Possibility of Discrete Symmetries in GR: Exploring CPT Symmetry

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• Tertius
In summary: This is not a physical mechanism; it's just a mathematical tool. The dual universe they create is not a physical entity; it's just a mathematical structure.In summary, the authors claim that they have found a way to create a dual universe that is physically identical to the original, except that the energy density is negative in the dual universe. This is not a physical mechanism; it's just a mathematical tool. The dual universe they create is not a physical entity; it's just a mathematical structure.
Tertius
TL;DR Summary
It seems the universe is not continuously symmetric with time because of a few factors, entropy, expansion, etc. But, is there a possibility of defining a sensible discrete temporal asymmetry for the universe?
Our current model (FLRW) is clear that the universe has a continuous temporal asymmetry. This is seen as the expansion factor grows with time, and thermodynamically with entropy.
A continuous transformation in the current model ##t \rightarrow t + dt## is not the same as ##t \rightarrow t - dt ##.
Because of general covariance, however, there is not an absolute notion of time. This mean we can basically shift any coordinate system some amount to avoid a discrete symmetry. This is the edge of my knowledge on this...

Is it possible to define some discrete temporal symmetry relating to the coordinate system on some manifold? Or does this ultimately amount to combining QFT with GR, so we can have a meaningful CPT symmetry for the universe?

Also: found this paper while researching the topic. Does this seem legit or is there a problem with the way the authors use discrete symmetries with the EFE? (Equation 1 shows the basis of what they are doing, with ##dx^\mu \rightarrow -dx^\mu##. Eventually, by extension from electrodynamics, they get to eq. 7, which is supposedly a CPT symmetrized EFE)
https://arxiv.org/pdf/1103.4937.pdf

The Bill
Tertius said:
Does this seem legit
I don't think so. The paper claims that the gravitational "charge" of an object is its energy-momentum 4-vector instead of its invariant mass. I have never seen that in any other textbook or paper on GR, and it is inconsistent with the general principle that the trajectory of a test object in GR depends only on its 4-velocity and the spacetime geometry. So no, I don't think the reasoning in this paper is valid.

The Bill, vanhees71 and Tertius
Tertius said:
eq. 7, which is supposedly a CPT symmetrized EFE)
That equation is not the problem; it's just the standard EFE with a particular sign convention. (There is no need to "CPT symmetrize" the EFE.)

The problem is equation (8), where they introduce a ratio ##m_{(\text{g})} / m_{(\text{i})}## out of nowhere. There is no such ratio in GR; it is inconsistent with the principle I described in my previous post, that the trajectory of a test object depends only on its 4-velocity and the spacetime geometry (which in equation (8) are represented by ##dx^\mu / d \tau## and ##\Gamma^\lambda_{\mu \nu}##).

The Bill, vanhees71 and Tertius
Tertius said:
Because of general covariance, however, there is not an absolute notion of time. This mean we can basically shift any coordinate system some amount to avoid a discrete symmetry.
I'm not sure what you mean by this. There is no need to invoke any such thing to show why the expanding FRW geometry that is used to describe our universe is time asymmetric. The time asymmetry is obvious from the fact that one can define a family of geodesic worldlines (those of the comoving observers) with an expansion scalar that is always positive (which corresponds to the fact that our universe, in our best current model, will expand forever). In a time symmetric spacetime that would be impossible (the expansion would have to stop at some instant and then reverse).

The Bill and vanhees71
Tertius said:
Is it possible to define some discrete temporal symmetry relating to the coordinate system on some manifold?
If the spacetime is time symmetric, sure: just choose appropriate coordinates and do the transformation ##t \to - t##. The solution will be invariant under such a transformation. (The fact that this cannot be done for the spacetime that describes our universe is another way of seeing that it is time asymmetric.)

The Bill and vanhees71
Tertius said:
does this ultimately amount to combining QFT with GR, so we can have a meaningful CPT symmetry for the universe?
CPT symmetry as it's standardly understood does come from quantum field theory. It is normally formulated in flat spacetime, i.e., using only SR, not GR. Flat spacetime is time symmetric.

The Bill and vanhees71
PeterDonis said:
I'm not sure what you mean by this. There is no need to invoke any such thing to show why the expanding FRW geometry that is used to describe our universe is time asymmetric. The time asymmetry is obvious from the fact that one can define a family of geodesic worldlines (those of the comoving observers) with an expansion scalar that is always positive (which corresponds to the fact that our universe, in our best current model, will expand forever). In a time symmetric spacetime that would be impossible (the expansion would have to stop at some instant and then reverse).

