return to homepage A Critique of _{μ} dx_{ν} Once again, these infinitesimals cannot be zero. They must not be.
But in equation 18a of section 8, Einstein gives us this:dτ _{0} = √-g(dτ) Where τ is the volume. And says, "If √-g were to vanish at a point of the four-dimensional continuum, it would mean that at this point an infinitely small 'natural' volume would correspond to a finite volume in the co-ordinates. Let us assume this is never the case. Then g cannot change sign….It always has a finite value." This means that the infinitesimal values must be zero. If they have no finite value then they must be zero. There is no third choice. Even Newton and Cauchy understood this. The epsilon delta proof understands it. Newton said, in effect, "any value you can name, I can name a smaller one: that is all that is necessary to my proof. My value is smaller than yours but it is not zero." The epsilon delta proof mirrors this argument with a bit more rigor. And yet the tensor calculus and Einstein proceed by disallowing the volume at the limit from having any finite value. In one place in the proof ds must not be zero, at another dτ must be zero. The tensor calculus, just like the regular calculus, cannot decide whether the infinitesimals have extension or not. They must have extension in the math, they must not have extension in the physical field. Another thing is curious here, and I cannot pass it by without comment. The very next equation in Einstein’s text is this one: Eq.19: |dx’ _{σ}/dx_{μ}| = 1This is found by letting √-g = 1, so that dτ’ = dτ. This is extraordinary, because if the volume of every infinitesimal is equal at the limit, then that means that everything is equal at the limit. Time and distance must be equal at the limit, which means that space is homogeneous at the limit. Not only is the field rectilinear at the four-vector limit, it is non-time-dilated and non-length-contracted and non-mass-increased. Einstein admitted above that SR works at the GR limit, meaning that the field is Galilean and Euclidean at that level. But he has not taken his analysis far enough. Volumes are not equivalent in SR, since lengths are not equivalent. Therefore his GR field at the limit must be an absolute. He almost admits this when he says, “The invariant √-g(dτ) is equal to the magnitude of the four-dimensional element of volume in the ‘local’ system of reference”. I draw your attention to that word “local”. He clearly has in mind what I have called the local system, which is the system measured locally, from no distance. Feynman also makes use of the local system in trying to explain some of the subtleties of Relativity. By using the two words “local” and “invariant” in the same sentence, Einstein has come as near as he ever did to the word “absolute”. He might as well have admitted it: according to the tensor calculus space is absolute at the limit. Meaning that it is invariant in every way. All local time and distance is equivalent, invariant, and absolutely equal to all other local time and distance. I don’t agree with his math, but I agree with his postulate. All local time and distance must be equal in order to measure relative differences. If you do not postulate local invariance you cannot derive transforms at all, neither in SR nor in GR. But of course this is extraordinary because it contradicts the current interpretation of Einstein, which insists that the local field does not exist. The standard interpretation likes to believe that Einstein somehow derived his transforms from nothing. It likes to insist that Relativity has no absolute postulates, except for the speed of light. It likes to insist that there is no absolute space, even at the limit, since this would seem to imply an ether. Nothing is more passé than the ether. I am not arguing for the reinstatement of the ether, but I think it is clear that we must postulate the equivalence of all local time and space, as I defined both of these in my SR papers. And I think it is clear that Einstein admits this, in his own way, precisely here. I have shown ^{2} that the reason this must be true is that making the speed of light absolute determines, by itself, that the local field must also be absolute. The two postulates are equivalent. If you assume that light is a constant, you must also assume that dt and dx are constants. That is to say, the local field is invariant. The speed of light is a clock setter. It is also a length setter.Perhaps the best way to show that the tensor calculus is superfluous is to simply solve a GR problem without it. The first problem Einstein solves with his new math and theory is the refraction or aberration of starlight by the sun. He has already posed as an axiom the fact that light moves curvilinearly in a gravitiational field. Now he only needs to calculate the amount of curvature in a real field. I will calculate in a completely different way. I dismiss his axiom as unproved. I do not accept that light must travel curvilinearly in a gravitational field, except under certain conditions. It will appear to travel curvilinearly to an observer in freefall or in orbit. We here on earth observing starlight passing the Sun are in a sort of freefall relative to the gravitational field of the Sun, so we would see starlight curve. But in this case, it is the acceleration of the Earth that makes the light appear to bend, not the pull of the Sun on the light. If you could stand on the Sun without burning up, you would be at rest relative to the central field and would not see starlight curve. The curvature of starlight is relative to position. It is not absolute, even within the field. It cannot be treated as an axiom. Therefore I must solve in a completely different way. To develop my method, we must return for a moment to the problem of the disk. Let the disk again be a merry-go-round, and let a child be on a horse on the periphery of the spinning ride. This child wants to throw me a ball. He can easily do so, and I will see the trajectory as a straight line. But the child will see the trajectory as a curve, provided he defines himself as stationary. This is all that Einstein means, or can mean. Replace the ball with a photon, and the child calculates that it has taken a curved path. But, once again, I calculate that it has arrived at my eye in a straight line.
