Einstein Field Equations: PDEs or ODEs? - Thomas

In summary, the conversation discusses the difficulty in solving Einstein's field equations, which are generally considered to be PDEs in 4-dimensional space. However, the lecturer in the video claims that they can be simplified to ODEs in certain cases, such as in FRW spacetimes used in cosmology.
  • #1
Thomas Moore
12
2
This past semester, I just took an introductory course on G.R., which translates to a lot of differential geometry and then concluding with Schwarzschild's solution. We really didn't do any cosmology. However, one of the themes that kept creeping up again and again is that in 4-dimensions, Einstein's field equations are partial differential equations, that are coupled, and generally, very difficult to solve. So, I was all fine with that. However, searching on YouTube, I came across these "newer" lectures that now claim around 6:00 that Einstein's equations are ODEs. Further, the lecturer discusses solutions of them. Can anyone elaborate? Are they PDEs or ODEs?

The lecture can be found here:

Thanks!
Thomas
 
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  • #2
Thomas Moore said:
Are they PDEs or ODEs?

Since in general the metric is a function of multiple variables, and the Einstein Field Equations are equations in the metric and its derivatives, they would have to be PDEs.

However, in a situation where the metric is a function of just one variable--as is the case in the FRW spacetimes standardly used in cosmology (everything is a function of the time coordinate ##t## only)--then the Einstein Field Equations will be ODEs. I believe that is the case being discussed in the video.
 

Related to Einstein Field Equations: PDEs or ODEs? - Thomas

1. Are Einstein Field Equations Partial Differential Equations (PDEs) or Ordinary Differential Equations (ODEs)?

The Einstein Field Equations are a set of ten PDEs that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. These equations are used in the theory of general relativity, which is a theory of gravity.

2. What is the significance of the Einstein Field Equations in physics?

The Einstein Field Equations are a cornerstone of modern physics, as they are used to describe the behavior of gravity on a large scale. They have been extensively tested and have been shown to accurately predict the behavior of objects in the universe, from the smallest particles to entire galaxies.

3. How were the Einstein Field Equations derived?

The Einstein Field Equations were first derived by Albert Einstein in 1915 as part of his theory of general relativity. He used complex mathematical techniques, including tensor calculus, to describe the relationship between gravity and the curvature of spacetime.

4. Can the Einstein Field Equations be solved analytically?

No, the Einstein Field Equations cannot be solved analytically. This means that there is no exact, closed-form solution to the equations. Instead, numerical methods and approximations are used to solve them and make predictions about the behavior of objects in the universe.

5. How have the Einstein Field Equations been tested and validated?

The predictions made by the Einstein Field Equations have been extensively tested and validated through observations and experiments. Some of the key evidence for the accuracy of these equations includes the bending of light around massive objects, such as stars, and the detection of gravitational waves, which were predicted by the equations. Additionally, the equations have been used to accurately predict the behavior of objects in the universe, such as the orbit of Mercury around the sun.

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