# Does antimatter curve spacetime in the opposite direction as matter?

According to the Dirac equation, antimatter is the negative energy solution to the following relation:

$$E^2 = p^2 c^2 + m^2 c^4.$$

And according to general relativity, the Einstein tensor (which roughly represents the curvature of spacetime) is linearly dependent on (and I assume would then have the same mathematical sign as) the stress-energy tensor:

$$G_{\mu \nu} = \frac{8 \pi G}{c^4}T_{\mu \nu}.$$

For antimatter, the sign of the stress-energy tensor would change, as the sign of the energy changes. Would this change the sign of the Einstein tensor, causing spacetime to be curved in the opposite direction as it would be curved if normal matter with positive energy were in its place? Or does adding in the cosmological constant change things here?

• Possible duplicates: physics.stackexchange.com/q/9371/2451 , physics.stackexchange.com/q/9375/2451 and links therein. Nov 3, 2013 at 21:49
• @Qmechanic Related, not duplicate.
– user28737
Nov 4, 2013 at 11:04
• If antimatter is the result of (very crudely) flipping some quantum numbers to the opposite value, why would you expect mass to be affected?
– jim
Jul 17, 2017 at 18:58
• @jim I find that I agree with your question. Also, I love your name. Something about it is very appealing. I hearby accept you as a subject and vassal of the great Jimpire
– Jim
Nov 8, 2019 at 13:31
• I want to give this link for experiment running at CERN to test the behavior of antimatter to gravity home.cern/news/news/experiments/… . Dec 7, 2019 at 6:59

Antimatter has the same mass as normal matter, and its interaction with gravity should be the same according to GR and QM.

That said, antimatter has only been created in tiny amounts so far and only few experiments have been performed to confirm there is no new physics involved.

The gravitational interaction of antimatter with matter or antimatter has not been conclusively observed by physicists. While the overwhelming consensus among physicists is that antimatter will attract both matter and antimatter at the same rate that matter attracts matter, there is a strong desire to confirm this experimentally, since the hypothesis is still open to falsification.

https://en.wikipedia.org/wiki/Gravitational_interaction_of_antimatter

• I want to give this link for experiment running at CERN to test the behavior of antiprotons to gravity home.cern/news/news/experiments/… . Dec 7, 2019 at 7:01

Currently there is no reason to believe/require antimatter has negative mass. It should therefore behave exactly the same in a gravitational field.

The matter-antimatter distinction is pretty arbitrary. We found protons/neutrons/electrons first, so particles of the same families that exhibit similar behavior are "matter", and those with certain properties (charge, baryon number, or something else, depending on the family) as opposite would be antimatter. We could call positrons as matter and electrons as antimatter and nothing would change except for our definition of lepton number (and the labels of the muon/tau).

When Dirac calls it a negative energy solution, he's looking at the case where we have a sea of ground state matter, and we excite one. The "hole" left behind by the excited particle behaves like the particle itself, but can recombine with an excited particle with no net energy change so one can view it as having a negative energy.

In this case, the hole does have negative mass because it is in a "sea" of positive-massed particles, and removing these leads to a hole with negative mass. And it behaves similarly from the POV as gravity.

In the general case, an antiparticle has the same energy as a particle.

• Then would this mean the negative energy solution isn't actually referring to antimatter per se, but rather to just a particle hole? Nov 3, 2013 at 23:47
• @abhishek when Dirac first discovered antimatter, he modelled it as holes inthe topmost filled state of negative energy states stretching infinitely down, which are already all filled by "ghost electrons" Nov 4, 2013 at 4:30
• @TobiasKienzler Ah, good point. I usually look at these thing from the theoretical POV, since in the standard model and all we do not ascribe negative energies to these. Then again, that doesn't entirely exclude a negative gravitational mass. Nov 4, 2013 at 9:09
• @TobiasKienzler Trudat. I don't see why it would be a far shot though, negative gravitational mass attracts negative gravitational mass if the inertial masses are the same. Pretty reasonable for clumping to occur, if the separation happened early on. Nov 4, 2013 at 10:06
• Neat - so we could theoretically have "antigalaxies" that are repulsed by "progalaxies", which might yield a contribution to the universe's inflation... I wonder if there are simulations/models of this Nov 4, 2013 at 11:05

Here's a naive argument for expecting antimatter and matter to both have "attractive" properties under gravity. General relativity describes gravity in terms of a valence 2 tensor $g_{\mu\nu}$. Upon quantization one would therefore expect a spin $2$ particle. The propagator might look something like

$$(g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho})\frac{i}{q^2+i\epsilon}$$

which in the nonrelativistic limit yields a universally attractive potential by comparing low energy scattering with the QM Born approximation. For more details see Peskin and Schroeder page 126.

