Kaluza-Klein Theory and 5D Einstein Equations

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  • #1
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Others are telling me the Einstein Field Equations can work in other dimensions other than 4D (3D space + 1D time). How true is it? So I'd like to ask for clarifications. I googled about it and found one reference for example:

Kaluza–Klein theory - Wikipedia

The five-dimensional (5D) theory developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Einstein in 1919,[2] and published them in 1921.[3] Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components. 10 components are identified with the 4D spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called the "radion" or the "dilaton". Correspondingly, the 5D Einstein equations yield the 4D Einstein field equations, the Maxwell equations for the electromagnetic field, and an equation for the scalar field. Kaluza also introduced the "cylinder condition" hypothesis, that no component of the five-dimensional metric depends on the fifth dimension. Without this assumption, terms are introduced that involve derivatives of the fields with respect to the fifth coordinate. This extra degree of freedom is such that the field equations of fully variable 5D relativity grow enormous in complexity. Standard 4D physics seems to manifest the cylinder condition, and the corresponding simpler mathematics.

<snip>
The field equations are obtained from five-dimensional Einstein equations, and the equations of motion from the five-dimensional geodesic hypothesis.
I assume the Einstein equations is same as the Einstein Field Equations (because when I clicked the highlight, the EFE came out).
If EFE can work above 4D. Up to what D can it be valid? Can it be made to work in the compactified dimensions too?

Remember Lisa Randall RS1 and RS2 seem to use General Relativity above 4D too (doesnt it?).
 

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  • #2
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Here is illustration of my questions:

efe2.JPG


efe_illus.JPG


"the contents of the universe tell spacetime how to bend, but conversely, the curvature of spacetime dictates how objects move. "

from
The Maths of General Relativity (7/8) - The Einstein equation - YouTube

How do you illustrate it when spacetime is 7 dimensional (large or infinite Randall RS wise) for instance? I know it is hard already to illustrate real 4D spacetime.

Can Newtonian gravity work at 7 large or infinite dimensions?

Or in BSM models with EFE higher than 4D, do they just use the basic EFE and add other terms in the basic EFE that don't have to do with curvature or matter contents?

Another reference about higher dimensional EFE.

Higher-dimensional Einstein gravity - Wikipedia

"Higher-dimensional Einstein gravity is any of various physical theories that attempt to generalise to higher dimensions various results of the well established theory of standard (four-dimensional) Einstein gravity, that is, general relativity. This attempt at generalisation has been strongly influenced in recent decades by string theory."
 
  • #3
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This thread partly stemed from what Peterdonis mentioned in message #41 of the thread "General Relativistic quantum theory".

I wrote:

"the EFE can handle many dimensions (not just 4)"

Peterdonis replied

"Not the EFE of GR, no. That EFE is specifically for 4 dimensions."

I knew I have read in books before the EFE was not specifically for 4 dimensions. So I started this thread and asked elsewhere too. What I learnt is this.

When you derive the EFE from the Einstein-Hilbert action,

20210113_074231.jpg


at no point do you need to restrict the number of dimensions d to 4 (note though that the coupling constant κ depends on d). Variation with respect to the metric leads to the usual field equations,

20210113_074309.jpg


Nowhere in the derivation do we need to explicitly fix d. So this is valid in whatever 3 + 1 dimensions.

For all intents and purposes, GR and EFE are basically synonymous (the EFE are a part of GR). Standard GR works with a 4-dimensional manifold, but it holds in any number of dimensions


Whatever, what is very intruiging about all this is that in other (possible) Multiverse where the world is also 3D with time. It didn't mean it is automatically ruled by GR. So the question to ask is, why is this universe ruled by GR? What laws did it come from?

Peterdonis continued with the following statements. I hope someone can clarify it as im still perflexed where the compactified Calabi Yau space is emergent from.

My understanding (which might possibly be mistaken; perhaps other experts on this forum can weigh in here) of how "spacetime emerges" in string theory is that the string mode that looks like a massless spin-2 field at low energy only affects the 4 dimensions we actually observe, not the others. If that is correct, then the dimensions contained in the Calabi-Yau spaces do not emerge that way; and that was what you were asking about.
Fra and other experts. Any insights on this?
 
