General metric with zero riemann tensor

In summary, the conversation discusses a metric consistent with interval and its transformation to the Minkowski form. It is found to be a special case of the FLWR metric and a transformation is suggested. Further discussion explores the regularity of the transformation and the solutions of the equation for Riemann's tensor. The conclusion is that all solutions of the equation are isomorphic with a tensor in the form of a diagonal matrix.
  • #1
archipatelin
26
0
A metric consistent with interval:
[tex]\mathrm{d}s^2=-\mathrm{d}\tau^2+\frac{4\tau^2}{(1-\rho^2)^2}\left(\mathrm{d}\rho^2+\rho^2\mathrm{d}\theta^2+\rho^2\sin(\theta)^2\mathrm{d}\varphi^2\right)[/tex]
get zero for riemann's tensor, therefor must be isomorphic with minkowski tensor.
But I don't find thus transformation of coordinates from [tex]\tau,\,\rho\rightarrow t,\,r[/tex]
so that after transformation is interval in minkowski form:
[tex]\mathrm{d}s^2=-\mathrm{d}t^2+\fmathrm{d}r^2+r^2\mathrm{d}\theta^2+r^2\sin(\theta)^2\mathrm{d}\varphi^2[/tex].

What's transformation?
 
Physics news on Phys.org
  • #2
Haven't checked carefully, but it looks like a Robertson-Walker metric with a(t)=t, [itex]\rho=r/2[/itex], and k=-1.
 
  • #3
bcrowell said:
Haven't checked carefully, but it looks like a Robertson-Walker metric with a(t)=t, [itex]\rho=r/2[/itex], and k=-1.

Yes, it is special case of FLWR metric. But, still don't know this transformation.
 
  • #5
George Jones said:

Thanks,
I found this transformation in form:
[tex]\rho\equiv{}\rho(t,r)[/tex]
[tex]\tau\equiv{}\tau(t,r)[/tex]
Solutions with respect to minkowski metric are:
[tex]\rho=\frac{2t}{r}\left(1\pm\sqrt{1-\left(\frac{r}{2t}\right)^2}\right)[/tex]
[tex]\tau=\frac{1}{2}r\frac{1-\rho^2}{\rho}[/tex]
But it's regular only for [tex]\left\|\frac{r}{2t}\right\|<1[/tex].
However minkowski metric is regulare everywhere. It's OK?
 
Last edited:
  • #6
I want to make sure that all solutions of the equation
[tex]R_{\mu\nu\varkappa\lambda}=0[/tex]​
for any dimension [tex]D[/tex] are isomorphic with tensor in form
[tex]g_{\mu\nu}=\mbox{diag}(\pm{}1,\pm{}1,\dots,\pm{}1)[/tex]​
Or are there other solutions?
 

FAQ: General metric with zero riemann tensor

What is a general metric with zero Riemann tensor?

A general metric with zero Riemann tensor refers to a mathematical construct in the field of differential geometry that describes a space where the Riemann curvature tensor is equal to zero. This means that the space is flat and has no intrinsic curvature.

How is a general metric with zero Riemann tensor related to spacetime?

In the theory of general relativity, spacetime is described by a general metric with zero Riemann tensor. This means that the curvature of spacetime is zero, indicating that the laws of physics are the same at every point in space and time.

What are some examples of spaces with a general metric and zero Riemann tensor?

Some examples of spaces with a general metric and zero Riemann tensor include Minkowski space, Euclidean space, and flat toroidal space. These spaces are used in physics to represent the absence of gravity or the gravitational field.

What implications does a general metric with zero Riemann tensor have in physics?

The presence of a general metric with zero Riemann tensor has significant implications in physics, particularly in the study of gravity and the behavior of matter in space. It allows for the formulation of the special theory of relativity and plays a crucial role in Einstein's theory of general relativity.

What are the practical applications of studying a general metric with zero Riemann tensor?

The study of a general metric with zero Riemann tensor has many practical applications, including in the fields of astrophysics, cosmology, and even engineering. It helps us understand the behavior of matter and energy in space, which is crucial for developing technologies such as GPS systems and satellite communications.

Similar threads

Replies
16
Views
3K
Replies
13
Views
2K
Replies
5
Views
776
Replies
12
Views
2K
Replies
12
Views
2K
Replies
2
Views
1K
Replies
11
Views
938
Back
Top