Obviously we have two choices in developing transformation equations here. From the point of the view of the child, we need a math that describes curved lines. From the point of view of the parent standing off the ride, we don't. We can use pretty simple math. But once a transform is arrived at, it will work in both directions. All we have to do is reverse it. So of course we should choose to develop the transforms from the point of view of the parent. Einstein claims that this is not always possible. There is not always a point of view that can be found where the paths of the light rays are straight. He gives as an example the gravitational field of the sun. There is nowhere one can go, he says, where one can calculate with straight light paths; therefore we need a new highly complex math. But this is simply not true. One may or may not be able to carry oneself physically to a place in space where the light rays passing the sun will appear straight, but one can easily carry oneself there conceptually. Just imagine that the sun is expanding at a constant rate, one that would perfectly offset the curvature of the light path. Then develop the transforms, and afterwards reverse the direction of the acceleration. You can do simple Euclidean math, meaning algebra, to describe the field, and then reverse it. You make all the acceleration vectors point toward the sun, instead of away from it. The field itself is exactly the same either way. If I am accelerating toward you, you are accelerating toward me. This obviates the need for curvilinear math altogether. And Einstein's own equivalence principle is what gives us permission to do this!Besides, the reason we cannot carry ourselves physically to a point where light does not curve relative to the field of the Sun is an experimental reason, not a theoretical reason. Meaning that if we could bear very high temperatures, we could go stand on the Sun and see light without curvature. In fact, in a slightly different case, we could actually do the measurement. Einstein calculates a bending of starlight by Jupiter, predicting .02 arc seconds. Well, we could land on Jupiter and see the light unbent by Jupiter’s field there. On the surface of Jupiter, we would be at rest relative to Jupiter’s gravitational field. We would not expect to see a bend of .02 seconds of arc from that position. Just think of the operator of the merry-go-round. If he were standing in the very center, and if he were not rotating, then he would not see the ball thrown off the ride as curved. He would see like the parents, not like the child. Even from the point of view of the child on the merry-go-round, I do not see the need for tensors. Einstein says that Euclidean math cannot be used since no Euclidean space can be found in a gravitational field from which to measure the curvature. There is no orthogonal space underlying the curved space. But this is simply false. It is false since curvature necessarily implies a relation to a non-curved background. The meaning of curvature is absolutely dependent upon the existence of a straight line. If the child on the merry-go-round sees a ball thrown off the ride—or calculates a photon emitted—to travel a curved path, he must see it with regard to a Euclidean background. That background is in fact the "space" of the parent off the ride. If the space of the parent were also curved, then the child could not see any curvature. Motion, whether linear or curved, is always defined relative to a background. A curve in curved space is not a curve. As proof of this, we know that starlight is refracted only by referring it to some Euclidean system. It is curved relative to the background we apply to it in order to see how curved it is. If the ray is refracted some measurable quantity, we know this quantity only by referring it to the path the light would have taken if the field of the sun were not present. This reference field of measurement is precisely the Euclidean field I am talking about. Whether it exists in reality is not