Can General Relativity and Curved Spaces Be Described in Complex Coordinates?

In summary, the conversation discusses the relationship between geometry and coordinates in different spaces, specifically in special and general relativity. It is noted that in pseudo-euclidean spaces, T is also real but the definition of distance is different. The concept of euclidean and Minkowsky spaces is also mentioned, with the difference being the orientation of a subset in a 4-dimensional complex space. The conversation also delves into the idea of curved spaces in general relativity and the possibility of working completely in the complex area. It is mentioned that in some cases, 86 dimensions may be required to accurately represent a 4-dimensional spacetime. An academic example is given, with the source being a discussion on a physics forum. The conversation ends
  • #1
Dmitry67
2,567
1
I am not familiar with that stuff, so please don't laugh.

I know some facts about the geometry when coordinates are real. In pseudo-euclidean spaces (like in special relativity) T is also real, just the definitions of a distance is different.

R^2 = X^2+Y^2+Z^2 - T^2

But we can say that it is an euclidean space, but T is imaginary. So, if we assume that coordinates are complex, the only difference between euclidaen space and Minkowsky space is an orientation of a subset in a 4-dimensional complex space.

So far I hope it is correct.

My question is, what about General relativity and curved spaces? I read that for 3space+1time curved space can be put into 86 dimensional manifold with 3 timelike dimensions.

What if we work completely in the complex area, so there is no difference between space and time dimensions?
 
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  • #2
Dmitry67 said:
I read that for 3space+1time curved space can be put into 86 dimensional manifold with 3 timelike dimensions.

uhh? 86? :confused:

did you get this from http://www.eng.uah.edu/~jacksoa/literature/MD_Int2.pdf? … 86 was just an academic example … it could have been 42 or 717 or … :wink:
 
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  • #4
it is from here: https://www.physicsforums.com/showthread.php?t=290098

George Jones said:
If the metric for the higher dimensional pseudo-Riemannian manifold is required to restrict down to the metric for 4-dimensional spacetime, then it could take a lot of dimensions.

Chris Clarke* showed that every 4-dimensional spacetime can be embedded isometically in higher dimensional flat space, and that 90 dimensions suffices - 87 spacelike and 3 timelike. A particular spacetime may be embeddable in a flat space that has dimension less than 90, but 90 guarantees the result for all possible spacetimes.

* Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian
manifolds," Proc. Roy. Soc. A314 (1970) 417-428
 

Related to Can General Relativity and Curved Spaces Be Described in Complex Coordinates?

1. What is the definition of a complex space in geometry?

A complex space in geometry refers to a space or object that contains both real and imaginary components. In other words, it is a space where the coordinates or measurements are expressed as complex numbers instead of just real numbers.

2. How is geometry in complex spaces different from traditional geometry?

Geometry in complex spaces involves the study of shapes and figures using complex numbers, whereas traditional geometry uses only real numbers. This means that in complex spaces, shapes can have both real and imaginary dimensions, and the rules of traditional geometry may not apply.

3. What are some applications of geometry in complex spaces?

Geometry in complex spaces has many practical applications, including in physics, engineering, and computer graphics. It is used to study and understand complex systems and phenomena, such as fluid dynamics, electromagnetism, and quantum mechanics.

4. How does the concept of complex conjugates relate to geometry in complex spaces?

The concept of complex conjugates is closely related to geometry in complex spaces. In geometry, complex conjugates are used to represent reflections and rotations of shapes in complex space. This allows for a deeper understanding of the symmetries and transformations of geometric objects.

5. Are there any challenges in studying geometry in complex spaces?

Yes, there are several challenges in studying geometry in complex spaces. One of the main challenges is visualizing and understanding complex shapes and figures, as they may not follow the same rules as traditional geometry. Additionally, the use of complex numbers can make calculations and proofs more complicated and abstract.

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