  • #4
We live in Minkowski space with Lorentzian signature with the dimensions of space and one dimension of time. Ignoring time, we live in flat Euclidean space. String theory postulates that there are actually 10 or 11 spacetime dimensions, with the extra dimensions compactified on an internal manifold, such as a Calabi-Yau threefold. Whether our space is flat or curved, general relativity does not assume that our universe is embedded or suspended within higher dimensional space. It would have been easier to formulate if we had assumed that. When physicists say space is “flat”, they mean something different than the average person’s usage of the word “flat”. It has to do with the fact that parallel lines to do not intersect. Also, there are theories such as brane world cosmology that assume that our universe is within higher dimensional space. According to the traditional Big Bang, the Big Bang was the beginning of time, so there was no such thing as “before the Big Bang” However, more recent theories such eternal inflation postulate that time extends infinitely backwards. A Calabi-Yau manifold is a Kahler manifold with vanishing first Chern class. We use Calabi-Yau with SU(3) holonomy, which is 6-dimensional. 10 - 6 = 4, which we identify with the four spacetime dimensions we observe. If you start with E8 x E8 superstring theory, and compactify the extra dimensions on the correct Calabi-Yau, you get something similar to the Standard Model. The details of the topology that the extra dimensions are compactified on determines the low energy physics. If you are asking, why are the extra dimensions compactified on that manifold, the answer is that we chose to compactify the extra dimensions on that manifold in order to get a result that is as close as possible to the real Universe. If you are not satisfied by that, you could say there is a multiverse with an infinite number of universes, and our universe is anthropically selected, so for example, universes where the extra dimensions are compactified on different manifolds have no observers.
 
  • #5
271
10
We live in Minkowski space with Lorentzian signature with the dimensions of space and one dimension of time. Ignoring time, we live in flat Euclidean space. String theory postulates that there are actually 10 or 11 spacetime dimensions, with the extra dimensions compactified on an internal manifold, such as a Calabi-Yau threefold. Whether our space is flat or curved, general relativity does not assume that our universe is embedded or suspended within higher dimensional space. It would have been easier to formulate if we had assumed that. When physicists say space is “flat”, they mean something different than the average person’s usage of the word “flat”. It has to do with the fact that parallel lines to do not intersect. Also, there are theories such as brane world cosmology that assume that our universe is within higher dimensional space. According to the traditional Big Bang, the Big Bang was the beginning of time, so there was no such thing as “before the Big Bang” However, more recent theories such eternal inflation postulate that time extends infinitely backwards. A Calabi-Yau manifold is a Kahler manifold with vanishing first Chern class. We use Calabi-Yau with SU(3) holonomy, which is 6-dimensional. 10 - 6 = 4, which we identify with the four spacetime dimensions we observe. If you start with E8 x E8 superstring theory, and compactify the extra dimensions on the correct Calabi-Yau, you get something similar to the Standard Model. The details of the topology that the extra dimensions are compactified on determines the low energy physics. If you are asking, why are the extra dimensions compactified on that manifold, the answer is that we chose to compactify the extra dimensions on that manifold in order to get a result that is as close as possible to the real Universe. If you are not satisfied by that, you could say there is a multiverse with an infinite number of universes, and our universe is anthropically selected, so for example, universes where the extra dimensions are compactified on different manifolds have no observers.
Thanks for the insights. Yesterday i was reading a thread about the simulation hypothesis with amusement. I googled to read what is the latest and i saw this paper.

https://arxiv.org/abs/1011.5499

"Background

A hundred years of research have validated quantum and relativity theories in sub-atomic and cosmic domains, yet they conflict at the core. The quandary is that:

1.Quantum theory assumes an objective space background, which relativity specifically denies. For quantum theory to satisfy relativity it must be background independent, i.e. not assume, as it currently does, that quantum states arise in a fixed space and evolve in a fixed time [3].

2.Relativity assumes objects exist locally, which quantum theory specifically denies. For relativity to satisfy quantum theory it must be foreground independent, i.e. not assume, as it currently does, that localized objects move relatively through space-time.

These two great theories contradict because each debunks an objective reality assumption the other still clings to. Quantum theory challenges the objective reality of foreground objects, but still assumes a fixed background. Relativity theory challenges space and time as objective backgrounds, but still assumes fixed foreground objects. Both theories rebelled against the idea of objective reality in different ways, so each exposes the other's conceptual baggage but ignores its own.

To reconcile, both theories must abandon entirely all objective reality assumptions, i.e. reject objective space, objective time, objective existence, objective movement and any similar ideas. The prime axiom here is that nothing in the physical world exists of or by itself, so while it seems substantial and self-sustaining, both its foreground and background arise from processing. One can't "half-adopt" this theory, so it has no fixed space or time that quantum states exist in, nor any fixed entities that move relatively. The only constant is information processing, with space, time, matter and energy just outputs. In this model, not only is matter-energy calculated by "space" as Zuse suggests [4], but space itself is also calculated. "

Just for sake of discussions. Lets say it were true. Does it mean there is no longer a need for quantum gravity? If still needed, why?
 
  • #6
The paper who quote is just talking about the "simulation hypothesis" implying that our universe is a simulation being run on a computer in another universe. Is that universe also a simulation being run on another computer in yet another universe, ad infinitum? It reminds me of religious people who say the universe was made by God. Well, was that god made by a different god, ad infinitum?

The following article talks about background independence in gauge theory.

https://home.uni-leipzig.de/tet/?page_id=2382

The following paper talks about quantum gravity.

https://arxiv.org/pdf/0907.4238.pdf
 

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