# Why does things travel in a straight line in inertial frames?

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?

• Consider mentioning how you define inertial frames in curved spacetimes. – Qmechanic Sep 1 '19 at 8:52

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.

• I understand that. My question was why does it follow the straight line? – Unnikrishnan Puthumana Sep 1 '19 at 10:05
• It’s a postulate! This is Newton’s first law. – Cham Sep 2 '19 at 3:19
• 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. – Michele Grosso Sep 2 '19 at 14:49