Earth Center of Mass: GR & Inertial Object Acceleration

In summary, in general relativity, there are no global inertial reference frames. The concept of an inertial frame is limited to a local region of approximately flat spacetime. In curved spacetime, objects follow space-time geodesics which can be affected by the curvature of spacetime, resulting in geodesic deviation. This can be visualized using space-time diagrams on curved surfaces. The Earth's center of mass is only inertial at the center, not at the surface, where the geometry can be approximated as a cone. To transition between the cone and the cylinder at the center and infinity, curvature must be introduced.
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Karl Coryat
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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.
 
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Karl Coryat said:
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.
 
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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?
 
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Karl Coryat said:
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.
 
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Thank you.
 
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Karl Coryat said:
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.
 
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Fascinating, thank you!
 
  • #9
Karl Coryat said:
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:

gr_space_time_01_crop.png
 

FAQ: Earth Center of Mass: GR & Inertial Object Acceleration

1. What is the Earth Center of Mass?

The Earth Center of Mass is the point in space where the entire mass of the Earth is considered to be concentrated. It is the average location of all the mass in the Earth, including its solid surface, oceans, and atmosphere.

2. How is the Earth Center of Mass related to General Relativity?

General Relativity is a theory that describes how gravity works in the universe. According to this theory, the Earth's mass and the distribution of that mass determine the curvature of space-time, which in turn determines the motion of objects around the Earth. The Earth Center of Mass plays a crucial role in this theory, as it is the point around which the Earth's mass is distributed and affects the curvature of space-time.

3. What is the significance of the Earth Center of Mass in understanding Inertial Object Acceleration?

Inertial Object Acceleration is the change in an object's velocity due to a force acting on it. The Earth Center of Mass is important in understanding this concept because it is the point around which the Earth's mass is evenly distributed, and thus it is the point where an object would experience no acceleration if it were not affected by any external forces. This concept is known as the center of gravity, and it helps us understand how objects move in relation to the Earth's mass and gravitational pull.

4. How do scientists determine the Earth Center of Mass?

Scientists use a variety of techniques to determine the Earth Center of Mass, including satellite measurements, gravitational field measurements, and mathematical calculations based on the Earth's shape and mass distribution. These methods allow scientists to pinpoint the exact location of the Earth Center of Mass with a high degree of accuracy.

5. Can the Earth Center of Mass change over time?

Yes, the Earth Center of Mass can change over time due to a variety of factors, such as the movement of tectonic plates, changes in the Earth's rotation, and the redistribution of mass due to melting ice caps or volcanic activity. However, these changes are relatively small and do not significantly affect the overall location of the Earth Center of Mass.

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