pertinent. It can be made to exist as a concept, and indeed it must be in order to measure the degree of aberration of starlight. This field is the field we should choose to do our math in, since it is a Euclidean field and requires no esoteric tensors or slipshod concepts. In fact, the curvature of the light defines the field at that radius. Since a gravitational field is not dependent upon the mass of the test particle, a photon will do as well as any other test mass. But again, the curvature or strength of the field could never be measured without referring it to the path a light ray would have taken without the gravitational tug. That is, a straight light path. This straight light path is precisely the orthogonal field we require: it is the Euclidean space underlying every non-Euclidean field.I have proved this beyond a doubt in my paper on aberration, where I found the same number for aberration near the sun as Einstein, using only one simple algebraic equation. In comparison, let us look first at Einstein’s math. In 1911, he provided this equation in his paper On the Influence of Gravitation on the Propagation of Light:a = (1/c ^{2})∫(kM/r^{2})(cosθds)where a is the aberration and the integral is taken from θ = π/2 to θ = -π/2. This equation took him pages of complex math and theory to derive. Unfortunately, it gave him .83 arc seconds, a number that history has conveniently forgotten. In 1916, in The Foundation of the General Theory of Relativity, Einstein gave us this equation [eq. 74]:B = ∫(∂γ/∂x _{1})(dx_{2})Where B is the aberration, the integral is taken from +∞ to -∞, and γ = √(-g _{44}/g_{22})So we have an integral of partial derivatives, one of which is the transform gamma expressed as the square root of the negative ratio of two tensors. This equation took him 50 pages of math and theory to derive. From all this Einstein gets the number 1.7 arc seconds. I get the same number from this equation: θ = tan ^{-1}(gt^{2}/2)/dWhere g is the acceleration of the earth and d is the distance to the sun. I derived this equation simply by turning all the acceleration vectors due to gravity around. Rather than have the acceleration vector g point at the center of the earth, I turned it around and had it point from the center of the earth outward. Conceptually, this had the effect of making the earth actually accelerate spherically. Mathematically, it had no effect at all beyond flattening out the field. Turning the acceleration vector around made the field a Euclidean field, a field I could do simple math in. Once I achieved my number for aberration, I could leave the vector turned or put it back to the way it was. If I put it back the way it was, I return to a curved field. But my number is the same either way. Just look: opposite = s = gt ^{2}/2t = time for light to travel from the tangent of the sun to the earth = light time from sun to earth + light time of the radius of the sun s = (9.78 m/s ^{2})(501s)^{2}]/2s = 1,230,000m tanθ = opposite/adjacent tanθ = 1,230,000m/1.51 x 10 ^{11}mθ = 1.68 seconds of arc To see why this number does not match the current number 1.75, go here. I show that gamma causes the current equations to fail. The number 1.75 is an outcome of the current equations, not of the newest measurements. You will say that I have ignored the gravity vector of the sun here, which is the main vector. But I showed in my previous paper that the operation of measuring aberration allowed us to ignore the sun’s field. If we turn the sun’s vector around in the same way we did the earth’s, then the sun will be expanding after the light has already passed the tangent. We do not care what happened before the light reached the sun, since the aberration did not take place then. We care what happened after the light reached the sun. But if we have turned the vector around in order to flatten out the field, then the expansion of the sun cannot affect the light. It seems odd, but it is the field of the earth that causes aberration, once the vectors are reversed.
