Matrix Formalism of GR: Good References for Writing Equations

[thehangedman]

Are there any good references out there for writing the equations of GR in matrix format? For example:

ds^2 = g_mn dx_m dx_n -> ds^2 = dx+ g dx

where the matrix version of g_mn (g) would be hermitian, dx+ is the conjugate...

covariant derivative:

Y_n||m = dY_n/dx_m - {n, km} Y_k -> Y||m = dY/dx_m - G_m Y

in matrix format:

dg/dx_m + G_m g + g G+_m = 0 is the vanishing covariant derivative of the metric.

This is just a different way to write the same mathematics. It seams it would be easier to work with, but I can't find any good references for it. Hasn't someone else done this already? I'm just looking for something that gives the original GR back, no new theories, just new notation...

 

[pervect]

I've seen one, but I can't locate it. I think it was aimed at electrical engineers, if I recall correctly. "Exploring black holes" might work, I don't know, I haven't read the whole thing, just some of the free downloads. Several chapters of this book are available on the internet. Much of the approach is very reminiscent of MTW's "Gravitation", without the high-level math. So it might not be quite what you asked for, but parts of it are free - and I'd check it out. Take a look at

http://www.eftaylor.com/general.html

However, there are good reasons for the serious student to not use matrix notation and to learn tensors. Rank 2 tensors can easily be represented in matrix form, so they aren't the issue. The problem is representing rank 4 tensors, such as the Riemann curvature tensor. I suppose you *could* think of a rank 4 as a general linear map from one matrix to another. This requires not a 2-d array, but a 4-d data structure. For space-time that's 4x4x4x4 = 256 numbers, which however are not all independent in the case of the Riemann, which must obey some identies (the Bianchi identites).

 

[Entropia]

There are several good references available for writing the equations of General Relativity (GR) in matrix format. One recommended resource is the book "Introduction to General Relativity" by John Dirks. It provides a comprehensive introduction to the mathematical foundations of GR, including a section on matrix formalism. Another useful reference is the paper "Matrix Formalism of General Relativity" by K. S. Thorne and F. J. Dyson, which discusses the advantages and limitations of using matrix notation in GR.

Additionally, there are many online resources available, such as lecture notes and tutorials, that cover the matrix formalism of GR. Some examples include the lecture notes by Prof. Sean Carroll from the California Institute of Technology and the tutorials by Prof. Leonard Susskind from Stanford University.

It is important to note that the matrix formalism of GR is just a different way of expressing the same mathematical equations. It may provide a more compact and elegant notation, but it does not introduce any new theories or concepts. Therefore, any good reference on GR should also cover the matrix formalism, even if it is not explicitly mentioned in the title.

In conclusion, there are many good references available for writing the equations of GR in matrix format. It is recommended to explore different resources and find the one that best suits your learning style and level of understanding. With practice and patience, you will be able to master this notation and use it to your advantage in studying and solving problems in GR.