If any coordinate system is valid in GR, I would assume you could replace ##t \rightarrow -t##, but wouldn't it be required to get the same physics? Like it wouldn't make the energy density negative, so it should be the same 'universe'.
I suppose I am really wondering about a bi-verse, and the symmetries required to make something like that, as in the recent publication: https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.4.L022027

They create a dual universe from an action (eq. 9), and a coupling (eq. 10). And overlay two FRW metrics on the same manifold (I assume). Is it possible to have two universes (each with their own metric) on the same manifold? It seems the coordinates would only be valid in a single universe, which is why I was curious about discrete transformation that could create the metric for the other universe.

PeterDonis said:
CPT symmetry as it's standardly understood does come from quantum field theory. It is normally formulated in flat spacetime, i.e., using only SR, not GR. Flat spacetime is time symmetric.
For some reason I didn't realize the QFT application was after the Lorentz invariance proof. Here is a very nice quote from wikipedia for anyone else reading about the history:
'The CPT theorem appeared for the first time, implicitly, in the work of Julian Schwinger in 1951 to prove the connection between spin and statistics.[3] In 1954, Gerhart Lüders and Wolfgang Pauli derived more explicit proofs,[4][5] so this theorem is sometimes known as the Lüders–Pauli theorem. At about the same time, and independently, this theorem was also proved by John Stewart Bell.[6] These proofs are based on the principle of Lorentz invariance and the principle of locality in the interaction of quantum fields. Subsequently, Res Jost gave a more general proof in 1958 using the framework of axiomatic quantum field theory.'

vanhees71
Tertius said:
If any coordinate system is valid in GR, I would assume you could replace ##t \rightarrow -t##, but wouldn't it be required to get the same physics?
No. We are not talking about an ordinary coordinate transformation, where the whole point is that all invariants are preserved--which means you have to transform all tensors together with the coordinates. We are talking about just replacing ##t \to - t## without changing anything else and seeing what happens. In general this kind of operation does not preserve invariants; but in the special case of time symmetric spacetimes where the original coordinates were chosen properly, it does.

Tertius said:
Like it wouldn't make the energy density negative
I have no idea what you are talking about here.

Tertius said:
, so it should be the same 'universe'.
I have no idea what you are talking about here either.

Tertius said:
I suppose I am really wondering about a bi-verse, and the symmetries required to make something like that, as in the recent publication: https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.4.L022027

They create a dual universe from an action (eq. 9), and a coupling (eq. 10). And overlay two FRW metrics on the same manifold (I assume).
No, that's not at all what they are doing. They are taking a condensed matter system and constructing a (hypothetical) analogy with what they explicitly call a "toy" model. (Also, this is a very advanced paper and you should not expect to be able to follow it without a graduate-level background in the subject matter.)

Tertius said:
Is it possible to have two universes (each with their own metric) on the same manifold?
No. This doesn't even make sense.

Tertius said:
discrete transformation that could create the metric for the other universe.
There is no such thing.

The Bill
PeterDonis said:
No. We are not talking about an ordinary coordinate transformation, where the whole point is that all invariants are preserved--which means you have to transform all tensors together with the coordinates. We are talking about just replacing ##t \to - t## without changing anything else and seeing what happens. In general this kind of operation does not preserve invariants; but in the special case of time symmetric spacetimes where the original coordinates were chosen properly, it does.I have no idea what you are talking about here.I have no idea what you are talking about here either.No, that's not at all what they are doing. They are taking a condensed matter system and constructing a (hypothetical) analogy with what they explicitly call a "toy" model. (Also, this is a very advanced paper and you should not expect to be able to follow it without a graduate-level background in the subject matter.)No. This doesn't even make sense.There is no such thing.
That first part makes sense. I see how you would need to transform all of the tensors with the ordinary coordinate transform.
I'm a nuclear scientist trying to learn GR and cosmology (on my own), so I definitely have gaps in my knowledge here.
I was trying to understand how having two metrics would be a useful model, or if it is mathematically coherent.
Thanks for the responses

Tertius said:
I'm a nuclear scientist trying to learn GR and cosmology (on my own)
What sources are you trying to learn from?

The Bill
PeterDonis said:
What sources are you trying to learn from?
I've mostly relied on Carroll, and have used Weinberg as a reference, and some MTW.
I think what is still not clear to me is the relationship between manifolds, metrics, and observers. For example, taking some cosmic situation, making a sensible coordinate system for it, and using that to generate what an observer will detect at various places on that coordinate system.

