Linearized Gravity: Why & How Does it Follow?

  • Thread starter kexue
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In summary, the conversation discusses how from the metric splitting of g_{ab}=\eta_{ab}+h_{ab}, it follows that g^{ab}=\eta^{ab}-h^{ab}. This is because when working with GR, the R is not zero despite the flat space-time metric splitting. By introducing a small parameter epsilon, it is shown that g^{ab} can be expressed in terms of linear and higher order epsilon terms, ultimately leading to the cancellation of all quadratic terms to maintain the identity g_{ab}g^{bc}=\delta_a^c.
  • #1
kexue
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Why and how does from g[tex]_{ab}[/tex]=[tex]\eta_{ab}[/tex]+h[tex]_{ab}[/tex] follow
g[tex]^{ab}[/tex]=[tex]\eta^{ab}[/tex]-h[tex]^{ab}[/tex]?

Don't get the latex right, first equation all indices are supposed to below, second equation all are supposed to be up.

thanks
 
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  • #2
I think that you will arrive soon at this property.

If you work with GR, such a metric splitting does not mean introducing a flat space-time since the R is not zero whatever variable change you do.
 
  • #3
kexue said:
Why and how does from g[tex]_{ab}[/tex]=[tex]\eta_{ab}[/tex]+h[tex]_{ab}[/tex] follow
g[tex]^{ab}[/tex]=[tex]\eta^{ab}[/tex]-h[tex]^{ab}[/tex]?

Don't get the latex right, first equation all indices are supposed to below, second equation all are supposed to be up.

thanks
Because then

[tex]
g^{ab}g_{bc} = \delta^a_c
[/tex]

up to first order. Maybe it helps to write

[tex]
g_{ab} = \eta_{ab} + \epsilon h_{ab}
[/tex]

then we would like that [itex]g_{ab}g^{bc}=\delta_a^c[/itex] . So up to first order in epsilon you can easily see that [itex]g^{bc}=\eta^{bc} - \epsilon h^{bc}[/itex]
does the job for you:

[tex]
g_{ab}g^{bc} = ( \eta_{ab} + \epsilon h_{ab}) (\eta^{bc} - \epsilon h^{bc})
= \delta_a^c + \epsilon(h_{ab}\eta^{bc} - h^{bc}\eta_{ab}) + O(\epsilon^2)
[/tex]

Now, you can raise the indices of h with eta because we are working at linear order in epsilon; every correction to it would give higher order epsilon terms. So we get

[tex]
g_{ab}g^{bc} = \delta_a^c + \epsilon (h_a^c - h_a^c) + O(\epsilon^2) = \delta_a^c + O(\epsilon^2)
[/tex]

You could say that the minus-sign in [itex]g^{ab}[/itex] is for cancelling the two factors linear in epsilon in order to get [itex]g_{ab}g^{bc}=\delta_a^c[/itex]. Ofcourse, if you go up higher in order epsilon, say epsilon squared, then all the terms quadratic in epsilon have to cancel to maintain the identity [itex]g_{ab}g^{bc}=\delta_a^c[/itex].
 
  • #4
thanks so much, haushofer! crystal-clear now
 

1. What is linearized gravity?

Linearized gravity is a mathematical approximation used to simplify the equations of general relativity, which describe the behavior of gravity in the universe. It is used in situations where the gravitational fields are weak, and the velocities involved are much smaller compared to the speed of light.

2. Why is linearized gravity used?

Linearized gravity is used because it simplifies the equations of general relativity and makes them easier to solve. It is also more accurate in situations where the gravitational fields are relatively weak, such as in our solar system.

3. How does linearized gravity follow from general relativity?

Linearized gravity follows from general relativity through a process called linearization. This involves expanding the equations of general relativity in a series of terms, and then keeping only the first-order terms. This approximation leads to the linearized equations of gravity.

4. What are the limitations of linearized gravity?

The main limitation of linearized gravity is that it is only accurate for weak gravitational fields and low velocities. It cannot fully describe the behavior of gravity in extreme situations, such as near black holes or during the early stages of the universe.

5. How is linearized gravity used in scientific research?

Linearized gravity is used in many scientific studies, particularly in astrophysics and cosmology. It is used to model the behavior of gravity in our solar system, as well as in larger scales such as galaxy clusters. It has also been used to study the effects of gravitational waves and to test the predictions of general relativity.

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