0
$\begingroup$

Since objects follow geodesics in spacetime, that is the locally shortest path, it would seem to me that unless objects move, they do not trace any path at all. In other words, if I'm stationary on Earth, I should only be affected by the curvature of time, since I always move through time. But on the flipside, photons do not travel through time at all, and so my question is: are they only affected by the spatial curvature?

$\endgroup$
5
  • 1
    $\begingroup$ unless objects move, they do not trace any path at all An object that is stationary in space is moving through spacetime. $\endgroup$
    – Ghoster
    Nov 23, 2022 at 18:54
  • $\begingroup$ the curvature of time Time does not have curvature. Spacetime has curvature, and the curvature tensor has components involving the time direction along with spatial directions. The $tttt$ component is zero. $\endgroup$
    – Ghoster
    Nov 23, 2022 at 18:57
  • 2
    $\begingroup$ photons do not travel through time at all They travel through the time of any observer. $\endgroup$
    – Ghoster
    Nov 23, 2022 at 18:57
  • $\begingroup$ @Ghoster Regarding your statement that time does not have curvature: It is often said that near Earth, 99.99% of gravity is due to time dilation and it is only near a black hole that space curvature accounts for 50% of gravity. Can you reconcile these for me? $\endgroup$ Nov 23, 2022 at 21:04
  • 1
    $\begingroup$ @foolishmuse near Earth, 99.99% of gravity is due to time dilation That’s talking about the $tt$ component of the metric tensor. It represents time dilation, and it is the only important component in the Newtonian limit of GR. It isn’t talking about the Riemann curvature tensor. $\endgroup$
    – Ghoster
    Nov 24, 2022 at 0:35

1 Answer 1

0
$\begingroup$

But on the flipside, photons do not travel through time at all, and so my question is: are they only affected by the spatial curvature?

Indeed it is a little tricky to analyze pulses of light. They travel on what are called “null geodesics”. With timelike geodesics you can parameterize them with the proper time, which is the invariant “length” of the worldline.

But for null geodesics the invariant “length” is zero, so the procedure described above results in a division by zero. So instead of parameterizing by proper time we use what is called an affine parameter.

For timelike lines proper time is an affine parameter, and for null world lines an affine parameter is not proper time but it does distinguish between different events on the null geodesic.

So null geodesics do trace a path through spacetime. That path is parameterized by an affine parameter that is not proper time. As that affine parameter varies, both the position and the time vary, so light is affected by both in any way that it makes sense to say curvature in space is separate from curvature in time.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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