Consider the 3 dimensional projective space minus the infinity point, and empty of matter.

We see that it has ADM mass. In other words, a perfectly fine geometry, orientable and asymptomatically flat, that has an unusual topology in our thought experiment, gives rise to mass in the absence of matter.

The question is why we don't consider the emergence of a sort of mass without matter a contradiction or a problem in the theory of general relativity, and instead accept the possibility of such a universe?

I'd like to see both physical and mathematical arguments if possible.

Please watch this lecture video for further details:


  • $\begingroup$ How exactly are you defining "matter"? We already know in many simpler examples that mass doesn't have to come packaged in little well-defined lumps. $\endgroup$ – knzhou Mar 20 at 20:29
  • $\begingroup$ A vanishing energy–momentum tensor? $\endgroup$ – VVM Mar 20 at 20:36
  • $\begingroup$ There are many cases in physics where things are locally zero but not globally zero. How about the Aharanov-Bohm effect, where the vector potential is always locally zero but a phase shift happens anyway? $\endgroup$ – knzhou Mar 20 at 20:50
  • $\begingroup$ Also, it's kind of a moot point because you're considering an incredibly unrealistic situation. Forget about questioning whether GR is true in the physical world -- question whether your intuitive ideas of "mass" and "matter" should make any sense whatsoever in this highly artificial setup. $\endgroup$ – knzhou Mar 20 at 20:52
  • $\begingroup$ What exactly is being contradicted here? If you fill a box with photons then the box has mass even though the photons do not. Does that seem contradictory to you? $\endgroup$ – Javier Mar 20 at 21:16

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