General Relativity: Is it Local?

In summary, spacetime is a differential manifold with Minkowski spacetime attached to each point. The laws of physics in this spacetime do not include gravity. However, gravity can be understood as the curvature of spacetime. To define the concept of curvature, it is necessary to evaluate at least one neighborhood of a given point. General relativity is considered to be a local theory, meaning that its Lagrangian depends on the fields and their derivatives at a single spacetime point. Overall, an infinitesimally small neighborhood is sufficient to define curvature, making general relativity a local theory.
  • #1
accdd
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Spacetime is a differential manifold and at each point is attached a Minkowski spacetime.
There the laws of physics are the usual ones without gravity.
Gravity is the curvature of spacetime. To define the concept of curvature do we need to evaluate at least one neighborhood of point P? Is general relativity local?
 
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  • #2
As all so far successful relativistic field theories also GR is local, i.e., it's Lagrangian depends on the fields and its derivatives at a single spacetime point.
 
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  • #3
accdd said:
To define the concept of curvature do we need to evaluate at least one neighborhood of point P? Is general relativity local?
An infinitesimally small neighborhood suffices for that purpose, so yes, it's local.
 
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Related to General Relativity: Is it Local?

1. What is the concept of local in General Relativity?

The concept of local in General Relativity refers to the idea that the laws of physics are the same in all reference frames. This means that an observer in one reference frame should be able to make the same observations and measurements as an observer in a different reference frame, as long as they are in the same local region of space and time.

2. How does General Relativity explain the phenomenon of gravity?

General Relativity explains gravity as the curvature of spacetime caused by the presence of matter and energy. The more massive an object is, the more it curves the fabric of spacetime, and this curvature is what we experience as gravity.

3. Can the principles of General Relativity be applied to the entire universe?

Yes, the principles of General Relativity can be applied to the entire universe. In fact, it is the most accurate theory we have for describing the large-scale structure and behavior of the universe.

4. How does General Relativity relate to Einstein's famous equation, E=mc²?

E=mc² is a fundamental equation in General Relativity, as it represents the relationship between energy, mass, and the speed of light. This equation is used to describe the equivalence of mass and energy, and it plays a crucial role in understanding the effects of gravity on objects with mass.

5. What are some real-world applications of General Relativity?

General Relativity has several real-world applications, including GPS technology, which uses the theory to account for the effects of gravity on satellite signals. It also helps us understand the behavior of black holes and has been used to make predictions about the expansion of the universe and the existence of gravitational waves.

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