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What is the difference between Timelike, Spacelike and Null geodesics? Does motion on a geodesic correspond to non accelerated motion? What kind of particles travel on these respective geodesics?
If we consider the Penrose diagram of Minkowski Spacetime, then we observe that timelike geodesics start from past timelike infinity and end at future time like infinity. Does that mean that any massive particle will follow this path?
Spacelike, null and timelike geodesics correspond to geodesics (in a spacetime with signature $-+++$) with a tangent vector $u$ of positive, zero or negative norm, $|u| = g_{\mu\nu} u^\mu u^\nu$. It does correspond to non-accelerated motion, in fact acceleration in general relativity is deviation from geodesic behaviour :
$$a^\mu = \ddot x^\mu + {\Gamma^\mu}_{\alpha \beta} \dot x^\alpha \dot x^\beta$$
Particles moving on those curves are particles of $p^2 = -M^2 < 0$ for timelike curves (generally called massive particles or tardyons), $p^2 = - M^2 = 0$ for null curves (generally called massless particles or luxons), or $p^2 = -M^2 > 0$ for spacelike curves (generally called tachyons).
Massive particles do not necessarily end at timelike infinity, but realistically they do. The only class of particles that can reach null infinity are constantly accelerated ones, such as Rindler observers.
