B Unifying Non-Gravity Forces & Spacetime Warping

[szopaw]

TL;DR Summary

Is there a theory interpreting all forces similarly to gravity, i.e. as warping of spacetime, or does some principle prevent it?

As far as I know, the grand prize of a Theory Of Everything is mathematically uniting of all forces in the conditions close to the big bang but one of the main problems from the GR end of things is that gravity is not actually a real force to be combined with anything.
All the most popular efforts are either string theory or quantising gravity, and I will not pretend to have more than "I've watched a lot of YouTube on the topic" level of understanding of those, but is there any attempt to try to view all the other forces from the perspective of GR, i.e. as warping of spacetime? Does string theory do that and I just don't understand that? Or is there an established reason for why that cannot be done?

The reason why I'm even considering the topic is a mention that an alternative to negative mass in Alcubierre Drive could be specially designed electromagnetic fields. That seems to imply that you can, but I do not know the source for that.

 

[PeroK]

You could start here:

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

 

[PeterDonis]

is there any attempt to try to view all the other forces from the perspective of GR, i.e. as warping of spacetime?

Not of ordinary 4-dimensional spacetime, since gravity itself already "uses up" all the warping that is possible there.

But there are theories that add more dimensions to ordinary 4-dimensional spacetime in order to explain other interactions by warping of those dimensions. The first such theory that was proposed that I'm aware of was Kaluza-Klein theory, which added a fifth dimension that was curled up into a tiny circle in order to explain electromagnetism as warping of this dimension. String theory currently claims to explain all of the known non-gravitational interactions by adding six or seven dimensions and explaining the interactions as warping of those dimensions.

None of these theories have made any testable predictions that go beyond the standard theories we already have for non-gravitational interactions, so they are all speculative at this point.

 

[PeterDonis]

The reason why I'm even considering the topic is a mention that an alternative to negative mass in Alcubierre Drive could be specially designed electromagnetic fields.

No, that won't work. Adding more dimensions and using warping in those dimensions to explain non-gravitational interactions does not change the fact that exotic matter (which is a better term than "negative mass") is required in ordinary 4-dimensional spacetime for an Alcubierre drive.

 

[Ibix]

None of these theories have made any testable predictions that go beyond the standard theories we already have for non-gravitational interactions, so they are all speculative at this point.

I thought Kaluza-Klein did predict something - a scalar field that we don't see. Hence it remains intriguing that you can get EM to drop out of it, but not practically relevant.

 

[PeterDonis]

I thought Kaluza-Klein did predict something - a scalar field that we don't see.

Ah, yes, that's right.

 

[haushofer]

I thought Kaluza-Klein did predict something - a scalar field that we don't see. Hence it remains intriguing that you can get EM to drop out of it, but not practically relevant.

What's also intruiging, is that in string theory you more or less have a "double strike": you get more spatial dimensions which need compactification, hence the possibility for Kaluza-Klein like unifications; but at the same time you already have photons in your string spectrum.

 

[haushofer]

Summary:: Is there a theory interpreting all forces similarly to gravity, i.e. as warping of spacetime, or does some principle prevent it?

It can be done, for example in Kaluza Klein theories. There, you add extra dimensions which are then compactified. The usual equations for the gravitational field then also give you the Maxwell equations, which is pretty amazing! Alas, the compactification is not "stable": compactifying a spatial dimension is 1 thing, but keeping it compactified is something else. This makes these kinds of theories quite contrived, which goes under the name of "moduli stabilization".

Neglecting extra dimensions and compactification: the reason why gravity is spacetime geometry, is the equivalence principle. All test masses show the same behaviour in a gravitational field. This is not true for e.g. electromagnetic interactions. E.g, the equation of motion for a charge Q in an electric field E reads

<br /> ma = QE \rightarrow a = \frac{Q}{M}E<br />

Hence the acceleration depends on the ratio of electric charge and mass. Compare this to a test mass in a gravitational field g:

<br /> ma = mg \rightarrow a = g<br />

 

[haushofer]

I thought Kaluza-Klein did predict something - a scalar field that we don't see.

Yes; a 5 dimensional metric contains 5*6/2=15 components, while a 4-dimensional metric contains 4*5/2=10 components. So a 5-dimensional metric can contain one 4-dim. metric, one 4-dim. vector and one scalar field
(15=10+4+1).

 

[kent davidge]

So a 5-dimensional metric can contain one 4-dim. metric, one 4-dim. vector and one scalar field
(15=10+4+1)

Going by this reasoning, a 4 dimensional metric can contain two vector fields and two scalar fields. What could they be?

 

[PeterDonis]

Going by this reasoning, a 4 dimensional metric can contain two vector fields and two scalar fields.

Where are you getting that from?

 

[kent davidge]

Where are you getting that from?

From @haushofer's #9.

 

[PeterDonis]

From @haushofer's #9.

How do you get from post #9 to what you said?

 

[kent davidge]

How do you get from post #9 to what you said?

... he says

a 5 dimensional metric contains 5*6/2=15 components, while a 4-dimensional metric contains 4*5/2=10 components

and then

So a 5-dimensional metric can contain one 4-dim. metric, one 4-dim. vector and one scalar field
(15=10+4+1)

So he is using the number of independent components to count the number of types of fields. Following that, since a 4-dimensional metric has 10 independent components, we have room for two vector fields, 4 components each, and two scalar fields, 2 components each.

 

[PeterDonis]

So he is using the number of independent components to count the number of types of fields.

No, he's not. He's not just randomly throwing numbers around like you are doing. He is describing a specific operation.

Take a symmetric 5 x 5 matrix (a 5-dimensional metric). You can split that up into a symmetric 4 x 4 matrix (a 4-dimensional metric), one 4 x 1 vector (a 4-dimensional vector), and one number (a scalar).

If you do the same kind of operation for a symmetric 4 x 4 matrix, you get a symmetric 3 x 3 matrix (a 3-dimensional metric), one 3 x 1 vector (a 3-dimensional vector), and one number (a scalar). This is what is done, for example, in the ADM formalism for GR: the 4-metric is split up into a 3-metric, a shift vector (3-vector), and a lapse function (scalar).

Following that, since a 4-dimensional metric has 10 independent components, we have room for two vector fields, 4 components each, and two scalar fields, 2 components each.

No, we don't, because you can't just randomly throw vectors and scalars together and say it's "the same" as a symmetric matrix.

 

[szopaw]

Thank you all for you insights, in none of the pop-science sources that talk about string theory etc. did anyone ever mention that compactifying higher dimensions is the equivalent of GR for other forces. That makes the introductiont of those higher dimensions suddenly make perfect sense, rather than being something spat out by an arcane mathematical machine.

Neglecting extra dimensions and compactification: the reason why gravity is spacetime geometry, is the equivalence principle. All test masses show the same behaviour in a gravitational field. This is not true for e.g. electromagnetic interactions. E.g, the equation of motion for a charge Q in an electric field E reads

<br /> ma = QE \rightarrow a = \frac{Q}{M}E<br />

Hence the acceleration depends on the ratio of electric charge and mass. Compare this to a test mass in a gravitational field g:

<br /> ma = mg \rightarrow a = g<br />

I don't think there is a better answer to "Is there a principle preventing it" than one easily presented at high school physics level.