You tell me which equation you would rather teach to students, or supply a full derivation for, or develop a theory to contain. The so-called free data in Einstein’s equations is considerable. There isn’t a speck of free data in my equation. My “field strength” is 100%. No doubt you will say you would prefer to teach an equation that is correct. But my equation is correct. It gets the right number. Modern mathematicians have been famous for claiming that you can apply any math you want to a problem, as long as you do it in the right way. And they are correct. Minkowski answered his critics by saying that his fields were a mathematical convenience: he did not need to prove their physical reality, he only needed to get the right answer. In this, he was right. Any math that does not contain false postulates is as good as any other math. ^{3} But modern mathematicians have used this as an excuse for test-driving all sorts of esoteric maths. I turn the tables on them by showing a simple algebraic solution, in a Euclidean field, that undercuts their tensor calculus. I have achieved the ability to calculate in Euclidean fields by making a single change: I have turned all acceleration vectors around. Simply reversed them. Einstein himself gave me the ability to do this. I am not contradicting his theory, I am just interpreting it in simpler language and math. The vector reversal is a direct outcome of the equivalence he spent so much time proving. His elevator car in space showed us that mathematically there is no difference between “gravity down” and “acceleration up.” If this is true, then there is nothing stopping us from assuming the earth is accelerating up. We assume it just to do our math and then drop the assumption afterwards if we like. It is a mathematical convenience and nothing more, precisely like the convenience of i (except that my simple math is a lot more convenient than i). This is what “equivalence” means: it means that the two situations give us the same number. In developing equations, we are seeking numbers, nothing more. We can house those equations later in any theory we like. If “gravity down” and “acceleration up” are mathematically equivalent, and “acceleration up” gives us a Euclidean field wherever we want it, why should anyone prefer non-Euclidean math? Only a show-off would prefer fifty pages of calculations where he could get by with one page. After all this, I have come to the conclusion that the Gaussian field and the tensor calculus were imported by Einstein for reasons that are strictly ones of public relations. They are impressive in their own right, in the way of all esoterica, and I think it is fair to say that the math of General Relativity is as famous as the theory itself. Einstein's genius is that he chose just the right gown in which to dress his theory for the ball. An avant garde math was sitting around just waiting to be asked out by some ambitious theory, and Einstein had an eye for such matchmaking. Neither the field nor the math were really necessary to an updated theory of gravity, but Einstein worked very hard to make them seem so. He created an incredible amount of confusion, in part by infusing into his writing a mishmash of German philosophy and big, imprecisely used words (like epistemology), in part by writing in a very opaque style—a style that was and is a commonplace in science writing. This faux-haut style fooled most of his fellow scientists of the time. Where a truly learned discourse would have sent them running for cover, or simply confused them, a stilted sentence structure with a quasi-mystical lilt thrown in now and again was just enough to impress. In the case of General Relativity, it was sufficient to obscure the straightforward nature of the problem at hand and to seem to recommend the equally obscure math of Gauss in order to solve it. Not as much new confusion was necessary with General Relativity as with Special Relativity, since the reader was still reeling from all the murky sentences and murkier equations of the latter and therefore did not need many new clouds in his sky. Besides, most of the obscurity of GR would be provided by the tensor calculus.Some feminists have recently tried to establish Einstein's first wife Mileva as an equal partner in his triumph if not his fame, and it may well be so. If it is, then she bears equal responsibility for the disaster of the tensor calculus, since, from what I understand, it was she who first recommended it. She was the professional mathematician. It may be that she was the one who saw the PR possibilites. However that may be, it seems fair to say that it was Einstein himself who thought to rewrite Special Relativity with the tensor calculus in order to make it shine at Princeton. This seems proof beyond anything that the math was used as the primary selling point, since there is no other argument that can be made for using the tensors to solve SR. If the argument for making GR a Gaussian field is tenuous, the argument for making SR one is non-existent. Einstein has already stated that the SR field must be Euclidean and Galilean, by definition. It can therefore hardly benefit from the imprecision (free data) and unwieldiness of matrices and field equations. For much more on this problem, you may now visit my multi-part analysis of the Einstein field equations, which revisit this problem in finer detail, including a line-by-line critique of Einstein's 1916 paper. ^{1}Section 5, paragraph 1^{2}See section 8 of my paper on Special Relativity. ^{3}Minkowski’s math contains several false postulates, as I show elsewhere. It contains the false postulates of Einstein (like x = ct) and it adds to them the false postulate that time travels at a right angle to x, y, z.Go to my correction of the perihelion precession of Mercury If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many |