. So Wikipedia is actually the perfect placed to go to critique the status quo.
If we go to the page on General Relativity, and seek the full mathematical explanation of GR rather than the simplified "Introduction to the Problem," we are treated to a section called, Why tensors? I quote it in full:
The principle of general covariance states that the laws of physics should take the same mathematical form in all reference frames and was one of the central principles in the development of general relativity. The term 'general covariance' was used in the early formulation of general relativity, but is now referred to by many as diffeomorphism covariance. Although diffeomorphism covariance is not the defining feature of general relativity, and controversies remain regarding its present status in GR, the invariance property of physical laws implied in the principle coupled with the fact that the theory is essentially geometrical in character (making use of nonRiemannian geometry) suggested that general relativity be formulated using the language of tensors. This will be discussed further below.
Of course, it isn’t really "discussed further below", except in the way it is discussed here—meaning the question "why tensors?" is aggressively avoided in all places. At no point in the paragraph above does the writer even begin to answer the question he or she asked. The first sentence is false: the principle which states that the laws of physics should take the same form from all frames is not the principle of general covariance, it is the principle of relativity, and this principle comes from Special Relativity, not General Relativity. Since Special Relativity originally had nothing to do with tensors, the whole first sentence is misdirection. The second sentence is not to the point: it is simply inserted to cause more confusion. The question was, "Why tensors?" not "What is diffeomorphism?" The third sentence, which admits that diffeomorphism has no necessary connection to GR or to tensors (then why mention it?), then claims that two things suggested the language of tensors: 1) the invariance property of physical laws, 2) the fact that GR is geometrical in character. Concerning 1), I thought the idea was covariance, not invariance. Which is it? The real answer: neither one, since neither covariance nor invariance require the use of tensors. Concerning 2), is the writer suggesting that anything that is geometrical in character requires tensors? The area of a triangle is geometrical in nature: does it require tensors?
And then we have that parenthetical addition about "making use of nonRiemannian geometry." That is just more confusing bombast, since it both false and not to the point. There are lots of geometries that are nonRiemannian, including highschool geometry, but GR doesn’t use highschool geometry. And using nonRiemannian geometry, meant as the writer no doubt meant it, begs the question, since it is as much to say that GR uses tensors because it uses a geometry that uses tensors. I don’t think that clears anything up for anyone.
I know it seems like a bad joke, but this kind of nonwriting and nonmath is the rule, not the exception. All of contemporary math and science reads like this. No one is capable of defining any of their terms clearly or answering a reasonable question in a reasonable way. It is all a nonsensical hodgepodge of precisely this sort, sprinkled heavily with fake equations and floating terms.
Here’s another example.
The intuition underlying the tensor concept is inherently geometrical: as an object in and of itself, a tensor is independent of any chosen frame of reference. [emphasis theirs]
First of all, I don't know what intuition has to do with defining a mathematical term. The "tensor concept" should be logical, not intuitive. And for the writer to suggest that a mathematical "object" is an object in and of itself is just ludicrous. I thought modern mathematics was supposed to be fleeing philosophy, but here we have a claim that a tensor is either a noumenon or an objet en soi. I honestly can't figure out if we are supposed to believe a tensor is an ideal object or an existential object here. But in no case can a mathematical term be an object in and of itself, not philosophically, mathematically, or otherwise. Especially, it cannot be independent of any chosen frame of reference. This is true regardless, but in the context of Relativity, it is so obvious that it takes a an astonishing degree of opacity to overlook it. The first principle of Relativity is that nothing is independent of any chosen frame of reference. Or, to say it another way, all things must be dependent on a chosen frame of reference. They are dependent on that frame of reference, since if anything goes outside its own frame of reference, it needs a transform. That is not independence, that is dependence. If it were independent, it would not need a transform, and would not be relativistic. So there could not be anything more inconsistent than to claim that the field equations of GR are measured by tensors, the equations being dependent upon a frame of reference, but the tensors being independent. That is like saying, "My body needs oxygen to live, except for my legs."
