###### Acceleration in Relativity. A Critical Introduction

Just as the first book came out, and just before the first conference I attended in 2008, there was a meeting of Time and Universe (tau) on the theme of the Unruh effect in Montreal, at the same venue as the spacetime ontology conference. Unruh himself was present. It was immediately obvious that there was a parallel between attempts to build a particle interpretation of standard quantum field theory ‘for’ accelerating observers and attempts to redefine EM radiation for accelerating observers. (In the classic case, the observers have eternal uniform acceleration in flat spacetime, so they are examples of Killing observers.) But when I wrote my first book, I had not heard about the Unruh effect.

At first glance, it seemed to me that the thesis of my book would also write off such attempts as physically pointless. The proposed particle interpretation (Rindler particles) depends on a choice of accelerating frame, but according to me, there are no physically natural accelerating frames. (The Unruh effect is still there, of course: when the Unruh-DeWitt detector accelerates uniformly through the standard Minkowski QFT vacuum, it registers, so it no longer functions as a Minkowski particle detector, since there are none in that field state. This is an interesting prediction about the standard QFT vacuum.) So the thesis in my first book would imply likewise that it was a very silly idea to try to attribute this or that point of view to accelerating observers. I was convinced deep down that I must be wrong, given the consensus that seemed to be reflected in that meeting. But there I was in my first ‘public’ appearance, unknown, with no credentials, and Angela Lahee had come from Springer hoping that I would promote my book!

Luckily nobody realised the connection so I had time to go home to our peaceful backwater in the Pyrenees and understand the basics of the Unruh effect. Ten months later, I felt reasonably sure my criticisms would carry over. The main arguments are presented qualitatively in the pdf file that summarises my talk at the Bad Honnef conference. But anyone familiar with the Unruh-DeWitt detector in the context of the Unruh effect will easily see the connection with my remarks above about accelerating detectors.

After Bad Honnef, and noting that nobody came to me with any major contradictions about the import of that talk, I set about writing a third book. The aim was twofold: first to provide a straightforward, physically comprehensible mathematical description (no fancy mathematical structures) of all semi-Euclidean frames adapted to accelerating observers in flat spacetime, with a complete classification of all rigid motions and a discussion of their relevance to such frames; and second to criticise what seem to me to be naive interpretations of physical quantities expressed relative to such frames.

A lot of other acceleration-related ideas got into the book, including criticism of the notion of Killing observer in general relativity and criticism of the idea that general relativity somehow explains inertia, along with more discussion of the way the existence of self-forces can in fact throw light on the notion of inertia. There was also an in-depth investigation of the clock and ruler hypotheses, as I understand them, in the light of Bell’s famous paper *How to Teach Special Relativity*, and some comments on Mashhoon’s papers, and in particular, his locality hypothesis, insofar as I understand it.

The title I chose was *Acceleration in Relativity. A Critical Introduction*. It was intended for anybody who had been through a first course in general relativity and understood it. Of the two referees found in extremis by Springer, one considered the subject interesting and the arguments cogent but felt that physicists would not be impressed by it (also that the notation was not modern enough). The other rejected it, we may say ‘out of hand’, as nonsense, dwelling heavily on my lack of credentials, which was rather disappointing. It implies that, without credentials, we are condemned always to remain without credentials!

I make the whole of this book available here in the original portable document format (1.6 MB). I am currently producing a shorter, sharper version, although not necessarily in any attempt to get it ‘peer reviewed’ by a publisher. If people are interested in it through this website and if I can distribute it even to a handful of such people in that way, it will be just as useful to me, since the sole aim, as always, is to get considered and rational criticism of the ideas. Note that Chap. 10 should be replaced by the pdf available here.

As I said above, it contains a down-to-earth physically comprehensible discussion of semi-Euclidean frames (with Euclidean constant time hypersurfaces) adapted to the motion of accelerating observers in flat spacetime, together with a complete discussion of the relevance of rigid motion, a complete classification of rigid motions in flat spacetime, and a few other relevant curiosities relating to frames in curved spacetimes. Apart from that, the main critical thesis is that, without introducing an acceleration symmetry into our theories of non-gravitational physics, there is little point trying to imagine what accelerating observers will think about their observations. All that matters is the way accelerating detectors will interact with fields, and this our theories tell us perfectly well without ever mentioning observers. So not to mince words, this implies that a considerable part of what Unruh theorists do is a pure waste of time from the point of view of physics, although the mathematics is quite wonderful, like all mathematics.