Antimatter is not the negative energy solutions to the energy-momentum relation! Even in the Dirac sea model it isn't. In the Dirac sea model, the negative-energy modes are all filled with electrons, and the absence of one of those electrons is a positron. Just as the absence of a positive-energy particle has (relatively speaking) a negative energy, so the absence of a negative-energy particle has (relatively speaking) a positive energy. So the Dirac sea predicts that positrons have the same positive energy as electrons.

We also know experimentally that positrons have positive energy and positive inertial mass. If positrons and electrons repel each other gravitationally, then positrons have a gravitational mass that is negative, and therefore not equal to their inertial mass. That violates the equivalence principle. Without the equivalence principle, the spacetime model of gravity doesn't work. There's no question at that point of positrons curving spacetime in the opposite direction, because the whole notion that gravity is spacetime geometry is dead. Needless to say, I don't consider this to be very likely.

The sign of the stress-energy tensor does not change for antimatter. There are various energy conditions (ANEC, WEC, etc.) that stipulate various bounds on the stress energy tensor, but the only things that violate them are small scale quantum effects such as the Casimir force, the scalar inflaton field, and dark energy (which we don't yet know what it is, but could be, for example, the cosmological constant).

The ALPHA experiment demonstrates that antimatter (in this case, anti hydrogen) behaves the same as matter in a gravitational field:

http://www.nature.com/ncomms/journal/v4/n4/full/ncomms2787.html

The answer to your question is currently not known. Based on GR, it is believed that antimatter will curve space in the same way as ordinary matter. But if antimatter and matter repel each other, then GR is not correct. Then fundamental principles are completely different, see for example Cabbolet paper: http://onlinelibrary.wiley.com/doi/10.1002/andp.201000063/abstract Hard to read, but introduction is accessible. So then you might get that antimatter curves spacetime in opposite direction. Sergey

• FWIW, we now have confirmation that antimatter falls in Earth's gravity field. We still haven't empirically verified that antimatter gravitationally attracts matter (or antimatter), assembling sufficient quantities of antimatter isn't easy. ;) Apr 10, 2020 at 4:45

Just a simple (trivial ?) argument against anti-matter having negative mass (or reversing curvature) :

You have one star (or a single particle, if you prefer) made of normal matter, and another one made of pure anti-matter (or an anti-particle). You push them one to the other (by force, or whatever), so they anhihilate each other completely, giving you a lot of electromagnetic radiation. What will happen to the curvature ? Flat spacetime ? (if the anti-matter star reverse the curvature) And yet, electromagnetic energy do curves spacetime too, like normal matter. So is anti-matter !

For antimatter, the sign of the stress-energy tensor would change, as the sign of the energy changes. Would this change the sign of the Einstein tensor, causing spacetime to be curved in the opposite direction as it would be curved if normal matter with positive energy were in its place?

The sign of the stress-energy tensor would not change. The hole interpretation has been mentioned in other answers, but I prefer the Feynman-Stuckelberg interpretation as it is more in keeping with mathematical structure, as well as physical observation, and works for all particles, not just fermions.

Energy is the time component of a 4-vector. A negative energy would be a particle going backwards in time. We would therefore see the creation of a negative energy particle as the annihilation of a positive energy antiparticle, and the annihilation of a negative energy particle as the creation of a positive energy antiparticle (exactly as expressed in the field operators).

It is unambiguously the case in that we observe antiparticles with positive energy.

For space time too have an indent in the opposite side it will need the opposite of matter. A large volume of free space. True free space with almost no quarks. This will also have a large potential energy just as mass but it would have negative pull.

While I believe the answer to this is UNKNOWN, I have some thoughts to add to the argument.

If you picture gravity as being a field that bends the fabric of space time, you can then picture what a reverse warp of space time could look like and how that would affect something like a photon passing close by.

In both cases the photon would have an identical lens effect and so it would appear as though gravity is behaving identically.i.e. The orientation of the warp is irrelevant.

If you now move two particles together - one antiparticle with an imagined reverse gravitational bend, the combined space time bending would reduce as they begin to cancel out. Eventually it will flatten and there will be nothing to hold the energy of each particle in check. You'd convert from a mass into energy as expected for a matter antimatter collision.

Taking this further, you can remove the need for a black hole to contain a singularity. If you consider all mass is energy held in place by fields including the space time distortion, nothing in the dimensions we perceive needs to be truly solid. Two electrons for example could then occupy the same space, their size being unchanged from an electron size but their charge being x2 and their space time gravitational effect being also double that of an individual electron.

All of this is conjecture as the maths and other theories are something beyond my knowledge. However, theories are just that and free thought is always the best start point and should start from a blank slate.