Spin and torsion of the connection

In summary, the conversation discusses the use of a connection with torsion in General Relativity to model particles with spin. It is mentioned that invariance under the pointcarré group implies conservation of spin, and that bosons can be seen as quantifications of the connection in Yang-Mills theory. The concept of Einstein-Cartan-Gravity is also brought up as a possible extension of GR in the presence of spin, where torsion couples to spin density but is non-propagating. The conversation concludes that while GR and ECT are different mathematically, they are indistinguishable in experiments due to the suppression of torsion effects.
  • #1
kroni
80
10
Hello,

I read in an article that in GR it is possible to use a connection with torsion to model ponctual particle with spin. The connection can be decomposed in a curvatue part and a torsion part. In an other part, we know that the invariance under pointcarré group imply conservation of spin of particle. Can i conclude with a lot of prudence that a particle with spin is a exitation of the connection in GR ?

I say this because boson are quantification of field that is, for yang mills theory, the value of the connection.

Thanks.
 
Physics news on Phys.org
  • #2
If there's spin, there no GR anymore. If there's a torsion, there's no GR anymore. So your first statement is inaccurate. What one needs to read is Hehl's work on Poincare gauge theory, which can be seen as a semi-classical extension of GR.
 
  • #3
I would propose to start reading a few papers regarding Einstein-Cartan-Gravity, which is the most simple and natural extension of GR in the presence of spin.

The interesting thing is that torsion couples to the spin density but is non-propagating; so there are no torsion-waves (gravitational waves are pure curvature-waves). In vacuum (vanishing energy-momentum / vanishing spin density) there is no torsion and the theory reduces to GR. Inside matter (non-vanishing energy-momentum / non-vanishing spin density) there can be torsion but its effects are highly suppressed.

So in essence GR and Einstein-Cartan-Gravity are different mathematically, but indistinguishable experimentally!
 
  • #4
I am so sad, because i work with someone who prove that rotational waves in matérials have a fermionic behaviour (standard compression waves are bosonic). So, i was thinking that, may be, fermions appear when a quantification of the torsion of the connection is done. But if torsion is non propagative, it fail. Thanks for your answer, i will read the paper you speak about.

Clément
 
  • #5
but within the material torsion could propagate b/c the spin current of the material does!

but sorry, I can't say anything regarding effective quasi-particle theories with fermionic d.o.f. + Einstein-Cartan
 
  • #6
So in essence GR and Einstein-Cartan-Gravity are different mathematically, but indistinguishable experimentally!
Are you sure about that statement? I thought Einstein-Cartan allows violation of conservation of angular momentum and of spin, conserving only the sum of the two. So it is certainly experimentally distinguishable from GR, which conserves angular momentum strictly.
 
  • #7
haael said:
Are you sure about that statement? I thought Einstein-Cartan allows violation of conservation of angular momentum and of spin, conserving only the sum of the two. So it is certainly experimentally distinguishable from GR, which conserves angular momentum strictly.

Inside matter ... there can be torsion but its effects are highly suppressed ... So in essence GR and Einstein-Cartan-Gravity are different mathematically, but indistinguishable experimentally!

I have to check the calculations but as far as I remember the effects are measurable in principle, not in practice. Either there is isolated spin (e.g. from elementary particles) and therefore GR / ECT effects are not measurable, or there are macroscopic GR / ECT effects due to angular momentum, but then the violations due to spin are suppressed. So as far as I can see there is no known experimental setup which can distinguish between GR and ECT.
 

1. What is the difference between spin and torsion of the connection?

Spin and torsion are two different mathematical concepts used to describe the behavior of connections. Spin is a measure of how much a connection "twists" or rotates a vector as it is parallel transported along a curve. Torsion, on the other hand, measures the extent to which a connection fails to preserve the lengths of vectors as they are parallel transported. In other words, spin describes the rotational aspect of a connection, while torsion describes the stretching or shearing aspect.

2. How are spin and torsion related to curvature?

Spin and torsion are related to curvature in the sense that they are both measures of how much a connection deviates from being "flat." Curvature, which is a more commonly known concept, measures how much a connection fails to preserve the shape of a small parallelogram as it is parallel transported around a closed curve. Spin and torsion are two additional measures that provide more detailed information about the behavior of the connection.

3. What are some real-world applications of spin and torsion?

Spin and torsion have various applications in physics and engineering. For example, in general relativity, spin and torsion are used to describe the behavior of particles in the presence of gravitational fields. In quantum mechanics, spin is a fundamental property of particles and is used to describe their intrinsic angular momentum. In engineering, spin and torsion are used to analyze the behavior of materials under stress and strain.

4. How do spin and torsion affect the behavior of connections?

Spin and torsion can have significant effects on the behavior of connections. In general, a connection with higher spin or torsion will result in a greater distortion of vectors as they are parallel transported. This can lead to phenomena such as frame-dragging in general relativity, where the spin of a massive object causes the space around it to twist and rotate.

5. How are spin and torsion mathematically represented?

Spin and torsion are represented mathematically using tensors. Spin is represented by the spin tensor, which is a rank-2 tensor that describes the rotational behavior of the connection. Torsion is represented by the torsion tensor, which is a rank-3 tensor that describes the stretching or shearing behavior of the connection. These tensors can be used to calculate the spin and torsion of a connection at any given point in space.

Similar threads

  • Differential Geometry
Replies
5
Views
2K
  • Special and General Relativity
Replies
10
Views
5K
  • Science and Math Textbooks
Replies
2
Views
2K
  • Special and General Relativity
Replies
1
Views
930
Replies
2
Views
2K
Replies
1
Views
1K
  • Special and General Relativity
2
Replies
40
Views
2K
  • Beyond the Standard Models
Replies
7
Views
2K
Replies
12
Views
6K
  • Quantum Physics
3
Replies
87
Views
5K
Back
Top