If you don't understand my point, let us look at a quote of Arnold Sommerfeld, an eminent physicist who translated some of the early work of Minkowski and Einstein. He said,
Einstein's "theory of relativity" is a widely misunderstood and not very fortunate name. Not the relativizing of the perceptions of length and duration are the chief point for him, but the independence of natural laws, particularly those of electrodynamics and optics, of the standpoint of the observer." [Schilpp, Albert Einstein, p.99]
I have shown in other places that Sommerfeld was always lost when trying to understand both Minkowski and Einstein, and this is another example. He has it precisely backwards. What he probably means is that the laws of physics are "invariant" across the transforms, which is true. But length and time cannot be "independent" of the standpoint of the observer. Relativity means "relative to," which means "dependent upon." The measurement of a length or a time is dependent upon who is doing the measuring. That is why we need a transform in the first place. If length and time were independent of the observer, then space would be absolute. We would be back to Newton, which I am not advising. I am simply advising that relativity be interpreted correctly.
In the same way, and for the same reason, the tensor cannot be independent of chosen frames of reference, since the tensor is the link between given frames of reference. The matrix is our transform between systems, and is thereby dependent upon them both.
Now let us look at the fundamental problem of the tensor field. This is how Wiki defines the tensor field:
In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences and engineering. It is a generalisation of the idea of vector field, which can be thought of as a 'vector that varies from point to point'.
And a vector field is:
In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space.
The geometric intuition for a vector field is of an 'arrow' attached to each point of a region, with variable length and direction.
Obviously, the main difference between a tensor field in GR and a vector field is that the GR field is not Euclidean. It is Gaussian or Riemannian or otherwise nonEuclidean. And we have a tensor at every point in the space, not a vector. But the problem of the tensor field is the same as that for the vector field: it is not properly defined relative to physical space. Einstein tried to apply his tensor space directly to physical space. He stated that the gravitational field expressed by his field equations was in fact equivalent to physical space. For Einstein, the field and the space were mathematically equivalent. So our first question must be to ask if this is possible. Is it logically possible to assign a tensor field directly to physically space?
And the answer is no, as I have shown in great detail elsewhere. You can’t assign any mathematical space, field or manifold directly to the physical world. There is always a dimensional difference between the mathematical space and the physical space. But with tensors, the dimensional difference is even greater. Any vector field must be several dimensions away from physical space, and I don’t mean that in any esoteric way. I mean simply that a point in a vector or tensor field has several dimensions, whereas a physical point has zero dimensions. A line has more dimensions than a point, and a vector has more dimensions than a line. A vector has direction, which implies motion, which implies time. So already we are three dimensions away from the physical world.
Look at this quote from the internet^{1}:
Points can simply be represented by their coordinates. A local description of a curve is given by the point coordinates, and its associated tangent or normal. A local description of a surface patch is given by the point coordinates, and its associated normal. Here, however, we do not know in advance what type of entity (point, curve, surface) a token may belong to. Furthermore, because features may overlap, a location may actually correspond to multiple feature types at the same time.
For some uses, this may be a valid description of the tensor, but in GR, points cannot be "represented by their coordinates," much less express "multiple feature types." This is because in GR, points are meant to refer directly to physical points in space. Physical points have no extension, no dimensionality, and no other possible physical attributes or features, including any time period. But "assigning coordinates" in the way of the tensor calculus automatically gives the point a mathematical dimensionality and extension. Notice that in the quote above, the point is being assigned multiple coordinates and multiple features. Coordinates, plural. By the necessary operations and axioms of the tensor calculus, this is to assign multiple physical attributes to the point. A point in GR is not a point on some graph, it is meant to be a point in space. The field is not an abstract field, is the fundamental field of existence. This being so, you cannot assign that point any extensions or other attributes or features, including density, force, pressure, etc. The point in GR is (or should be) untouchable and incapable of mathematical representation of any sort, precisely because it is a point. It is a zero in the field and nothing else. The only way you can assign "coordinates" to it is if those coordinates stand for potential ordinal numbers. But the coordinates normally assigned in a tensor field are not potential ordinals, they are potential cardinals. Counting numbers. Einstein made the central error in the history of mathematics, and history has let him get away with it. He has assigned extension to the zero. He has assigned features to the point.
