David:

Thank you for your extensive reply to my post about the twin clock scenario.

I was trying to create a fresh way of grasping the twin clock scenario, of getting an intuitive common sense take on “Einstein’s time dilation”, a fresh angle on its “reality”.

When I had done that, I saw that the twin clock scenario holds unexpected lessons that I had not originally seen in it, and I tried to indicate what they are. In a nutshell, the twin clock scenario has the core of the reason why the theory of gravity must be built on Minkowski geometry.

I chose to use Minkowski diagrams for the demonstration of the twin clock scenario.

I first offered a list of candidate intuitively simple explanations of the difference in duration registered in the twin clock scenario, in terms of clock rate, clock acceleration, and clock speed, and I invented a neutral observer for whom differences in these were all null and therefore not explanatory of difference.

I rely on the principle that an intuitively reasonable explanation should work for all observers, and I produce an observer who finds that none of the above three simple explanations work. By suggesting a higher dimensional picture than my two-dimensional Minkowski diagram, Burt’s comment brings it to our attention that there are countless other observers for whom they don’t work. Therefore none of the simple three, at least alone, is an intuitively reasonable explanation. This is an intuitive motive for looking for more subtle candidates, still with a common sense or intuitive slant.

My candidate explanation goes like this:

The rate of a clock, the speed of movement, and the acceleration are all differential conditions. They are rates. Besides intensive variables, densities, conveniently represented by tensors, physics is about extensive variables, global quantities. A physical explanation needs an integration of differential conditions, to produce definite results. We want some kind of definite integral.

There are various ways of integrating given differential conditions. One can do a line integral, or an area or volume integral, the result being a number. Or one can find a curve, for example an integral curve of a field of vector rates. There are various ways of setting definitive conditions for that. One can say that the curve must pass through certain points on the boundary or in the interior of the field, or that it must have derivatives with certain values in specified places. Or one might produce a surface as the result of the integration. And so on.

I chose the non-technical word adventure to try to suggest the general intuitive meaning of this variety of ways of setting the definitive conditions for integrating.

Of course it will need to be made precise for each scenario. For example, your choice of established terminology, “spacetime paths”, sometimes also called world lines when they are pervaded by an enduring particle. The mathematically fitting notion of spacetime paths provides an elegant and apt explanation in abstracto, but is bloodless, wanting allusion, not enough for a feeling of intuitive understanding. And I am still working here at a more general intuitive level; that is why I chose to speak of adventure.

Your reply contains the maxim:

“Just nice local measurements, and an accumulation thereof (anything with a memory, like clocks).”

This maxim is in agreement with the local aspects of adventures, but it also suggests a celebrated line of thinking, a line that is often taken to lead to the “general theory of relativity”.

Speeds and accelerations and relative clock rates are local, differential, notions. The maxim leaves open the method of accumulation.

The “general theory of relativity” is a statement of local, differential, conditions, with no free parameters, as you have noted, a fact most admirable and illustrious, even egregium. But it leaves open the method of accumulation, in its particular case, the boundary conditions. This is where it is ambiguous. For the setting of boundary conditions, the use of the “general theory of relativity” requires some theoretical supplementation, and, to be valid in practice, that needs a global Minkowski geometry.

The “general theory of relativity” thinks differentially, in terms of countless local tangent spaces each with its own private Minkowski geometry, its own nice local measurements, but it doesn’t think of putting them together into a single global geometry. So it neglects causality, which requires a global Minkowski geometry, as proved by Robb in 1913.

The twin clocks scenario, unexpectedly to me, turned out to contain the bones of Robb’s discovery, which is sometimes nowadays called the Alexandrov-Zeeman theorem.

In using an accelerated reference frame for the twin clocks scenario, one finds flagrant violations of causality. One finds clocks disappearing into nowhere, appearing from nowhere, and appearing in duplicate at different places at the same time. Indeed, no matter how small the non-zero relative speed of the away-clock, there will be such violations of causality in the accelerated frame.

Violations of causality are physical nonsense. The “general theory of relativity” just carries on as if this didn’t matter, and allows accelerated frames without a fuss. Consequently, it needs to get its boundary conditions from an external source, being alone in itself ambiguous about them.

“… all must be invariant to general coordinate transformations…” This statement expresses an admirable and important concept. For the orthodoxy it has been mesmeric. It is a very general statement, and I think it needs some provisoes.

It is also an admirably true and important statement that “the phenomena of nature care absolutely nothing for what coordinates we may assign to the points of space and time.” This is a valuable and powerful principle of physical explanation. I have used just that principle above for the twin clock scenario. For the orthodoxy I also has been mesmeric.

Neither of these important principles, however, is a law of nature. They are just methodological guidelines, advice about how to write laws of nature. It might also be said that nature doesn’t even care if we don’t use any coordinate system at all, or that nature doesn’t care if we don’t study physics. Half echoing a famous man: For a valid empirical science, laws of nature must take precedence over aesthetics of formalism.

The second statement as it stands is about so-called admissible coordinate systems, that is to say, coordinate systems of spacetime that respect the difference between space and time (but perhaps it was, or perhaps it was not, intended to be more general, and to refer also to coordinate systems that mix up space and time coordinates). It does not state that that just any old arbitrary system of equations, that accurately describe the phenomena of nature, reveals the laws of nature, that is to say, is form-invariant under transformation of spacetime coordinates. The accurate description might have been a lucky accident that does not reveal what we might think of as laws of nature such as we like to see. It took Einstein and Hilbert’s special skills and genius to find a very special just-right system of dynamical equations for gravity that would bring about form-invariance under transformation of spacetime coordinates. Form-invariance does reveal laws of nature. We want our dynamical equations to reveal laws of nature.

