No. of Independent Components of Riemann Tensor in Schwartzchild Metric

In summary, the Riemann tensor in a general 4d space time has 20 independent components. However, in a more symmetric metric, the number of independent components may reduce. For the Schwartzchild metric, there is no independent components of the Riemann tensor since the metric completely determines it. The counting of 20 independent components does not consider the additional constraints needed to "integrate" the Riemann tensor and obtain a metric. These constraints are discussed in Stephani's Relativity, stating that in general, such a system will not have solutions.
  • #1
ObsessiveMathsFreak
406
8
In general 4d space time, the Riemann tensor has 20 independent components.

However, in a more symmetric metric, does the number of independent components reduce? Specifically, for the Schwartzchild metric, how many IC does the corresponding Riemann tensor have?

(I think it is 4, but I cannot find a source to confirm this)
 
Physics news on Phys.org
  • #2
Once the metric is specified, there are no independent components of the Riemann tensor, since the metric completely determines the Riemann tensor.

Also, the counting of 20 independent components of the Riemann tensor doesn't take into account the requirement that one should be able to "integrate" the Riemann tensor to get a metric. These additional constraints are listed in Stephani's Relativity: an introduction to special and general relativity, on p143 in the section beginning with "The determination of the metric from a specified curvature tensor amounts ... to the solution of a system of twenty second-order differential equations for the ten metric compoenents ... In general such a system will posess no solutions ..."
 
Last edited:

1. What is the Riemann tensor in Schwartzchild metric?

The Riemann tensor in Schwartzchild metric is a mathematical object that describes the curvature of spacetime in the presence of a non-rotating, spherically symmetric mass, such as a black hole. It is a key component in Einstein's theory of general relativity.

2. How is the Riemann tensor related to the Schwartzchild metric?

The Riemann tensor is derived from the Schwartzchild metric, which is a solution to Einstein's equations of general relativity for a non-rotating, spherically symmetric mass. The Riemann tensor contains information about the curvature of spacetime in this metric.

3. What are the independent components of the Riemann tensor in Schwartzchild metric?

There are six independent components of the Riemann tensor in Schwartzchild metric. These are known as the Ricci rotation coefficients and are denoted by the symbols $\Gamma^r_{\theta\theta}$, $\Gamma^r_{\phi\phi}$, $\Gamma^\theta_{r\theta}$, $\Gamma^\theta_{\phi\phi}$, $\Gamma^\phi_{r\phi}$, and $\Gamma^\phi_{\theta\phi}$.

4. Why is the number of independent components of the Riemann tensor important?

The number of independent components of the Riemann tensor is important because it allows us to fully describe the curvature of spacetime in the presence of a non-rotating, spherically symmetric mass. These components provide important information about the strength and direction of gravitational forces in this type of spacetime.

5. How is the Riemann tensor used in the study of black holes?

The Riemann tensor is used in the study of black holes to understand the effects of gravity on the curvature of spacetime near these massive objects. It helps us to predict the behavior of matter and light as they approach and interact with a black hole, and it is a key component in calculating important quantities such as the event horizon and the singularity of a black hole.

Similar threads

A GW Binary Merger: Riemann Tensor in Source & TT-Gauge
    • Special and General Relativity
Replies
4
Views
967
I Calculate Ricci Scalar & Cosm. Const of AdS-Schwarzschild Metric in d-Dimensions
    • Special and General Relativity
Replies
1
Views
1K
A Schwarzschild metric in Fermi normal coordinates about a radially infalling timelike geodesic
    • Special and General Relativity
Replies
13
Views
638
I Help Understanding Metric Tensor
    • Special and General Relativity
Replies
12
Views
2K
I Calculate Gaussian Curvature from 4D Metric Tensor
    • Special and General Relativity
Replies
14
Views
2K
I Invariant Mass-Energy in FRW Spacetime
    • Special and General Relativity
Replies
19
Views
1K
I Number of independent components of the Riemann tensor
    • Special and General Relativity
Replies
6
Views
3K
I What are the independent components of the Riemann tensor
    • Special and General Relativity
Replies
7
Views
3K
I Computing Riemann Tensor: 18 Predicted Non-Trivial Terms
    • Special and General Relativity
Replies
1
Views
917
I Components of Riemann Tensor: 4 Indexes, 16x16 Matrix
    • Special and General Relativity
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top