You cannot have a point on a zerodimensional mathematical space, since no such space exists. A point on a onedimensional mathematical space (a number line) already has one dimension, by definition. That point is a distance from zero, and a distance has one dimension. So you are already one dimension divorced from reality. A point on a twodimensional space (like a simple Cartesian graph) has two dimensions. A point on a three dimensional space has three dimensions. A line in threedimensional space has four dimensions. A pointvector in threedimensional space has either four or five dimensions, depending on how it is defined and used. But GR has four dimensions—plus curvature, which adds a dimension—so a GR vector would have at least six dimensions, and possibly seven. But we still haven’t found a total count of dimensions, since
On a smooth curved surface such as a torus, the metric tensor (field) essentially defines a different inner product of tangent vectors at each point of the surface. [Wiki]
What this means is that, depending on the rank of the tensor, we can have several more dimensions at a single point in a manifold. A tensor is not just one vector, it can be many, all at the same point. As used in GR, a tensor is a matrix, an array of terms that transform one quantity into another. For example, if we have two tangent vectors at a point, then each vector may have all the dimensions of the manifold. The product would then have as many as 14 dimensions. This usually doesn’t happen, since the vectors are related; but the product will certainly add at least two dimensions, giving us a minimum of eight.
And yet, as I have shown, we find Einstein trying to apply his field equations to mass points. Meaning, for his equations to work properly, he needs to represent mass points in his field or space. I hope you can see this is going to be very hard for him to do, since he has no mass points in his space. He not only has no physical points in his space, he has no physical lines, no physical planes, and no physical solids. His possible objects start at six dimensions, so there is no place to put the world we know it into his mathematical space. His space is much too complex, even before he starts doing calculations. At best, he can calculate volumes doing complex things in his manifold, but he cannot represent simpler motions of simpler objects. And he certainly cannot talk of physical points. A point would have to have at least six different unrelated motions to even poke its head into his field with a rank zero tensor, and this is impossible by definition. A point in motion draws a line in any manifold, so it is no longer a point.
This is true of all other modern fields and spaces, not just GR. This is the problem with phase space, vector space, QED space, string space, and every other space being used now, without exception. The number of dimensions that these various spaces are divorced from physical space varies, but none of them are within three dimensions of reality, and none of them know it. This is why they all require some kind of renormalization, and why none of them can figure out why.
And this is why Einstein often found himself getting into renormalizationtype problems, which he preferred to call "field strength" problems. He began encountering these problems even before he began trying to merge GR into unified field theories. Every problem in GR that has required some difficult mathematical fix can be traced back to this problem, without exception, including all the problems encountered using the field equations on black holes and on the big bang. In my paper on the ether, I showed the problem with singularities in field equations, but every other problem of GR is founded in this dimensional confusion—a confusion caused by the cavalier attitude that modern physicists and mathematicians have to defining their fields and terms. They simply can't be bothered to define anything clearly, stating that this is philosophical or metaphysical. But it is neither. Failure to clearly define and assign terms and variables and fields and spaces is a mathematical error, and it leads to precisely the sort of mathematical and physical errors that we have seen over and over in the 20th century.
Now let's look at another fundamental problem, one that may be as important as the dimensional one, or moreso. Wiki says,
Many mathematical structures informally called 'tensors' are actually 'tensor
fields'—an abstraction of tensors to field, wherein tensorial quantities vary from point to point.
The problem lies in that last part. A field that varies from point to point is a physical contradiction. In abstract mathematics, this is not necessarily true—you can have an abstract field that varies from point to point, since this field is not tied to "reality." And in engineering this may also be true, since the field may be more or less abstract—it is not the physical field directly, but may be a representation of a dependent field. But a field that varies from point to point cannot be applied to physical space itself, since space must be assumed to be invariant. This was implicitly true with Newton, and it is even more true with Einstein, since Einstein stated it explicitly. Some have thought that Relativity allows space to vary, but this is not true. Measurements vary, but space does not vary. Measurements on time and distance and mass vary, but local time and distance does not vary.