While nature does not care what coordinate system we use, nature does care very much not to violate causality, and this is a law of nature. It is a real physical principle, not just a methodological guideline. Methodological guidelines can be validly sidestepped in favour of other valid methodological guidelines. But laws of nature cannot be sidestepped.

Again, while nature does not care what coordinate system we use, we care very much that we can physically empirically verify our statements about nature. This is not a mere methodological guideline: it is an absolute mandate of empirical science. We need coordinate systems that we can use empirically.

These two concerns, causality and empirical verifiability, constrain our choices of coordinate systems. We should distinguish between reference frames and coordinate systems. We may allow a coordinate system that seems to show violations of causality provided that coordinate system has a transformation into another that does not. We will distinguish coordinate systems, that do not violate causality, and can be empirically measured, as reference frames, and use them as bases to generate more general coordinate systems. We may solve a particular problem in terms of an aptly chosen general coordinate system, but we will need the reference frame to guide us out of perplexities of causality violation. Also we will need our reference frame to be physically empirically verifiable, abbreviated by the term admissible. We can physically empirically verify things in a coordinate system only if it respects the difference between time and space. That is to say, it must respect the difference between clocks and measuring rods, for those are our facilities for physical empirical verification. Only a restricted class of coordinate systems for spacetime provide for those facilities.

The following paragraph is very closely based on pages 17-18 of Anatoly Logunov’s Lectures in Relativity and Gravitation: A Modern Look, translated from the Russian by Alexander Repyev, Nauka and Pergamon, first English edition, 1990; some sentences are word-for-word.

The “general theory of relativity” is importantly driven by the fact that the quantities that determine gravity are densities that can be expressed as tensors with respect to spacetime. Consequently we have access to a methodological notion of covariance: “An equation [relating unknown functions of the spacetime coordinates] is said to be covariant [under some arbitrary differentiable transformation of the spacetime coordinate system just when] its new unknown functions expressed in terms of the new [spacetime coordinates] satisfy equations of the same form as the old functions in terms of [the] old [spacetime coordinates].” This notion of covariance is a mighty help in the work of finding a just-right system of equations. But it doesn’t do the whole job; more is needed to achieve form-invariance of dynamical equations. “Form invariance for a metric under some transformation [(…)] is a more stringent requirement than the covariance of equations. This requirement is a constraint on the class of frames of reference: they must be such that when transformed into another the functional form of the metric tensor of spacetime would remain unchanged.”

Putting together these constraints on our coordinate systems and reference frames, we will find that Minkowski geometry is irremovably built into any correct and acceptable system of differential equations for the dynamics of gravity. This is denied by the orthodoxy of the “general theory of relativity”, which thereby invalidates itself. It is respected by the Logunov relativistic theory of gravity.

From a more directly empirical viewpoint, one would ask about the above-mentioned maxim about nice local measurements, do the neutral observer’s sightings through his telescope come within the definition of nice local measurements? Are astronomical observations nice local measurements? If not, would the maxim bar them? Why?

I would like to revise the last paragraph of my post that your reply refers to. “Time is not an event, nor a process, and so it cannot be caused, nor can it be a cause. It does not make sense to say that velocity or acceleration affect time, because such a statement seems to imply that time can be subject to, or patient of, causal agency. Time is in a sense part of the abstract theoretical receptacle that encompasses events and processes, which are causes and effects.”

Moving on from Newton’s theory of time as independent of space, we 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 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.

It is from causality that we construct the notion of spacetime. We find from causality that if we want a geometry for spacetime, it must be the Minkowski geometry. I have some comments on this in another post http://www.scienceblog.com/cms/einsteins-time-dilation-and-length-contra.... Once we have Minkowski geometry we are in a position to set up arbitrary coordinate systems and consider arbitrary coordinate transforms, according to desirable methodological guidelines, and respecting the laws of nature. This gives us a valid basis for a relativistic theory of gravity, with a geometrisation that sees test particles moving along geodesics and so forth.

The orthodoxy of the “general theory of relativity” proposes to take Minkowski geometry into account just as an afterthought, apparently plucked out of nowhere, not as a foundation because it is the most fundamental law of nature. The orthodoxy of the “general theory of relativity” says “Oh, we can take Minkowski geometry adequately into account by making all relevant tangent spaces have it, and tying them together by using a scale factor that makes the speed of light in a vacuum the same for them all.” But this is too late in the development, and it doesn’t provide the foundation of causality. Just the scale factor is not enough to put them all together into one underlying Minkowski geometry, such as is required by the fundamental law of nature that we call causality.

The ‘equivalence’ story of Feynman and Weinberg is not correct. They get a relativistic theory with general coordinate freedoms from a foundation of Minkowski geometry, but they do not show the converse, getting global Minkowski geometry from an orthodox “general theory of relativity”.

Einstein’s lofty, epochal, original, and brilliant insights into gravity can make physical sense only in the setting of an overall Minkowski geometry. His “special theory of relativity” is an expression of Minkowski geometry, which expresses all the relativity theory that is needed for the study of gravity, and indeed all the relativity theory that there is. Properly speaking, there is no “general theory of relativity”. Whitehead drew Einstein’s attention to this long ago, but Einstein rejected it, and the orthodoxy has followed Einstein. The proposal that Minkowski geometry drops out of the operative differential equations of gravity is mistaken. To come to terms with all this, perhaps a good start may be to read Logunov’s Lectures, cited above.

Christopher