The Bill
Tertius said:
I've mostly relied on Carroll, and have used Weinberg as a reference, and some MTW.
Ok. All good sources.

Tertius said:
I think what is still not clear to me is the relationship between manifolds, metrics, and observers.
The quick five cent summary:

A "manifold" is a set of points with some topology. For example, ##\mathbb{R}^4## is the manifold of 4-tuples of real numbers with the standard topology.

A "metric" is a 2nd rank covariant tensor field on a manifold, i.e., a function that assigns a 2nd rank covariant tensor to each point in the manifold, with some requirements of continuity, etc. whose details can be fleshed out as needed. The physical meaning of the metric is that, when contracted with a displacement (i.e., an infinitesimal change from one point in the manifold to a nearby point), it gives the "squared length" along the displacement.

The term "observer" is used with a number of different meanings, but to me the most useful one is that an "observer" is a timelike worldline--a continuous timelike curve in the manifold (where "timelike" means that the "squared length" of displacements along the curve, given by the metric, is of the right sign to be the [square of the] proper time measured by a clock that has the curve as its worldline) that describes the observer's history through spacetime--along with a set of 3 mutually orthogonal spacelike vectors, all carried along the worldline such that they are always orthogonal to it, which realize the three "spatial axes" that the observer uses to make measurements.

Tertius said:
taking some cosmic situation, making a sensible coordinate system for it, and using that to generate what an observer will detect at various places on that coordinate system.
IIRC Carroll gives some good examples of this. Is there a particular one you're having a problem following?

The Bill

Follow up on manifolds/metrics - from the formulation of the maximally symmetric deSitter spacetime (in Carroll pg. 324 or Weinberg), you define 4D coordinates (the metric) on the surface of (or embedded in) a fictitious 5D Minkowski space. What specifically is the manifold in this example? Is it ##\mathbb{R}^4## or ##\mathbb{R}^5##? And, for deSitter, the curvature is defined to be positive everywhere, doesn't that imply that the manifold on which the coordinates lie has a 'shape'? Maybe that answers the question, that the manifold must then be ##\mathbb{R}^4## because the underlying 5D system is Minkowskian.

Observers - timelike worldlines are a very helpful way to think about it. I had only read previously that observers were a locally flat part of spacetime, so the basis vectors are defined such that locally the observer always sees Minkowski spacetime. For me, this is where the clarity ends. We have curved coordinates over some area of spacetime, and wherever an observer is, they will see the same physics, which I believe is the equivalence principle.

A good simple example I found in Carroll was pg 116-117, and it is easy to follow. A basic expanding universe (spatially time asymmetric, as we discussed), a co-moving observer(at rest), and you calculate the energy of a photon the observer detects. Equation 3.87 is an affine parameter solution of the geodesic equation ##dt/d\lambda = \omega/a##, showing the cosmological redshift for any comoving observer at time t. The inverse of this equation should define the 'coordinate' velocity of light. I noticed in Einstein's original paper, he computed the deflection of light in a gravitational field using a change in the radial velocity of light (he even used the word speed). So, even though the coordinate velocity of light can change in various coordinate systems, the observer always measures light to travel at velocity c. My main question from this example - in the cosmological model, the light is redshifted because space is expanding. Couldn't we model this as a change in the coordinate speed of light, that previously the light was moving slower, and it has to 'speed up' so we observe it red-shifted (from ##E=hc/\lambda##)? I know the energy calculation in the example shows that the actual energy decreases, is there a version of this computation that preserves the energy of the photon and redshifts with a time-dependent change in 'coordinate speed'?

Tertius said:
from the formulation of the maximally symmetric deSitter spacetime (in Carroll pg. 324 or Weinberg), you define 4D coordinates (the metric) on the surface of (or embedded in) a fictitious 5D Minkowski space.
Yes, but this is just a convenience; the 5D Minkowski space is not part of the manifold, as the word "fictitious" makes clear.

Tertius said:
What specifically is the manifold in this example? Is it ##\mathbb{R}^4## or ##\mathbb{R}^5##?
The underlying manifold for de Sitter space is ##mathbb{R}^4##.

Tertius said:
And, for deSitter, the curvature is defined to be positive everywhere, doesn't that imply that the manifold on which the coordinates lie has a 'shape'?
No, it implies that we have put a metric with constant positive curvature on the manifold ##\mathbb{R}^4##. Curvature is a property of the metric.

