How does Curvature Relate to 'Gravitation'?

In summary, the concept of curvature in general relativity can be translated into the Newtonian idea of gravitation through the definition of a new type of acceleration known as "G-for-geometry-acceleration." This allows for a geometric theory of gravity, where gravity curves space and freely falling particles move along straight lines in curved spaces called geodesics. However, this aspect of general relativity is not emphasized enough in most texts on the subject.
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Karl G.
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I have searche many general relativity texts and have not found an answer to the following question: How does curvature translate into the Newtonian idea of gravitation? For example, how is Newton's law of gravitation, where all matter attracts all matter, an approximation to the idea of curvature? I vaguely know that the idea is related to the equation of geodesic deviation, but could somebody explain this more clearly?

Thanks!
 
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  • #2
The handwavy idea is to define a new sort of acceleration. In Newtonian physics, a force produces an acceleration. This can be measured by an accelerometer. The funny thing is that when an object is acted on by gravity alone, ie. when it is falling freely, an accelerometer attached to it reads zero, even though the acceleration is not zero. If we define a new sort of "G-for-geometry-acceleration" that is what the accelerometer reads, then we end up with a geometric theory of gravity. In this view, gravity curves space, and freely falling particles move with zero G-acceleration which means they move on "straight lines" in curved spaces called geodesics.
 
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Thanks! Unfortunately, all GR yests I have pored over do not emphasize this aspect of GR enough, in my opinion.
 
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Sorry! Substitute 'texts' for 'yests'. To say my keyboard is antiquated would be quite an understatement.
 
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Karl G. said:
Sorry! Substitute 'texts' for 'yests'. To say my keyboard is antiquated would be quite an understatement.
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1. What is curvature and how does it relate to gravitation?

Curvature refers to the bending or warping of space-time due to the presence of mass or energy. It is a fundamental concept in Einstein's theory of general relativity and is directly related to the force of gravitation. This means that the distribution of mass and energy in the universe determines the curvature of space-time, and this curvature in turn determines the motion of objects under the influence of gravity.

2. How does the curvature of space-time affect the motion of objects?

The curvature of space-time affects the motion of objects by altering the path they follow through space-time. Objects with mass or energy will follow the shortest, or geodesic, path through curved space-time, which is what we experience as the force of gravity. This means that the curvature of space-time determines the trajectory of objects, including the planets in our solar system and the motion of light.

3. Does the curvature of space-time only affect large objects like planets and stars?

No, the curvature of space-time affects all objects, regardless of their size. Even small objects like particles or photons are affected by the curvature of space-time. However, the effects may be very small and difficult to measure for smaller objects, which is why we typically only observe the effects of curvature on larger objects.

4. How does the curvature of space-time explain the concept of gravity?

The curvature of space-time explains gravity by showing that what we experience as the force of gravity is actually the result of objects following the shortest path through curved space-time. This means that the force of gravity is not a force at all, but rather a result of the curvature of space-time caused by the presence of mass and energy. This is a fundamental shift from Newton's theory of gravity, which described gravity as a force acting between objects.

5. Can the curvature of space-time be observed or measured?

Yes, the curvature of space-time can be observed and measured through various experiments and observations. For example, the bending of starlight around the sun during a solar eclipse is a direct observation of the curvature of space-time. Other experiments, such as the Gravity Probe B mission, have also provided evidence for the curvature of space-time. However, measuring the curvature accurately can be challenging and requires advanced technology and techniques.

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