# Motion without matter

Suppose you have a vacuum spacetime with non zero cosmological constant then you can show that two test particles will move toward each other or apart depending on whether it is negative or positive (AdS or dS) i.e. we have "motion without matter".

However, surely this is present in other vacuum spacetimes (even those with vanishing cosmological constant). For example, in Schwarzschild, the Ricci tensor and Ricci scalar may well both be zero, but the space is still curved. We still have, for example, a non-zero Kretschmann scalar and so two test particles will still move towards each other as they fall into the black hole i.e. we still have motion without matter.

Why then, did de-Sitter make such a big deal of his universe with positive cosmological constant having motion without matter?

Thanks.

• The Schwarzschild geometry has a non-zero ADM mass. You can argue that the spacetime contains no matter because the matter is all at the singularity and the singularity isn't part of the spacetime. However this seems excessively pedantic. I'm sure de Sitter would have scoffed at the idea that a Schwarzschild spacetime contains no matter, and I kind of sympathise with him. Oct 26, 2015 at 17:47
• @John Rennie Thanks very much for your reply. I take your point that Schwarzschild does indeed have "motion with matter" However, I thought that pure AdS also has a non-zero "ADM" mass. I put that in quotation marks since I'm under the impression we need to modify the normal ADM prescription in asymptotically AdS spaces (see under eqn 61 of this paper: arxiv.org/pdf/hep-th/9902121.pdf). How can it be considered "motion without matter"? Oct 27, 2015 at 10:38
• The ADM formalism dates from 1959, some forty years after de Sitter found his solution. It would have been "obvious" to de Sitter that his universe contained no mass. In any case you can't apply the ADM formalism to de Sitter space, which is after all what you are asking about. Oct 27, 2015 at 10:55
• @JohnRennie Sorry I still don't understand the situation with AdS. I accept that we can't apply ADM to AdS space but in the paper I mentioned above, don't they use some modified version in order to define mass on a space that isn't asymptotically flat and consequently discover that even vacuum AdS has some "mass"? Or are you simply saying that de Sitter's staement of "motion without matter" was because these future developments had not yet been discovered and so he still thought of pure AdS as having no mass? Thanks again. Oct 27, 2015 at 12:23
• You asked why de Sitter thought the de Sitter spacetime contained no mass, and I've answered that. If you're now asking what is the physical significance of the ADM mass in anti-de Sitter spacetime then that's a different question and you should ask it by posting a new question. Oct 27, 2015 at 12:28