# I Is there an analog to Einstein's field equations for 2D?

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1. Sep 14, 2017

### Cathr

I am not familiar with tensors and I would like to know if it's possible to understand GR without using them. I imagine we use them to describe four-dimentional space-time, because a regular vector or matrix wouldn't be enough.

Is there an analog of Einstein's equations for a 2D space (plane) and time, therefore a 3 dimensional space-time?

2. Sep 14, 2017

### Staff: Mentor

You can only understand it in a conceptual sense without tensors. The common but flawed approach is the rubber sheet analogy with a bowling ball as some gravitating body and a marble that rolls around it, orbits it and eventually hits the bowling ball as its orbit decays.You can't read too much into other than geometry an dictate how an object moves.

Here's a book by Benjamin Crowell on General Relativity that initially lays out the theory without any heavy math and can give you a good understanding of what's covered by it:

http://www.lightandmatter.com/genrel/

3. Sep 14, 2017

### Ibix

4. Sep 14, 2017

### Orodruin

Staff Emeritus
We use tensors because it is not sufficient with vectors due to the concepts we wish to describe, not due to the dimensionality of the space. In fact, the dimensionality of the space has very little to do with things. We use tensors because they describe the concepts necessary to handle the geometry of the manifold, such as its metric and curvature. For example, the Riemann curvature tensor is a type (1,3) tensor regardless of the dimensionality of the manifold (in one dimension it will automatically vanish, but that is besides the point).

5. Sep 14, 2017

### George Jones

Staff Emeritus
Einstein's equation has the same mathematical form in 3-dimensional spacetime as in 4-dimensional spacetime. There is, however, a big difference in solutions. In 4-dimensional spacetime, there exist vacuum solutions of Einstein's equation (without cosmological constant) for which spacetime has non-zero curvature, e.g., Schwarzschild. In 3-dimensional spacetime, all vacuum solutions of Einstein's equation (without cosmological constant) have zero curvature.

6. Sep 14, 2017

### Cathr

7. Sep 14, 2017

### Cathr

@Orodruin @George Jones Thank you, I didn't know about that. It definitely means I should boost my math skills. Do you know any good soorces that explain tensors for beginners?

8. Sep 14, 2017

### Staff: Mentor

9. Sep 14, 2017

### Orodruin

Staff Emeritus
Fundamental tensor analysis should be available in any book covering mathematical methods in physics. Of course, there are also books focusing om tensors and delving deeper into differential geometry. It depends on what your level and goals are.

10. Sep 14, 2017

### Cathr

Thank you!

11. Sep 14, 2017

### Staff: Mentor

12. Sep 14, 2017

### Cathr

Thank you!

13. Sep 14, 2017

### PAllen

Also, in two dimensions, curvature is fully described by a scalar, though, of course, the curvature tensor is defined as well; its components all being derivable from scalar curvature.