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