1
$\begingroup$

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?

| cite | improve this question | | | | |
$\endgroup$
  • $\begingroup$ Consider mentioning how you define inertial frames in curved spacetimes. $\endgroup$ – Qmechanic Sep 1 '19 at 8:52
1
$\begingroup$

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
$(D/d\lambda) (dx^\mu/d\lambda) = (dx^\nu/d\lambda) \nabla_\nu (dx^\mu/d\lambda) = 0$
where:
$D/d\lambda$ 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.

| cite | improve this answer | | | | |
$\endgroup$
  • $\begingroup$ I understand that. My question was why does it follow the straight line? $\endgroup$ – Unnikrishnan Puthumana Sep 1 '19 at 10:05
  • $\begingroup$ It’s a postulate! This is Newton’s first law. $\endgroup$ – Cham Sep 2 '19 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. $\endgroup$ – Michele Grosso Sep 2 '19 at 14:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.