You will ask me where Einstein says this explicitly, so I will tell you: it is here, "The speed of light is constant for all frames of reference." That is enough by itself to set local time and distance. Nor does the curvature of space overthrow this, as has also been argued. As I showed in my first paper on GR, Einstein tells us outright that his own gravitational space becomes Euclidean at the limit. In small enough spaces, it is uncurved, and SR pertains. Well, this is precisely what local space is. It is space measured from no distance. It is ones own field, where transforms—either SR or GR—are unnecessary. I showed in my papers on Feynman where he corroborates this, and it is corroborated most every time a contemporary physicist speaks of "proper time." Proper time is local time, light's own time, and it does not vary. If it varied, we would have no way of measuring anything, or of calculating dilation or contraction or curvature.
Therefore, the physical field should not vary from point to point. The fundamental manifold that underlies all manifolds cannot vary. Measurements within it may vary, but the manifold itself must be homogeneous. If the fundamental manifold is not homogeneous, then none of the differentials in it have a basis. They cannot be defined. To put it another way, a fundamental manifold that varies from point to point is arbitrary. It is freefloating. You can prove absolutely anything with such a manifold, since it is nonaxiomatic. It has no assumptions, so you can change assumptions as you go.
Wiki even admits this:
We don't want our notion to depend on the particular method of mapping the surface. It should exist independently of latitude and longitude, or whatever particular 'cartographic projection' we are using to introduce numerical coordinates.
Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine.
The usual mathematical way involves defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed.
Mathematicians convinced Einstein that he could freefloat his coordinate system, hanging his tensor arrays from sky hooks, but, regarding physical space, they were and are wrong. This is why Einstein gets into trouble with his equations, even in the middle of his famous GR paper. As I have already shown, he can't figure out if his volumes in space have extension or not. His tensor field varies from point to point, and sometimes he likes this and sometimes he doesn't. He likes it when it allows him to push his equations in the direction he wants to go, but he doesn't like it when it throws him up against some axiomatic wall. To be specific, he doesn't like it when his volumes sometimes have extension at the limit and sometimes they don't. He can't seem to decide if his differentials are points or tiny volumes, whenever he goes to the limit. He doesn't like it because it forces him to come up with little ad hoc patches that don't make sense. Most people overlook these verbal patches, but I can tell that they bothered him regardless.
All this is caused by the fact that Einstein is trying to assign his tensor field to the physical field, but this can't work. He is actually ditching the idea of space, as well as the fundamental field, and claiming that his tensor field is all that exists. In this way he is a precursor to QED, which did the same thing. For both, the mathematical field is all that exists: only the math is defined, never the physics.
But to define the amount of curvature at each point in his tensor field, he has to assume an underlying rectilinear field, of an absolute sort. Curvature has no meaning except related to a straight line, and this straight line underlies his curve in every problem. But the tensor calculus won't allow him to admit this. He wants to claim that the tensor field is the fundamental field, but such a claim is illogical. A varying field can only be a dependent field, it cannot be the fundamental or defining field. The variations are variations in assigned parameters (like the length of an object in the field, the mass of an object in the field, the time on a clock in the field), but they cannot be fundamental parameters (like the manifold’s own length and time). If the manifold has no length and time, then you have no manifold. The coordinate system or frame of reference must have its own parameters. A frame of reference without its own absolute assignments is not a frame of reference. What are you referring your measurements to, if you are not referring them to the frame's own parameters? That is what "reference" means: referring back to standard parameters, comparing new measurements to accepted ones.
There is simply no such thing as a mathematical field applied to the physical world that is free of constraints, or that is "independent of our method of mapping," or that is "routine." Nor can there be a fundamental field that varies from point to point. The tensor field, existing as it is claimed to exist in GR, is not really a field. It is a matrix that exists without a space. It is an array of differentials with no frame of reference. It is the accelerations without the system. It is basically floating numbers, unassigned to any definite space. It is the variations without any postulate as to what they are varying with regard to. If they are varying regard to eachother, then there is really no space or manifold; there is only the matrix.
Some may find this clever, even stated this way, but it is not. Assigning a freefloating tensor "field" to physical space is not clever, it is definitely unclever. It creates a whole slew of unsolvable problems and tends to recommend to the brightest minds that they spend years trying to renormalize these problems with pages of fake equations.
The best thing that could happen is if physics were completely stripped of complex mathematics for a century or two. This would allow us to see beneath the esoteric rubbish and locate the major foundational holes that trip us up every time we cross the threshold.
^{1}"Tensor Voting," by Gérard Medioni, USC
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