Earth Center of Mass: GR & Inertial Object Acceleration

[Karl Coryat]

TL;DR Summary

Taking Earth's center of mass as our reference frame, how does GR account for an inertial object near the surface approaching with an acceleration of G?

Super-basic question that I'm embarrassed to ask. It's just what the summary says:

Taking Earth's center of mass as our reference frame, how does GR account for an inertial object near the surface approaching with an acceleration of G?

I assume (perhaps incorrectly) that this is an inertial reference frame. In that frame, the object seems to be acted on by a force.

 

[Dale]

Here is an insights article I wrote a while back about understanding GR in “everyday” scenarios:

https://www.physicsforums.com/insights/understanding-general-relativity-view-gravity-earth/

 

[PeroK]

I assume (perhaps incorrectly) that this is an inertial reference frame.

There are no global inertial reference frames in GR. Specifically, there are none in curved spacetime.

 

[Karl Coryat]

So the error is in assuming that if we're in flat spacetime, that we can make measurements of an object in curved spacetime that are consistent with GR?

 

[PeroK]

So the error is in assuming that if we're in flat spacetime, that we can make measurements of an object in curved spacetime that are consistent with GR?

Spacetime is either flat or curved. It can't be both. If you are in a local region of approximately flat spacetime, then you can use SR within that local region. But, any spacetime outside that region cannot be part of your local inertial reference frame.

 

[Karl Coryat]

Thank you.

 

[pervect]

Summary:: Taking Earth's center of mass as our reference frame, how does GR account for an inertial object near the surface approaching with an acceleration of G?

Super-basic question that I'm embarrassed to ask. It's just what the summary says:

Taking Earth's center of mass as our reference frame, how does GR account for an inertial object near the surface approaching with an acceleration of G?

I assume (perhaps incorrectly) that this is an inertial reference frame. In that frame, the object seems to be acted on by a force.

Consider a couple of free-falling objects both falling directly towards the center of the Earth. One is further away from the center of the Earth than the other, so the objects have different accelerations.

In Newtonian mechanics, this differeng acceleration would be the the result of tidal forces.

In GR, both objects are traveling along space-time geodesics. These geodesics have a number of defining properties, one is that they maximize (more precisely, extremize, but we'll slightly oversimplify it to say maximize) proper time. The space-time geodesics in GR separate from each other. This is called "geodesic deviation".

Over small distances, one might be able to ignore the tidal forces, which in GR is better described as geodesic deviation. But they doesn't actually vanish - one just ignores it, as it's a second order effect. Ignoring these second order effects is the closest that one can come to an "inertial frame" in GR.

GR basically says that geodesic deviation is equivalent to space-time curvature, and is also equivalent to the Newtonian ideal of "tidal forces". These three concepts are basically the same phenomenon, expressed in different paradigms. This is very slightly oversimplified, but that's the basic idea.

One way of visualizing this is to draw a space-time diagram on a curved surface. The "straight lines" (geodesics) on the curved surface don't stay a constant distance apart. For instance, if you draw "straight lines", curves of shortest distance, on the spatial surface of a sphere, they are great circles, and they don't stay a constant distance apart, but in fact intersect at some points, while they diverge at others.

People seem to be reluctant to draw space-time diagrams for some reason, but they're very helpful. One of the posters here, AT, has posted a rather nice diagram many man times, a 2d graphic showing 'straight lines' exhibiting geodesic deviation.

So, one way of understanding GR as a visual aid is to draw 2d space-time diagrams on 2d curved surfaces. This only handles 1 space and 1 time dimension, so mathematics (and not just these visual aids) are necessary to deal with a curved 4 dimensional space time.

It's not actually necessary for space-time to be the surface of some higher dimension object for it to be curved - curvature can be treated without such concepts via mathematics. But it's very convenient to imagine these extra, not-necessarily detectable, dimensions to get some insight into what's going on. It's possible, and even desirable, to treat the curvature abstractly and mathematically, but that's the subject of a textbook, not a short forum post.

 

[Karl Coryat]

Fascinating, thank you!

 

[A.T.]

I assume (perhaps incorrectly) that this is an inertial reference frame. In that frame, the object seems to be acted on by a force.

In curved space-time you have only locally inertial coordinates. The rest frame of the Earth is only inertial at the center, but not at the surface, where the geometry can be approximated as shown below:

Locally at the Earth's center the cone shown above would be a cylinder, so the apple would stay where it is. The get a transition between the cone at the surface, and the cylinder at the center and infinity, you must introduce curvature, as shown below: