Memory then and now

I’ve recently given a talk about the memory effect at the Australasian Society for General Relativity and Gravitation. The society celebrated 30 Years of Gravity Research in Australasia, and hosted a conference at ANU devoted to Past Reflections and Future Ambitions.

This seemed like a good opportunity to talk about the memory effect which was discovered, mathematically, around 30 years ago, and will likely be discovered, experimentally, in far less than 30 years. (Many of the other talks were in fact more about the experimental aspects of gravitational wave experiments, see conference program.)

Most of my talk was devoted to an explanation of Christodoulou’s 1991 PRL paper Nonlinear nature of gravitation and gravitational-wave experiments, which was also for me, personally, my first encounter with mathematical general relativity.

I remember fondly walking into Demetri’s office, having expressed my interest in writing my diploma thesis with him, when he told me

You are a physicist right? Now I have this paper, and the physicists tell me they find it difficult to read. So maybe you can explain?

In following months I had the opportunity to interact with him, and learn about his work; the result was 48 page exposition of his 4 page paper, which — looking at it now — is still a good companion for anyone who sets out to read his paper in detail: Diploma thesis (2008).

For an interesting anecdote about the involvement of Kip Thorne in the composition of this PRL, I recommend listening to Demetri’s farewell lecture (2018), (where also the above photo is taken from).

One of the reasons his paper is difficult to read is that it builds up on 500 page proof of the stability of Minkowski space (1990) by D. Christodoulou and S. Klainerman. While citations do not really mean anything in mathematics, it is nonetheless indicative of the influence of their work, to note that citations have roughly doubled in every decade since its publication:

Some of these citations can be attributed to the research activities of their students, and their respective students, of whom there are now more than a hundred, me being one of them. While few have read this book cover to cover, it has been an inspiration to many, and in particular set a bar in terms of what is possible in mathematical general relativity.

In the talk, I then spent some time describing an idealised gravitational wave experiment (as it would be carried out in space), and introduced the notion of memory simply as: we say a gravitational wave has memory if looking at the test masses that make up the detector, the configuration finally, namely after the passage of a gravitational wave train, differs from the initial configuration.

The point is that in linear theory (namely treating the Einstein equations say as system of linear wave equations for the curvature on Minkowski space) and using a slow motion approximation (for the sources, viewed as given and slowly varying) it has been known for a long time that a memory occurs for example in a binary merger if the final mass is moving relative to the initial centre of mass frame. Since this is typically not the case, the memory effect was deemed negligible.

For a detailed derivation of the linear theory along these lines, see for instance my lecture notes from the AMSI Summer School (2024).

Christodoulou’s contribution is that even in the setting when the linear theory predicts no overall change in the configuration of the test masses, there is a permanent change, a non-linear memory, whenever gravitational waves have been radiated. Moreover this effect is non-local, and non-linear, meaning that in order to understand the displacements that occur in a given direction from the source, the radiated energy in all directions has to be taken into account; and the shear enters quadratically into the following formula:

As the following photo from the talk conveys, the point I am trying to make is that the final displacement of test masses is obtained from an integral over all directions, against a certain kernel, of the a function which measures the total energy radiated in a given direction:

This contribution is not negligible. (See discussion in the paper.)

In the remainder of the talk, I outlined some of the more recent developments in mathematical relativity, and restricted myself to topics which closely related to this topic. For example, it won’t escape the reader that Christodoulou’s argument, unlike in linear theory, does not rely on the validity of the quadrupole formula; in fact, no reference is made to any matter system, and the memory observed here is purely due to the propagation of gravitational waves, and their non-linear interaction, in vacuum.

In the last decade a number of extensions of the stability of Minkowski space, to Einstein-matter systems have been achieved; however, to study the outgoing radiation from matter system — and only the radiation emitted from an isolated system — one needs to implement the so-called no radiation condition. That leads to a scattering problem, for which there has also been exciting progress.

While the quadrupole formula is not invoked in Christodoulou’s argument, it is nonetheless interesting to investigate its validity in the context of a fully non-linear theory. A good starting point is the 1976 paper of Ehlers et al lamenting the state of affairs:

The precise form of the no-incoming radiation condition was not known at the time of writing of Ehlers’s critique, but is now understood, again due to the results of Christodoulou and Klainerman, who also established the Bondi mass loss law in non-linear theory.

Looking forward, remembering that this was a workshop on past reflections, and future ambitions, I concluded the talk with a discussion of an open problem:

The discussion of this problem, and recent developments related to it, would be the subject of another talk. Instead, I’m going to end this post with an amusing commentary of the above paper of Ehlers, to be found in a book by Poisson and Will, Chapter 11:

But in a letter published in the Astrophysical Journal in 1976, Jürgen Ehlers, Arnold Rosenblum, Joshua Goldberg, and Peter Havas argued that the quadrupole formula could not be justified as a theoretical prediction of general relativity.

They presented a laundry list of theoretical problems that they claimed had been swept under the rug by proponents of the quadrupole formula. Among them were these: people assumed energy balance to infer the reaction of the source to the flux of radiation, but there was no proof that this was a valid assumption; no reliable calculation of the equations of motion that included radiation reaction had (in their opinion) ever been carried out; many “derivations” of the quadrupole formula relied on the linearized theory, which was clearly wrong for binary systems; since higher-order corrections had not been calculated, it was impossible to know if the quadrupole formula was even a good approximation; even worse, higher-order terms were known to be rife with divergent integrals. There was considerable annoyance among holders of the “establishment” viewpoint when this paper appeared, mainly because it was realized that its criticisms had considerable merit. As a result many research groups embarked on a program to return to the fundamentals and to develop approximation schemes for equations of motion and gravitational radiation that would not be subject to the flaws that so disturbed Ehlers et al. Among the noteworthy outcomes of this major effort was the fully developed post-Minkowskian formalism that forms the heart of this book. Toward the end of his life, Jürgen Ehlers, one of the great relativists of his time, admitted to one of us (after some prodding, to be sure, and only up to a point!) that the justification of the quadrupole formula was in much better shape than it was in 1976.

Experimentally, the situation was not at all controversial. By 1979, Taylor and his colleagues had measured the damping of the binary pulsar’s orbit, in agreement with the quadrupole formula to about 10 percent; by 2005, the agreement was at the 0.2 percent level. The formula has also been beautifully confirmed in a number of other binary-pulsar systems.