'Bad artists copy. Good artists steal.'Picasso

It is not like this,it is so complex I will lie if I say I see it,but simple is a key to picture multiple dimention as c,wave...at the same data entry for a 3d human.no

Again I am sorry I did not until now know that one can download for free from the web at http://aps.arxiv.org/PS_cache/gr-qc/pdf/0210/0210005v2.pdf a copy of Logunov's monograph The Theory of Gravity, translated into English by G. Pontecorvo, Nauka, 2001.

Christopher

In Part I of his book Principles of Relativity Physics, Academic Press, New York, London, 1967, James L. Anderson, at pages 1-101, gives an account of the methods of tensors and covariant derivatives and the like.

In Chapter 4, “Structure of Space-Time Theories”, Section 4-1, “The Elements of a Physical Theory”, at page 74, Anderson distinguishes for us between kinetically possible trajectories and dynamically possible trajectories. The formalism, the mathematical linguistic syntax and semantics, of the theory defines the kinetically possible trajectories. The physical content of the theory picks out from the kinetically possible trajectories those that the theory asserts can be found in nature, and they are the dynamically possible trajectories. In Section 4-2 Anderson tells us that covariance refers to the kinetically possible trajectories. Since he is simply writing about a change of coordinates, trivially this must be right. Anderson asserts not only this, but also that covariance requires that the covariance group take each dynamically possible trajectory to another dynamically possible trajectory. Since he is talking about mere changes of coordinates, again trivially he must also be right about this. But I think Anderson is here not being clear about the distinction between a change of coordinates with respect to a given frame of reference, which is in a sense a trivial change, and a change of reference frame. A change of reference frame is not trivial, for it may require the artefactual mathematical construction of "inertial forces". This happens with changes of non-inertial reference frames; it is a change in the form of the theory; with such a change, the phenomena are described in fundamentally different conceptual terms. Covariance of a theory is a formal mathematical requirement. In the present context, it ensures that the quantities of the theory are functions of point-events, with respect to a particular reference frame, and are therefore physical quantities, not just lucky mathematical functions of some lucky coordinate system. This formal requirement is important, and indeed is a main part of the reason for Einstein’s belief that his use of tensors had revealed new laws of nature.

But covariance is only a restriction on the desirable form for the expression of laws of nature. It says nothing specific about what those laws might be, yet it was mistakenly believed by Einstein, and continues to be mistakenly believed by the orthodoxy, that it reveals the laws of gravity. More importantly it refers only to various coordinate systems within a particular frame of reference, and does not attend to various possible frames of reference. A frame of reference in this context has a large influence on the appearance of proposed laws of nature. For example, in inertial frames of reference, the laws of mechanics have no "inertial forces", but in accelerated frames of reference, there appear "inertial forces" such as Coriolis forces. Of course this is important for physics, and is part of the reason that we know that the geometry of the physical spacetime in which we live is Minkowski.

In Part III, "Dynamical Space-Time Theories", Chapter 10, “Foundations of General Relativity”, Section 10-3, “The Principle of General Invariance”, at page 338, Anderson writes “We now come to the third principle that led Einstein to the general theory, the principle of general invariance, which is usually referred to as the principle of general covariance. There is still a good deal of confusion concerning just what content Einstein implied by this principle, due in part to his own writing on the subject.” Anderson goes on to note that Erich Kretschmann (Annalen der Physik, 53: 575-614, 1917, in German and unreadable to me, can you point to a translation for me?) that general covariance makes no assertion about the content of the laws, and that Einstein (ibid. 55: 241 (1918)) concurred with this view. Anderson is not alone in this reading. At http://arxiv.org/PS_cache/gr-qc/pdf/0603/0603053v1.pdf ,Vladimir S. MASHKEVICH writes: “However, Kretschmann argued that equations originally written in any coordinate system may be extended to all coordinate systems and thus made covariant; therefore the principle of general covariance involves no physical content. Einstein concurred with the argumentation.” Thorne, Lee, and Lightman (Phys. Rev. D 7: 3563-3578 (1973)) parenthetically at page 3568 seem to accept it too, at least in part, writing: “An argument due to Kretschmann shows that every spacetime theory possesses generally covariant representations.”

In summary, covariance is a guideline of formalism that does not make a specific assertion of a law of nature. For a law of nature, more is needed, and Einstein seems to have partly understood this when he wrote of “general” covariance.

The Hilbert-Einstein equations comply with the formal requirement of covariance. They are essential parts of both the orthodox “general theory of relativity” and the Logunov relativistic theory of gravity. Their discovery was largely due to the physical genius and epochal originality of Einstein. But they are not the whole of the theory of gravity. Four more equations are needed. For this, the orthodoxy proposes what it calls “coordinate conditions”, but these fail to express the physics needed to make up a proper whole theory of gravity. More is needed.

