Raising Indices with Minkowski Metric: Solving Weak Field Approx

In summary, the conversation discusses the expression ##g_{ab}=\eta_{ab}+\epsilon h_{ab}## and how the indices on ##h^{ab}## are raised with ##\eta^{ab}## to give ##g^{ab}=\eta^{ab}-\epsilon h^{ab}##. The conversation also mentions the use of the covariant and contravariant delta, and how the sign changes between them for an infinitesimal value of ##\epsilon##. This results in the contraction remaining unchanged to order ##\epsilon##.
  • #1
binbagsss
1,254
11
I have the expression ##g_{ab}=\eta_{ab}+\epsilon h_{ab}##,

The indices on ##h^{ab}## are raised with ##\eta^{ab}## to give ##g^{ab}=\eta^{ab}-\epsilon h^{ab}##

I am not seeing where the minus sign comes from.

So I know ##\eta^{ab}\eta_{bc}=\delta^{a}_{c}## and ##h^{ab}=\eta^{ac}\eta^{bd}h_{cd}## ,
And that ##\eta_{ab}## has signature ##(-,+,+,+)##,

Any help greatly appreciated.
 
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  • #2
If ##\epsilon## is an infinitessimal then the sign changes between the covariant and contravariant delta so that the contraction is unchanged to order ##\epsilon##
 
  • #3
Mentz114 said:
If ##\epsilon## is an infinitessimal then the sign changes between the covariant and contravariant delta so that the contraction is unchanged to order ##\epsilon##

Thanks .
 

1. What is the Minkowski metric?

The Minkowski metric is a mathematical tool used in the theory of special relativity to describe the geometry of space and time. It defines the distance between two events in a four-dimensional spacetime.

2. How does the Minkowski metric relate to raising indices?

The Minkowski metric is used to raise and lower indices in special relativity. This means that it can be used to convert between covariant and contravariant components of a tensor, which are necessary for solving problems in weak field approximation.

3. What is "weak field approximation" and why is it important?

Weak field approximation is a method used in physics to simplify complex systems by assuming that the gravitational field is weak. This is important because it allows for easier calculations and can provide useful approximations for real-world situations.

4. How is the Minkowski metric used in solving weak field approximation problems?

The Minkowski metric is used to calculate the gravitational potential in weak field approximation problems. By raising indices with the metric, the gravitational potential can be related to the energy-momentum tensor, which describes the distribution of matter and energy in spacetime.

5. Are there any limitations to using the Minkowski metric in weak field approximation?

Yes, there are limitations to using the Minkowski metric in weak field approximation. It is only applicable to problems in special relativity and cannot be used in situations where the gravitational field is strong, such as near massive objects like black holes.

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