# How can time run differently for different objects in models where the equivalence principle is broken?

In these three papers, the effect of gravity on anti-kaons, antiprotons, and positrons, is tested by how quickly time moves for them in the presence of Earth's (or the Sun's) gravitational field.

Tests of the Equivalence Principle with neutral kaons: Says that if anti-kaons responded differently to gravity, the kaon/antikaon mixing would be different throughout the year, as we move in the sun's potential, because the gravitational time dilation would vary through the year in a different way for the different species

Gravitational mass of relativistic matter and antimatter: Says that the positron would do synchrotron radiation at a different rate than the electron if they experienced the earth's gravity differently

A 16-parts-per-trillion measurement of the antiproton-to-proton charge–mass ratio: Says that the cyclotron frequency of the antiproton would be different than that of the proton if time was running differently for antiprotons in the earth's gravitational field.

Obviously, General relativity includes, and relies on, the equivalence principle: that the gravitational effect of a body is proportional to it's mass (inertia) and the movement of a test particle in a gravitational field is along geodesics, and therefore independent of what that body is made of. But these papers are referring to some GR-like model without an equivalence principle.

I know that in SR, time can run differently for differently moving objects, but in GR, my understanding was that time runs differently because of a property of the spacetime manifold near a massive body. So it seems to me that for the two different bodies to respond differently to gravity, they would have to be moving along different manifolds so that spacetime could be curved differently for particles vs antiparticles. This doesn't seem sensible to me - to have different objects moving along different manifolds. Isn't spacetime just one thing?

Forgive me, I know very little GR. Am I totally wrong about the nature of time dilation in GR? Is there some coordinate system where you don't need to interpret this as matter vs antimatter moving in different, independent spacetimes?

There are two points to be made here:

1. Since GR is a generalization of SR it automatically incorporates the time dilation from SR. In other words, GR adds time dilation affects caused by the curvature of spacetime, but it still includes the time dilation experienced by observers moving relative to each other.
2. Even within GR we have objects that respond to gravity in different ways. Massless particles necessarily travel along null geodesics while massive particles do not. This can give them very different orbital properties when moving near or around a massive object without requiring two different spacetime manifolds.

This being said, these experiments you reference are probing particles with the same mass in the same situation and thus GR would expect them to behave in the same way. The point of these experiments is to try and see if there are some small effects from new physics we don't know about yet that would treat them differently. Essentially experimental physics has two jobs: try to observe predicted results of our physical theories in order to support them, or, try to observe results that would break those theories and make us rethink things. Both of these are very important tasks but we now live in a time where we have two theories, GR and the standard model, that collectively explain most observed phenomena at reachable energy scales. We know they must break eventually, as one is classical and the other is quantum, but we need experimental results to tell us where they go wrong and lead us in the right direction.

EDIT:

Im adding on in hopes of fully answering the original question. These studies do not seem to make any attempt to replace GR, they just check if the weak equivalence principle (WEP) is violated. But the WEP is the thing that tells us the metric solution to Einstein's equations describes a spacetime manifold and massive test particles travel along the time-like geodesics of that manifold. If it was discovered that for some reason the inertial mass of antimatter did not equal its gravitational mass this nice image would no longer hold, test antiparticles would now take non-geodesic curves in spacetime, but the metric would remain unchanged and thus the manifold would be the same; there would not be two manifolds.

• Thanks for your response - was it mostly meant to confirm what I was stating about these experiments? I understood that these experiments are testing for violations of GR. I also understood that because they were comparing test masses on the same trajectory, the effect couldn't be just from SR time dilation. I didn't mean to ask "why are they testing this." I was hoping for someone to address the question: is their analysis modelling spacetime as being somehow two different manifolds, one for matter and one for antimatter, or is there some nicer coordinate system that fixes this? Apr 3, 2023 at 14:31
• I see, I think I misunderstood your question. But I believe my point still stands. All they are doing is trying to determine if something that GR predicts i.e. "particles of the same mass but opposite charge should be affected by gravity in the same way" is in fact true. I have not read the analyses in question but my point is that it is not generally experimentalists job to replace the old standing theory if they discover it doesn't hold up, so their analysis likely isn't modeling spacetime in any particular way but rather just seeing if GR is violated in this context. Apr 3, 2023 at 15:15
• Based on the abstracts my statement seems to be correct. They are just checking if the violation happens in nature, they are not speculating on what theory would replace GR. If your question was "what theory would explain this if the results violated GR?" then that is more of a philosophical discussion, but the answer is that you would somehow have to modify gravity to couple to antiparticles differently from particles, in which case the image of particles rolling along a universal spacetime manifold doesn't really make sense. Apr 3, 2023 at 15:22
• I see. Valid point. They may just be going "can we resolve the effect of GR time dilation on a system made of antimatter... yes." And they may not be necessarily worrying that the type of violation they're sensitive too would be VERY strange - GR being essentially correct in the way spacetime curves but somehow also different for different things. Apr 3, 2023 at 17:05
• Note: I accepted the answer in response to the edit. I like the description that we're testing whether or not antimatter follows geodesics - assuming that GR is otherwise valid for normal matter and for spacetime (an idea that many measurements support). Apr 5, 2023 at 10:31

The point is to answer the age old question, "Does antimatter fall up?" (https://cds.cern.ch/record/2243009/files/vol57-issue1-p039-e.pdf). An experimental test at CERN uses (used?) anti hydrogen, which is hard to make, and then observe a ballistic trajectory (which is hard to measure). This method measures a direct coupling to the local gravitational field.

Gravitational time dilation does not depend on local field strength, rather it arises from a difference in gravitational potential between two clocks. So the theory behind these experiments is to look at precise time-sensitive measurements between similar matter/antimatter systems. Very clever.

I don't think separate manifold are required for matter and antimatter. If equal masses experience different gravitational forces, then they'll have different potentials (per unit mass), and thus have different time-dilations. I am speaking in the weak field limit. How this would be coupled into a fully geometric theory is a problem for theorist.

In practice, one would expect to put an upper limit on difference between matter and antimatter in a gravitational field. A positive result would, of course, be revolutionary.

• I think unfortunately "how this would be coupled into a fully geometric theory" is actually the only thing I really wanted to know about. I know that Newtonian mechanics, and a description in terms of a potential, is a weak-field limit of GR. But I thought that even that is just a not-so-strong curvature of spacetime. Also, I work for one of those CERN experiments (ALPHA experiment). Which is why I was trying to get a better understanding of the clock based WEP tests. And the correct phrasing is "three different groups at CERN intend to use antihydrogen then observe a ballistic trajectory" Apr 3, 2023 at 17:01