I defined a covering map ℝ^2 → S^1 x ℝ in order to work with the manifold.(adsbygoogle = window.adsbygoogle || []).push({});

1) How can I find lorentzian metrics (=metric tensors) on S^1 x ℝ (2-dimensional manifold)?

I know that the diagonal matrix (2x2 matrix) of such a lorentzian metric must have signature 1. and there are some famous metrics like g= -dx^2 + dy^2. but how can i find/ define lorentzian metrics myself?

Approach: I know that a lorentzian metric has signature 1. This means the matrix elements of the normal form are g11= +1, g22= -1, g12= g21= 0.

But how can I find DIFFERENT lorentzian metrics? And since they all have the same normal form with signature 1, how can I distinguish them?

2) How does a timelike closed curve looks like on S^1 x ℝ ?

Let's take the metric above g= -dx^2 + dy^2.

How can I figure out if a curve on the cylinder with this metric is closed (and timelike; see examples below)?

Approach:

I defined some curves on the cylinder (= I define them in the ℝ^2 since the cylinder is a 2-manifold). What does "closed curve" mean in this case? Is a curve c: [0, 2∏ ] → ℝ^2 closed when c(0) =c(2∏)?

Even if in the ℝ^2 they do not look closed? but they are closed on the cylinder? Is this right?

Are the following ideas correct?

i)

c(x) = (x, sinx) and c'(x)= ( 1, -cosx)

=> <c'(x), c'(x)> = -1 + cos^2 (x) ≤ 0

but this means the curve is not timelike and not spacelike? How is this possible?

ii)

d(x) = (0,x) and d'(x)= (0,1)

=> <d'(x), d'(x)> = 1 >0

this means this curve is timelike

iii)

e(x)= (x,x)

but this curve is not closed? Right?

iv)

f(x)= (cosx, sinx) and f'(x)= (sinx, -cosx)

=> <f'(x), f'(x)> = -sin^2(x) + cos^2(x)

what does this mean? is f(x) not a closed curve?

etc.

It would be very helpful if somebody could just show me all steps using a simple example.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Lorentzian metrics w/ timelike closed curves on cylinder that is a 2-manifold

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for Lorentzian metrics timelike |
---|

I Calculating Killing vectors of Schwarzschild metric |

I Nonlinear relation between coordinate time and proper time |

A Rotating metric |

I Defining the components of a metric |

A Any 2-dimensional Lorentzian metric can be brought to this form? |

**Physics Forums | Science Articles, Homework Help, Discussion**