What is the history of the ADM formalism in quantum gravity?

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In summary, the conversation discusses the confusion surrounding the Shift vector in the ADM formalism in the field of Quantum Gravity. The individual is unsure why the contraction of N^a with VaI does not vanish in the first equation of (2.29) while it does in the third equation. They also question the relation between N^I and N^a and which map allows for a transformation between coordinate systems. The conversation also references the original ADM paper and suggests looking at Rovelli's book "Quantum Gravity" for further context and understanding.
  • #1
louva
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Hi! I am novice in the Quantum Gravity field, so it was logical for me to start with the ADM formalism, but I am very confused about the Shift vector, specially in the Peldan paper http://arxiv.org/abs/gr-qc/9305011.

In the first equation of (2.29): why the contraction of N^a with VaI does not vanish as it was the case for the third equation of (2.29), the N^a and N^I aren't the same vector but viewed by two different coordinate systems?

I am already apologizing for my bad english :)
 
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  • #2
I'm not sure, but the equations look different to me. The superscript for N matches the first and second subscripts of V in the first and third equations respectively.
 
  • #3
atyy said:
I'm not sure, but the equations look different to me. The superscript for N matches the first and second subscripts of V in the first and third equations respectively.

I mean the right hand side of the first equation of (2.29): N^a contracted with V^aI, isn't supposed to vanish as the third equation on (2.29) where the projection of N^I with V^aI is zero?

:S
 
  • #4
louva said:
I mean the right hand side of the first equation of (2.29): N^a contracted with V^aI, isn't supposed to vanish as the third equation on (2.29) where the projection of N^I with V^aI is zero?

:S

It looks like in one case the contraction is with "a" and in the other case with "I", which are indices that have different positions on VaI.

Also, the lower case Roman indices "a" appear to take possible values {1,2,3} (space), whereas upper case indices "I,J,K.." appear to take values {0,1,2,3} (local Minkowski basis). He also uses lower case Greek indices "α" which take values {0,1,2,3} (spacetime coordinates).
 
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  • #5
atyy said:
It looks like in one case the contraction is with "a" and in the other case with "I", which are indices that have different positions on VaI.

- Thank you :), I must confess that i have some difficulties with "this" Shif Vector and his behavior.
 
  • #6
Another question :)

Is there any relation between N^I and N^a, are they the same vector but viewed by different coordinate systems?

If Yes, which map allows us to pass from one coordinate systems to another?
 
  • #7
louva said:
Another question :)

Is there any relation between N^I and N^a, are they the same vector but viewed by different coordinate systems?

If Yes, which map allows us to pass from one coordinate systems to another?

I don't think so. It looks like if we take Peldan's tetrad basis vectors to be coordinate basis vectors, then http://arxiv.org/abs/gr-qc/9305011" [Broken] Eq 4.31.

So Peldan's and Gourgoulhon's N are the same. Peldan's NI is Gourgoulhon's n. Peldan's Na are the components of Gourgoulhon's β.
 
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  • #8
Just as a general (historical?) reference you might be interested in looking at the original ADM paper:
http://arxiv.org/abs/gr-qc/0405109

This dates back to around 1962. I think it was a chapter in a book compiled by Louis
Witten and published in 1962. You are probably familiar with arxiv. If not just click where it says "pdf" for a free download.

Another free online source that might be useful as context is the draft version of Rovelli's book "Quantum Gravity". It is not the final version that was published by Cambridge U. Press in 2004, but it is pretty close to final as you might expect in the early chapters coverning standard material.

The link is posted here
http://www.cpt.univ-mrs.fr/~quantumgravity/
The pdf link, for download, is this
http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
 

1. What is the ADM formalism?

The ADM formalism, also known as the Arnowitt-Deser-Misner formalism, is a mathematical framework used to describe the dynamics of space and time in Einstein's theory of general relativity.

2. How is the ADM formalism different from other formulations of general relativity?

The ADM formalism differs from other formulations of general relativity in that it expresses the theory in terms of the spacetime metric and its conjugate momentum, rather than the curvature of spacetime.

3. What are the advantages of using the ADM formalism?

One advantage of using the ADM formalism is that it allows for a decomposition of spacetime into space and time, making it easier to analyze the dynamics of gravitational fields in a more intuitive way. It is also useful in numerical simulations of general relativity.

4. How is the ADM formalism used in cosmology?

The ADM formalism is used in cosmology to study the large-scale structure of the universe and its evolution over time. It allows for the calculation of gravitational waves, which can provide valuable information about the early universe.

5. Are there any limitations to the ADM formalism?

While the ADM formalism is a powerful tool for studying general relativity, it has some limitations. It is not applicable to all types of spacetime, such as those with singularities or black holes. It also does not take into account quantum effects, which are important in extreme conditions such as the early universe.

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