Question

In a lorentzian manifold how are these formulas related?

Answer 1

Sorry, I do not know, but two things, one, what class is this for? and 2, it looks like your question is missing information

Answer 2

\[ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\]and\[(A,B)=g_{\mu \nu}A^{\mu}B^{\nu}\]

Answer 3

@FibonacciChick666 I'm sorry for taking so long to finish my question.

Answer 4

Oh, it's ok, what class is this and I'll see if I can find you a good reference

Answer 5

I'm just studying General Relativity. I'm really interested in differential geometry and topology. I would also like to know how this formula from tensor calculus\[g_{ij} = e_{i}.e_{j} \]Where the e's are basis vectors defined as the partial deriatives of the position vector with respect to the coordinates. To the differential geometry equation\[ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\]

Answer 6

Oh these appear to both be representing a dot product... I'm not entirely familiar with tensors and their notation however so you might have to explain some things to me for me to help you more. But I believe g_{mu nu} is the metric tensor which essentially boils down to the kronecker delta which makes that top formula become something along the lines of ds^2=-cdt^2+dx^2+dy^2+dz^2

Answer 7

Sorry, I've never seen this, but I can find you some good resources

Answer 8

Yeah, that is what the formula is for Cartesian coordinates. Non the less, when you work on a lorentzian manifold you allow space to be curved. This rules out any kind of affine coordinates (for example Cartesian) since you can't put affine coordinates on curved space. For curved space the metric tensor can take any values (keeping its determinant negative) and is not necessarily diagonal.

Answer 9

Here is a differential geometry book http://www.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

Answer 10

and another http://people.math.aau.dk/~raussen/I4/08/book.pdf

Answer 11

i think you will like this one the best http://www.astrohandbook.com/ch10/differential_geometry.pdf

Answer 12

Page like 40ish is the section on this topic

Answer 13

Yeah, I can't really help you there. I'd be interested in learning but I don't even know the difference between covariant and contravariant vectors.

Answer 14

I'm in the same boat, I haven't had differential geometry. I opted for graph theory instead

Answer 15

Well, if you wanna have a really quick introduction to this concepts you may look at the first twenty pages of Paul Dirac's General Theory of Relativity.

Answer 16

If you're still looking for the answer to this question, the second equation defines the inner product of two four-vectors A and B. That also defines the norm of the vector \( (A,A) = A^2 = g_{\mu \nu} A^\mu A^\nu \) which is essentially the vector's "length" (though in this particular space this is a bad term because it could be negative as well as positive).

If you consider an infinitesimal four vector \(dx\), then \(ds\) is the scalar "length" of that four vector, and is given by its inner product with itself - \(ds^2 = (dx,dx) = g_{\mu \nu} dx^\mu dx^\nu \)

Answer 17

@Jemurray3 Oh thank you. First of all I would like to know how you input equations within your paragraphs. Second of all, since a covariant vector is essentially a one form. i.e.\[A_{\mu}:\mathbb{R}^n \rightarrow \mathbb{R} \leftarrow \rightarrow A_{\mu} \in (\mathbb{R}^{n})^*\]And the basis of the vector space is \[B((\mathbb{R}^n)^*)=(dx_{1},.....,dx_{n}) \]where\[dx_{i}(v)=v_{i} \leftarrow v \in \mathbb{R}^n\]Considering integration how does one interpret the quantity \[dx^{\mu}\]? Should we put\[dx^{\mu}=g^{\alpha \mu}dx_{\alpha}\]where the covariant differential element is interpreted as above.

Answer 18

To input the math in paragraphs, use \ ( rather than \ [ (I had to put a space in between the slash and the open paren so it would be visible, but obviously you wouldn't put one there).

Yes, that's right. However, you should remember than \(dx^0 = dt, dx^1 = dx, dx^2 = dy, dx^3 = dz \) in more familiar notation. The Greek subscripts and superscripts are just a shorthand. Do you have a particular integral in mind?