"We study the nonlinear evolution of a weakly perturbed anti-de Sitter (AdS) spacetime by solving numerically the four-dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant. Our results suggest that AdS spacetime is unstable under arbitrarily small generic perturbations."
Piotr Bizoń, Andrzej Rostworowski
Notice that this study was done in 3+1 dimensional AdS4, but the authors claim (in the conclusion) that they observed "qualitatively the same behavior" for the 4+1 dimensional AdS5.
But with all the activity about AdS/CFT, it is it hard to believe that nobody checked the stability of AdS in classical GR before.
I posted this puzzle a while ago on wbmh, but subsequently it got deleted and I think it is worth reposting it here. Please write a comment if you have an answer (or remember the discussion on wbmh) and win the famous golden llama award.
Perhaps you have seen the video clip of astronaut David Scott dropping a hammer and a feather on the surface of the moon, repeating Galileo's famous (thought)experiment.
But strictly speaking in a high precision experiment the two objects will in general not hit the moon at the exact same time.
Why is that?
Let me clarify a few things.
We assume that the shape of the objects makes no difference (we assume the spherical cow approximation is valid) and the surface of the moon is perfectly smooth. Further we assume that there is no trace of an atmosphere on the moon and no electrical charges (attached to the objects). We assume that the presence of the astronaut (and his gravitational field) can be neglected and we assume the sun, earth and the other planets are sufficiently far away. We ignore quantum theory and assume that the many-worlds interpretation and any other conspiracy theories can be neglected...
added later: Furthermore we assume that all objects and observers move slowly compared to the speed of light, so that we can use the standard St. Augustine definition of simultaneity.
update: Akshay Bhat is the proud winner of the famous golden llama award ...
... he will enjoy a free subscription to this blog for a whole year. Congratulations!
If you want to see my own solution to this puzzle click here.
In this puzzle we consider a high precision re-enactment of the famous Galileo (thought)experiment and the claim is that hammer and feather dropped simultaneously from a height h will not hit the ground at exactly the same time.
In order to see this, we increase the height h and consider the full 3-body problem with feather (F), hammer (H) and moon (M) approximated as spheres (the famous spherical cow approximation).
Next we increase the distance between F and H and we increase the mass of the hammer H significantly. Therefore the moon will move towards H by a certain displacement a and thus the hammer has to travel the distance h - a until it collides with the moon M, while the feather F has to travel the increased distance sqrt( h^2 + a^2 ), which suggests that F will indeed collide with M slightly later than H.
But we have no full proof yet (notice that the feather is attracted by moon+hammer while the hammer is slightly less attracted by moon+feather and this could compensate for the different distances).
So in order to obtain full proof of our claim (without using too much math) we move the hammer H even further away from the feather F and we increase its mass until it exceeds the mass of the moon M significantly (perhaps it is easier to decrease the mass of the moon until it is more like a hammer).
But now we have transformed this thought experiment into a configuration where F and M are dropped on H, but from very different heights h and 2h. In other words, comparing with the original configuration, the assumption that the three bodies will collide at the same time is disproved by reductio ad absurdum.
I would like to make three more remarks:
1) The equivalence principle is an idealization (notice that in the general theory of relativity we consider test bodies to have infinitesimal mass and distances are small compared to the radius of curvature).
2) If we make the spheres small enough they will in general not collide at all (I leave it as an exercise for the reader to run a 3-body simulation and check this claim), except for symmetric initial configurations like the one in the last picture.
3) The contemporary opponents of Galileo could have made this reductio ad absurdum to counter his argument and discredit his physics. It is interesting to contemplate how science would have progressed in this case ...
We do know how to do the calculations, we can determine the probabilities for various experiments, real or imagined. In fact, an amazing machinery of methods and tools has been developed over decades for this purpose; A cornerstone of modern physics and science in general.
And yet, important foundational questions remain unanswered.
I am talking, of course, about statistics and our theories of probability.
Recently I found this:
"This book is about one of the greatest intellectual failures of the twentieth
century - several unsuccessful attempts to construct a scientific theory of
probability. Probability and statistics are based on very well developed
mathematical theories. Amazingly, these solid mathematical foundations
are not linked to applications via a scientific theory but via two mutually
contradictory and radical philosophies. One of these philosophical theories
(frequency) is an awkward attempt to provide scientific foundations for
probability. The other theory (subjective) is one of the most confused
theories in all of science and philosophy. A little scrutiny shows that in
practice, the two ideologies are almost entirely ignored, even by their own supporters."
But I guess before I buy the book I will browse the blog a bit more...
added later: ... and read some reviews: negative and positive. (I thank Jonathan for the links.)