Question involving curvature tensor

In summary, the conversation revolves around the mathematical relations between Yang-Mills theory and GR, specifically in the case of D=4 dimensions. The speaker is struggling with understanding the constraints imposed on the Riemann curvature tensor when the vacuum Einstein equations are not applied. They have proven that the combination of equation (1) and the second bianchi identity (B2) implies the Einstein equations, but they are wondering if the reverse is true and what additional constraints must be applied. The other person in the conversation is also interested in this topic and asks for more information and resources.
  • #1
Kalimaa23
279
0
Greetings,

I'm working out some of the mathematical relations between Yang-Mill theory and GR. I'm having difficulty working out a somewhat trivial thing, I was wondering if anyone here could help me.

To keep things concrete, I'll stick to the case D=4, but I'd like to be able to generalise to higher dimensional cases afterwards.

Consider the Riemann curvature tensor. It has 20 independent components. The vacuüm Einstein equations amount to setting the Ricci tensor to zero, which imposes 10 constraints on the Riemann tensor. The additional 10 components are then described the the Weyl-tensor.

Now imaging that we do not impose the Einstein equations. How many constraints would the equation

[tex]D^{\mu} R_{\mu\nu\sigma\tau}=0[/tex] (1)

impose, together with the second bianchi identity

[tex]D_{\left[\rho\right.}R_{\left.\mu\nu\right]\sigma\tau}=&0[/tex] (B2)
?

The reason I'm asking is because I've proven that (1)+(B2) implies Einstein equations. I'm wondering if the reverse if true, and if not, what additional constraints must be applied.

Thanks in advance.
 
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  • #2
Greetings! I'm also trying to understand the relation between Yang-Mills theory and GR. It sounds like you are trying to understand how many constraints the equation D^{\mu} R_{\mu\nu\sigma\tau}=0 (1) and the second bianchi identity D_{\left[\rho\right.}R_{\left.\mu\nu\right]\sigma\tau}=&0 (B2) impose on the Riemann curvature tensor when the vacuum Einstein equations are not imposed. I'm not sure if I can help you with this question, but I am interested to find out more. Could you please explain your proof that (1)+(B2) implies Einstein equations? Do you have any resources or references that you would recommend to a beginner in this field? Thanks!
 
  • #3


Hi there,

That's a great question you have brought up. Let's see if we can work through it together.

First, let's break down the equation D^{\mu} R_{\mu\nu\sigma\tau}=0. This is essentially the covariant derivative of the Riemann curvature tensor, which means it is a measure of how the curvature changes as we move along a certain direction. In this case, the direction is given by the index \mu. So, this equation is essentially saying that the curvature does not change along this direction, or in other words, the curvature is constant in this direction.

Now, let's look at the second Bianchi identity, D_{\left[\rho\right.}R_{\left.\mu\nu\right]\sigma\tau}=&0. This is a bit more complicated, but essentially it is a measure of how the curvature changes as we move along two different directions simultaneously, \rho and \mu. This equation is saying that the curvature is symmetric with respect to these two directions, or in other words, the curvature is the same regardless of which direction we move along first.

So, when we combine these two equations, we are essentially saying that the curvature is constant in one direction and symmetric in two directions. This means that there are only 6 independent components of the Riemann curvature tensor left, which is exactly the number of components in the Weyl tensor. So, the equation (1)+(B2) effectively imposes the same constraints as the vacuum Einstein equations, and therefore, it implies the Einstein equations.

However, the reverse is not necessarily true. There may be other constraints that need to be applied in order to fully determine the Riemann curvature tensor. These additional constraints may come from other physical considerations or mathematical arguments. So, while (1)+(B2) implies the Einstein equations, the reverse may not always be true.

I hope this helps! Keep up the good work on your research.
 

1. What is the curvature tensor?

The curvature tensor is a mathematical object used to describe the curvature of a space. It is a four-dimensional tensor that contains information about how the geometry of the space changes in different directions.

2. How is the curvature tensor calculated?

The curvature tensor is calculated using the Riemann curvature tensor, which is a measure of how the curvature of a space changes from point to point. It takes into account the second derivatives of the metric tensor, which describes the geometry of the space.

3. What is the significance of the curvature tensor in general relativity?

The curvature tensor is a fundamental concept in general relativity, as it describes the curvature of spacetime caused by the presence of matter and energy. It is essential in understanding how gravity works and predicting the motion of objects in the universe.

4. Can the curvature tensor be visualized?

While the curvature tensor itself cannot be directly visualized, its effects can be seen in the bending of light and the motion of objects in the presence of a massive body. It can also be represented mathematically using diagrams and equations.

5. Are there any practical applications of the curvature tensor?

Yes, the curvature tensor has many practical applications in fields such as cosmology, astrophysics, and engineering. It is used to model the behavior of black holes, gravitational waves, and the structure of the universe. It also has applications in the design of spacecraft trajectories and the construction of bridges and buildings to withstand gravitational forces.

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