Part of that more is form invariance (at pages 17-18 in A. Logunov, Lectures in Relativity and Gravitation, A Modern Look, translated into English by A. Repyev, Nauka, Pergamon, 1990; see also at pages 59 and 88 in http://arxiv.org/PS_cache/physics/pdf/0408/0408077v4.pdf). Form invariance is a requirement on frames of reference. In particular, for the theory of gravity, the frames of reference must reflect the global Minkowski geometry of spacetime. In contrast with covariance, form invariance has physical content, and reveals laws of nature. This is not recognised in the orthodoxy with its failed “coordinate conditions”, but it is expressed in their replacement in the Logunov relativistic theory of gravity by the four field structure equations (3.5.31 in Chapter 3 of the Lectures). The Logunov theory recognises that the validity of inertial frames of reference, distinct from the forces of gravity, is a law of nature, but the “general theory of relativity” fails to do so, and thereby omits essential facts about gravity.

Failure to understand the difference between covariance and form invariance is part of the muddled theoretical state and part cause of the consequent impossibility of empirical verification of the orthodoxy of the “general theory of relativity”, with its fatal confusion between the Minkowkski geometry of spacetime and the Riemannian geometry of the dynamical manifold.

Christopher

I am sorry I have not previously known of the following web reference to a free full length (254 pages) textbook. http://arxiv.org/PS_cache/physics/pdf/0408/0408077v4.pdf
It gives a .pdf file of a book with an easily grasped, thorough, clear, and physically reasonable account of the principle of relativity of spacetime in physics.

This is the relativity theory that underlies a proper account of the laws of gravity. It starts from the most elementary conceptions, and leads logically, with physical insight, to the principle of least action that is used for the construction of the Hilbert-Einstein equations, in terms of covariance and form invariance. It does not go so far as to give an account of the laws of gravity, but it gives the basic knowledge of relativity theory that is needed to understand them.

For a proper account of the laws of gravity, one needs this, but also more: one needs also an understanding of the modern field theory that has been developed since about 1965, to characterise the gravitational field, unknown to Einstein, obviously enough. There are reasons to believe that the laws of gravity are best stated in terms of a rank 2 tensor field of spin magnitude 2 and 0. This latter information is expressed in the 'field structure equations' that together with the Hilbert-Einstein equations make up the laws of gravity as they are at present best stated. The field structure equations also distinguish between inertia and gravity.

The reference above is a good start to understanding the laws of gravity. It is a free textbook.

Christopher

Physical causality is the very basis and foundation of empirical science, of the experimental method. An overthrow of causality would be an overthrow of empirical science.

If the orthodoxy of the “general theory of relativity” were to establish its overthrow of causality, it would thereby invalidate its own status as an empirical science. “No reasonable definition of reality could be expected to permit this.” (Phys. Rev. 47:777-780, 1935.)

(Newton’s mechanics was deterministic but did not respect causality, and this rightly worried Newton. Determinism is a different animal from causality, though some muddled mentalities confuse the two.)

Christopher

My slow and crumbled intellect has crept a little forward into understanding the theory of gravity. I am making a comparison between on the one hand the authoritatively established orthodoxy of the Einstein “general theory of relativity”, and on the other hand the relativistic theory of gravity, the latter as investigated by various authors over the years before and since the epochal discoveries of the genius of Albert Einstein. Some of those authors include Henri Poincaré, Hermann Minkowski, Alfred Arthur Robb, Alfred North Whitehead, Nathan Rosen, Robert Harry Kraichnan (who died on 26 Feb 2008), Vladimir Alexandrovich Fock, Suraj N. Gupta, Walter E. Thirring, Richard Phillips Feynman, Steven Weinberg, J.C.W. Scott, Norbert Straumann, and Anatoly Alexeyevich Logunov. I have been guided by the approach of Alfred North Whitehead and Anatoly Alexeyevich Logunov. In particular, I have been reading:
A. Logunov and M. Mestvirishvili, The Relativistic Theory of Gravitation, translated from the Russian by Eugene Yanovsky, Mir, Moscow, 1989.
Anatoly Logunov, Lectures in Relativity and Gravitation: A Modern Look, translated from Russian by Alexander Repyev, Nauka, Pergamon, 1990.
A.A. Logunov, The Theory of Gravity, translated by G. Pontecorvo, Nauka, Moscow, 2001. This can be downloaded from the web for free at http://aps.arxiv.org/PS_cache/gr-qc/pdf/0210/0210005v2.pdf.

Anatolii A. Logunov, Relativistic Theory of Gravity, Nova Science, Huntington, 2001.

