# 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 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?

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@Qmechanic Related, not duplicate. –  Waqar Ahmad Nov 4 '13 at 11:04
I hope so, anti-gravity propulsion if so, right? –  deed02392 Nov 4 '13 at 16:02

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

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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.

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Then would this mean the negative energy solution isn't actually referring to antimatter per se, but rather to just a particle hole? –  abhishek Nov 3 '13 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" –  Manishearth Nov 4 '13 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. –  Manishearth Nov 4 '13 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. –  Manishearth Nov 4 '13 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 –  Tobias Kienzler Nov 4 '13 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.

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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

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