Is a still object in 4D spacetime lightlike?

Question

Every particle in the universe is moving in spacetime: a massive "still" one (in the 3D sense) is moving in a purely timelike direction, a massive "moving" one in a direction with both timelike and spacelike components (but obviously timelike in total), a photon in a lightlike direction. We even theorized the tachyons that would move in a spacelike direction. But what about something that just doesn't change its coordinates, never, remaining in the same point of a manifold?

• Can it have a consistent explanation in Relativity?
• Would it be interesting?
• Would it be described as lightlike since its norm would be, well, zero?

Note: I'm not asking whether this case is physical: I know that it isn't. Tachyons aren't physical either, but they are mathematical objects that can be studied in Relativity.

Answer 1

Can it have a consistent explanation in Relativity?

If I understand it right, what you talking about is a single point $p$ in spacetime, may call it an event. So yeah, it is well defined as $p\in M$. However, it is unintresting as it does not posess any dynamics per se. As you said, it remains at the same point of the manifold.

Would it be described as lightlike since its norm would be, well, zero?

The "norm" is not defined on the spacetime but on its tangent spaces. Thus, you cannot call a single point/event light-, space- or time-like.

One thing I might add is that you can interpret the Green's function of a differential equation, take for example the wave equation $$(\partial_t^2 - \Delta)\psi = 0, $$ as the solution of the equation for a point like source (possibly in spacetime), i.e. a solution of $$(\partial_t^2 - \Delta)G(x,t) = \delta(x,t).$$ This might fits a little what you are searching for. However, it is just an interpretation of a mathematical tool and should not be taken too serious.

To come to a conclusion, a object sitting on a single point in spacetime is probably best described by that point $p$ but unintressting as it does not have any dynamics. Tachyons in turn have dynamical properties that can be studied. I hope this somehow helps you. Cheers!

Answer 2

But what about something that just doesn't change its coordinates, never, remaining in the same point of a manifold?

The world lines of particles with non-zero restmass are timelike curves within the lightcone. If at rest or not doesn't matter. A particle at rest moves through spacetime, which is all time in this case.

"At rest" in spacetime requires to stop time. You can't stop time for a particle though. If you could there would be no worldline und thus questioning of timelike vs spacelike etc. would be meaningless. You can connect two points on a spacelike hypersurface to obtain a spacelike wordline, but that's another story.