# Are a tachyonic particle a mathematical impossibility (not just physical)?

I recently learned from a helpful SE user that, in general relativity, the "law of geodesic motion" is actually a mathematical law, not a physical one. That is, a "test particle" (infinitesimal blob of spacetime curvature that does not disperse), when surrounded by "vacuum" (zero Ricci curvature) is mathematically constrained to follow a geodesic path.

Now I'm not familiar with the proof of this fact, but it got me thinking. A tachyon (in the original sense of a superluminal particle) should likewise be mathematically constrained to follow a spacelike geodesic.

But spacelike geodesics do not exist in the conventional sense of a maximal-interval path. That's because you can always deform a spacelike path to make it longer in space, or you can deform it to make it "walk back and forth more in time" (rotate the local tangent vectors closer to the light cone), thus reducing its interval.

I guess spacelike geodesics still exist in the sense of paths that parallel transport their own tangent vector, though I don't really have the intuition for how that works in this case.

But I figured maybe the failure of spacelike geodesics to be proper maximal paths might also prevent them from being paths that a localized blob of curvature could follow, even hypothetically. Can someone weigh in here?

• How can I be physically impossible? Of course I am possible, I am here and talking to you. Commented Jul 25, 2021 at 12:42
• Does thinking about the spacelike geodesic as a stationary-interval path help? Commented Jul 25, 2021 at 14:43
• @m4r35n357 Would you mind elaborating a bit? I thought stationary meant either maximal or minimal under infinitesimal variations. A spacelike path is neither, since you can either grow or shrink its interval in the ways noted. Commented Jul 25, 2021 at 15:48
• @m4r35n357 Oh, I get it! As in this post. Thank you! Commented Jul 25, 2021 at 15:54
• That is all I have, I just noticed an assumption in the OP and questioned it ;) I did look briefly here en.wikipedia.org/wiki/… but was no wiser for it. Your link explains things better! Commented Jul 25, 2021 at 16:04

Maybe there’s some context to this statement that somehow saves it, but taken alone it looks wrong:

the "law of geodesic motion" is actually a mathematical law, not a physical one

There’s some freedom in what’s assumed and what’s derived, but this can be one of the starting assumptions to motivate the deviation of the Einstein equations. I find it difficult to then say it is mathematical and not physical. For example Schutz calls this Postulate IV and then immediately generalizes it to the strong equivalence principle.

Some related technical notes here, including in the comments under my answer about what’s assumed and what’s derived by various authors: Christoffel symbol and covariant derivative

As regards tachyons, you certainly can draw space-like curves, but, as noted, the structure that free particles follow time-like geodesic gets built into the structure from the start.

• I see, that's interesting. So I guess the general idea is that the field equations and the geodesic principle are not independent, so they don't need to be postulated separately. I think that's what's most amazing about GR: it strips away so many assumptions, such that what must be assumed is not much more than the existence of the Lorentzian spacetime manifold itself! Commented Jul 30, 2021 at 20:42

the "law of geodesic motion" is actually a mathematical law, not a physical one

I am not sure , this statement is completely accurate. It is true only in the sense, that most laws are mathematical laws because they describe the behaviour of some quantities of a model in some mathematical way.

But, all physical objects in the real world, actually follow this law, so saying it is not a physical law does not seem correct to me.