For myself, I have found that the relativistic theory of gravity of Logunov and his colleagues is currently the best available guide to understanding the physics of gravity. I would be grateful for further guidance.

In a nutshell: Because of causality, the geometry of physical spacetime is strictly Minkowski, not general Riemannian. Physical spacetime is flat, not curved. This can be understood as follows.

The orthodox “general theory of relativity” of Albert Einstein is expressed in two sets of simultaneous equations, the Hilbert-Einstein equations, and the so-called “coordinate conditions”, taken conjointly. This conjunction is a failure. It fails to provide unambiguous predictions, and cannot be tested empirically. This is because the “coordinate conditions” are arbitrary and fail to express physical laws which are necessary for the physical meaning for the Hilbert-Einstein equations. The orthodox “general theory of relativity”, by itself, without arbitrary supplementary assumptions, does not lead to the parametrised post-Newtonian (PPN) approximation.

For a valid theory of gravity, built on the Hilbert-Einstein equations, the “coordinate conditions” must be replaced by what I like to call the field structure equations, formulated by Logunov and his colleagues as in the above references. These equations are not at all arbitrary, but rather, they express physical laws that are necessary to give physical meaning to the Hilbert-Einstein equations. The conjunction of the Hilbert-Einstein and the field structure equations makes definite and testable predictions. In particular it rigorously and unequivocally leads to the PPN approximation for empirical tests within the solar system. These have strictly and significantly empirically verified the relativistic theory of gravity as formulated by Logunov and his colleagues.

In short, the orthodox “general theory of relativity” and the Logunov relativistic theory of gravity formally share the Hilbert-Einstein equations, but they give entirely different meanings to them. Because of the failure of its “coordinate conditions”, the orthodox “general theory of relativity” is left behind as a half-baked empirically untestable ideology, while the Logunov relativistic theory of gravity, because it uses the correct field structure equations, is sound physics.

The orthodoxy with its nonsensical dogma of curved spacetime weakens the intellectual coherence of scientific discourse in general.

Christopher

Moving on from Newton’s theory of absolute time as independent of absolute space, we take causality as absolute, and this leads us to take up Minkowski geometry. For physics, we demote time from a principal fundamental notion, a notion that together with space defines the eternal absolute Newtonian-Platonic receptacle; we demote time to an abstraction that measures concrete adventures. The eternal absolute receptacle is no longer a primary abstraction for us.

What now for physics is conceptually primary and absolute in its place?

Causality, the absolute causal structure of adventures, of causes and effects, of processes, and, abstractly, of point-events, is the most fundamental concept and the most general law of nature.

Christopher

Causality-preserving transforms constitute a group called the causality group by E.C. Zeeman, J. Math. Phys. 5(4): 490-493 (1964), and Topology 6: 161-170 (1967). The causality group is composed of the Poincaré group and the scale changes (dilations). The Poincaré group is well known, and is composed of
(1) the translations of spacetime origin (four parameters);
(2) the rotations of space about the time axis (three parameters);
(3) the Lorentz boosts (three parameters).

The principle here is that causality is enough to generate Minkowski geometry.

Mathematically, by causality is meant that we start with a set R^4, merely coordinatised by the real numbers, with no topology, no concept of continuity, no metric, no pseudo-metric, no concept of linearity, no congruences, no geometry; not yet a spacetime, just a coordinatised set of point-events; and then we postulate just a certain partial ordering on this set, defined in terms of timelike (future and past and so ordered) and spacelike (and so not ordered) separations.

From the partial ordering and the coordinatising alone, we find that if we want a geometry for the set, it must be the Minkowski geometry. With no assumption of continuity or linearity, we find the causality group, with its subgroup, the Poincaré group, and its subgroup, the Lorentz boost group, all continuous and linear.

This was perhaps first discovered by Alfred Arthur Robb. His first publication of it was in 1913 (Heffer and Sons). Easier to find in the library is a version by the Cambridge University Press, 1914, A Theory of Time and Space. A second edition of the latter was entitled Geometry of Time and Space, Cambridge, 1936. Robb’s proof of 1914 takes up a whole book with 373 pages, because it works from simple principles and does not use the facilities of modern group theory, analysis, and topology. Robb also wrote The Absolute Relations of Time and Space, Cambridge University Press, 1941, but till now I have not been able to borrow a copy of it.

This was (perhaps independently) re-discovered by A.D. Alexandrov (1953), in Russian and not read by me (see http://www.cs.utep.edu/vladik/1998/olg98-4.ps.gz); and again by E.C. Zeeman (1964), as above. The Zeeman article is cavalierly written and its abstract fails to make it clear that the coordinatising of R^4 and the causal ordering are enough to construct the Minkowski geometry; the abstract gives the false impression that Minkowski geometry is assumed. The partial ordering was called the conical ordering by Robb, and the causal ordering by Zeeman. A.D. Alexandrov (Canadian Journal of Mathematics 19: 1119-1128 (1967) discusses transformations that preserve light cones, but his assumptions are much more restrictive, less general, than Robb's.

