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

[louva]

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 :)

 

[atyy]

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.

 

[louva]

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

 

[atyy]

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 V_aI.

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).

 

[louva]

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 V_aI.

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

 

[louva]

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?

 

[atyy]

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" Eq 4.31.

So Peldan's and Gourgoulhon's N are the same. Peldan's N_I is Gourgoulhon's n. Peldan's N^a are the components of Gourgoulhon's β.

 

[marcus]

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