# On metric and connection independence

• A
Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast.

I will be very thankful if someone can provide reference to such a paper.

Thanks.

inspired by the main theme of spacetime fabric

There is no "spacetime fabric"...

I don't know much about it, but it's called the Palatini formulation of GR:

https://en.wikipedia.org/wiki/Palatini_variation

Yes, I have read that article but it does elaborate much on it.

There is no "spacetime fabric"...

I am looking for ideas and suggestion which are accepted in the scientific literature.

PeterDonis
Mentor
I am looking for ideas and suggestion which are accepted in the scientific literature.

You were the one who mentioned "spacetime fabric" in the OP; that's why @weirdoguy pointed out that that's not an accepted concept.

haushofer

For example in one version of Platini formulation of gravity the metric and connections are taking as independent quantities. I know there are many other. But I am interested in understanding the physical motivation of such a step?

You were the one who mentioned "spacetime fabric" in the OP; that's why @weirdoguy pointed out that that's not an accepted concept.

Hi,

In the classical GR space time is manifold. Hence the use of the word spacetime fabric. Other ideas where spacetime is treated as discrete points is not what I was talking about. I assumed (incorrectly may be) weirdoguy is talking about some fictional ideas that spacetime as a continuous manifold does not exist. I guess, my bad?

Orodruin
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In the classical GR space time is manifold. Hence the use of the word spacetime fabric
I do not see the implication. "Fabric" has nothing to do with being a manifold, it is something typically made up by popular science descriptions.

dextercioby
atyy
http://space.mit.edu/LIGO/more.html
Gravitational Waves: Ripples in the fabric of space-time
Albert Einstein predicted the existence of gravitational waves in 1916 as part of the theory of general relativity. In Einstein's theory, space and time are aspects of a single measurable reality called space-time. Matter and energy are two expressions of a single material. We can think of space-time as a fabric; The presence of large amounts of mass or energy distorts space-time – in essence causing the fabric to "warp" – and we observe this warpage as gravity. Freely falling objects – whether soccer balls, satellites, or beams of starlight – simply follow the most direct path in this curved space-time.

@stevendaryl gave the correct answer in post #2. A useful reference might be Sotiriou, 6+1 lessons from f(R) gravity, https://arxiv.org/abs/0810.5594.

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PeterDonis
Mentor
I assumed (incorrectly may be) weirdoguy is talking about some fictional ideas that spacetime as a continuous manifold does not exist. I guess, my bad?

Yes. While the term "spacetime fabric" does appear in some sources (such as the one @atyy linked to), it's generally best to avoid using that term, precisely because it makes it unclear whether you are talking about standard GR or about other speculative ideas about spacetime. @weirdoguy was not talking about such speculative ideas himself; he was unclear about whether you were.

haushofer
For example in one version of Platini formulation of gravity the metric and connections are taking as independent quantities. I know there are many other. But I am interested in understanding the physical motivation of such a step?
That's motivated mathematically, as far as i know,not physically. In the end the connection still depends on the metric.

robphy
Homework Helper
Gold Member
Possibly useful:

https://www.physicsforums.com/threa...ic-compatible-connection.199500/#post-1512119

https://www.physicsforums.com/threa...a-metric-compatible-affine-connection.761642/

https://physics.stackexchange.com/questions/212298/how-to-measure-torsion-and-non-metricity

[i haven’t read this:] “The role of nonmetricity in metric-affine theories of gravity”
http://iopscience.iop.org/article/10.1088/0264-9381/31/4/045006
https://arxiv.org/abs/1308.1642

One reason there may be interest in alternative formulations is that they may be easier to generalize for a quantum theory of gravity, which may or may not respect torsion-free metric compatibility (assuming such structures exist there).

samalkhaiat
Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast.

I will be very thankful if someone can provide reference to such a paper.

Thanks.
Search for "First order formulation", there are countless number of paper on the subject. In GR, first order formalism is also known as the Palatini formulation.

martinbn