Minkowksi geometry encodes that there is a universal maximum speed of propagation of causal agency. Physically, in spacetime, light in a vacuum propagates at that speed, but is slowed down when it passes through a material with a high refractive index, such as diamond, and when it passes through the gravitational field of a heavy object, with a virtual “refractive index”, as seen in the Shapiro delay. Though its speed has not been directly measured, I am inclined to guess that gravity also propagates in a vacuum at that speed, but some theories suppose that it is carried by a “particle” with a very tiny non-zero rest mass, and therefore travels slightly less fast. It is not clear to me whether gravity is slowed down as it passes through the gravitational field of a heavy object. The “general theory of relativity” makes these measurements of the speed of gravity conceptually meaningless; it assumes them away. To give them physical meaning we need Minkowski geometry.

Zeeman 1964 points out that in “two-dimensional spacetime”, “causality” is not enough to generate “Minkowski” geometry. A higher dimensionality is needed for causality to do that, a remarkable fact. I do not know of an explanation of why physical causality requires precisely one time and three space dimensions; it is just the delivery of experience. A 1967 paper of Zeeman derives the Minkowski geometry and the Lorentz transformations from a simple topological premise, using the fine topology, defined again in terms of timelike (future and past) and spacelike separations.

This may be regarded as one good explanation of why Minkowski geometry is so useful for physics, which is largely concerned with causality. Minkowski geometry is just R^4 with congruences defined by the Poincaré group, that is to say, the causality group confined to a definite fixed scale. The congruences make the point set R^4 into a spacetime with a geometry.

A less fundamental explanation of why Minkowski geometry is so useful for physics was offered by von Ignatowski and later by Whitehead, by way of a universal maximum speed of propagation of causal agency, and another kind of explanation by Logunov who relates it to the motion of ponderable matter. It is an utterly inexplicable bedrock explanatory principle of physics that causal agency includes the experimenter’s powers to move and feel the motion of ponderable matter, and to make and see light. Thus these apparently different kinds of explanation of the usefulness of Minkowski geometry are related in physical terms.

(Very often causality and determinism are conceptually conflated, commingled, confused, or confounded, but logically they are very different animals, and nothing here is concerned with determinism. The conceptual difference is wide and deep and therefore I will not distract attention to or examine it here.)

The notion of physical causality is one of the strongest deliveries of experience. Physical causality is the ultimate abstraction of physical experiment, the most fundamental and universal law of nature. Occasions of experience are the ultimate concrete actually existing realities, the empirical entities from which we start our work of theoretical abstraction.

Christopher

In Part I of his book Principles of Relativity Physics, Academic Press, New York, London, 1967, James L. Anderson, at pages 1-101, gives an account of the methods of tensors and covariant derivatives and the like.

In Part III, “Dynamical Space-time Theories”, in his introduction to Chapter 10, “Foundations of General Relativity”, at pages 329-330, Anderson tells us about Einstein’s assumptions for his “general theory of relativity”. Anderson writes “…Einstein succeeded in actually eliminating geometry from the space-time description of physical systems …” Anderson goes on to clarify “While it is still convenient to use geometrical terminology in discussing the general theory, nowhere is it necessary to use the gravitational field tensor g mu nu explicitly as a metric (for example, in an expression for the distance between two neighbouring points).”

Christopher

In Part I of his book Principles of Relativity Physics, Academic Press, New York, London, 1967, James L. Anderson, at pages 1-101, gives an account of the methods of tensors and covariant derivatives and the like.

In Part III, “Dynamical Space-time Theories”, Chapter 10, “Foundations of General Relativity”, Section 10.4, “The Einstein Field Equations”, at page 347-348, Anderson tells us that “Any system of local differential equations will in general admit a large number of physically inequivalent solutions. To decide which one of these solutions are actually realized in nature, it is customary to supplement the equations with boundary conditions.” Anderson then notes that Einstein “suggested that one should not give boundary conditions to supplement the field equations [of his ‘general theory of relativity’], but rather should require that all solutions lead to geometries that are spatially closed.” Anderson goes on to say that “The question of boundary conditions is still to a large extent an open one.”

Although Anderson does not draw the following conclusion, and does not seem to even consider it, these remarks of his are in effect a polite way of saying that Einstein’s “general theory of relativity” is indefinite, ambiguous, or non-specific as to some physical predictions that might be asked of it, and is therefore in those respects not empirically falsifiable. It is only as to differential relations that it is falsifiable.

Christopher

Good luck with your new business

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