Why does physical entities travel in straight paths in a flat space-time and in geodesic in curved spacetime? Is it due to Inertia? If it is, then why does waves also follow the same pattern?

  • $\begingroup$ Consider mentioning how you define inertial frames in curved spacetimes. $\endgroup$
    – Qmechanic
    Sep 1, 2019 at 8:52

1 Answer 1


A free-falling particle follows a straight line independently whether a spacetime is flat or curved. What matters is how you define a straight line!

A straight line is defined as a curve which parallel transports its tangent vector.

In a flat spacetime (Minkowski) that definition matches with the intuitive concept of a straight line as the path of a light ray in our ordinary three dimensional space. In a curved manifold it is a geodesic obeying to the parallel transport constraint.

The geodesic equation is
$(D/d\lambda) (dx^\mu/d\lambda) = (dx^\nu/d\lambda) \nabla_\nu (dx^\mu/d\lambda) = 0$
$D/d\lambda$ directional covariant derivative
$dx^\mu/d\lambda$ tangent vector
$\nabla_\mu$ covariant derivative
$\lambda$ affine parameter

Note: Both a massive particle and a massless one (photon) follow a geodesic.

  • $\begingroup$ I understand that. My question was why does it follow the straight line? $\endgroup$ Sep 1, 2019 at 10:05
  • $\begingroup$ It’s a postulate! This is Newton’s first law. $\endgroup$
    – Cham
    Sep 2, 2019 at 3:19
  • $\begingroup$ It is a definition. A straight line is the path of a particle not subject to any force. In flat spacetime it is Newton's first law, as reported by Cham. In a curved manifold is the geodesic equation. Note that in general relativity gravity is not described as a force, but as curvature. $\endgroup$ Sep 2, 2019 at 14:49

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