Tertius said:
I had only read previously that observers were a locally flat part of spacetime
I'm not sure where you read that, but either the source was wrong or you misinterpreted it. This is why we always like to have specific references for these things instead of just saying "I read somewhere". Local flatness, or more precisely being local Lorentzian, is a property of the metric, not of observers.

Tertius said:
the basis vectors are defined such that locally the observer always sees Minkowski spacetime.
Again, either your source was wrong or you misinterpreted it. See above.

Tertius said:
We have curved coordinates over some area of spacetime, and wherever an observer is, they will see the same physics
This is true in de Sitter spacetime because the curvature is constant, so the local geometry is the same everywhere. It is not true in a general spacetime.

Tertius said:
which I believe is the equivalence principle.
No. The equivalence principle is true in any spacetime, even one in which the curvature varies a lot from point to point. The EP is really another way of describing the local flatness of the metric.

Tertius said:
in the cosmological model, the light is redshifted because space is expanding.
More precisely, light signals traveling between comoving observers are redshifted as the universe expands. Yes, that's correct.

Tertius said:
Couldn't we model this as a change in the coordinate speed of light
If you pick appropriate coordinates, sure. But the coordinate speed of light in the standard cosmological coordinates does not change. The redshift is due to the change in the scale factor.

Tertius said:
I know the energy calculation in the example shows that the actual energy decreases
More precisely, the energy measured by comoving observers decreases. Energy is observer dependent.

Tertius said:
is there a version of this computation that preserves the energy of the photon and redshifts with a time-dependent change in 'coordinate speed'?
No, because you can't just arbitrarily change what "the energy of the photon" means.

The Bill and vanhees71
Ok, the curvature is a property of the metric, that makes sense. Does it describe the curvature of the manifold? From https://en.wikipedia.org/wiki/Curva...xpress_the_curvature_of_a_Riemannian_manifold it would seem manifolds have 'curvature'. But I suppose it is a meaningless statement without a metric?

An accurate (I think) definition of the equivalence principle is that gravitational and inertial mass are the same. I assumed this meant that an observer would always see the same 'physics'. Precisely, that local measurements could not differentiate between an accelerated frame, and a frame at rest in a gravitational field. But, thinking more, if an observer is on the surface of a planet, they will see an upward burst of light at a particular frequency, but their friend in space will see that same burst of light at a lower frequency from the gravitational red-shift. So, obviously their measurements show different physical quantities, is this what you meant that in general spacetime, the 'physics' is not always the same?I may have misinterpreted the statement I read on another answer online (not the best place for sourcing, I know) It is apparently describing the precise meaning of observer in GR (i couldn't find in Carroll or Weinberg, or maybe it is just not a great question). 'At each point of the curved manifold, it is possible to construct frames (vierbein), consisting of four orthonormal set of vectors (one time-like and three space-like vectors), i.e. it is possible to construct a frame bundle on the manifold. Now, an observer is, precisely, a smooth section in the frame bundle. A section in the frame bundle is an integral curve of the time-like vector field.' Perhaps I thought incorrectly that a frame bundle was the same thing as the tangent space of a manifold.

PeterDonis said:
If you pick appropriate coordinates, sure. But the coordinate speed of light in the standard cosmological coordinates does not change. The redshift is due to the change in the scale factor.

Here is something I am not understanding -> if we rearrange the FRW metric for a null path, it is quick to see that ##\frac{dr}{dt} = \frac{c}{a(t)}##. Doesn't this mean the coordinate speed of a null path is decreasing with the expansion of the universe? Wouldn't this change in speed cause a red-shift?
Better wording -> how would you make a system of coordinates where the null paths change speed, if you cannot do this for FRW metrics?
What is it about the FRW coordinates that makes this way of thinking about it nonsensical

Tertius said:
the curvature is a property of the metric, that makes sense. Does it describe the curvature of the manifold?
When the term "curvature of a manifold" is used, as in the Wikipedia article you linked to, what is really meant is "curvature of a manifold with metric". "Manifold with metric" is the precise term for the thingies we have been talking about (a spacetime in GR is a manifold with metric). The curvature is described, as the Wikipedia article notes, by the Riemann curvature tensor, which is computed from the metric.

Tertius said:
An accurate (I think) definition of the equivalence principle is that gravitational and inertial mass are the same.
Sort of. The term "equivalence principle" has been used to mean a number of different things (there are at least three versions of it in the GR literature, the weak, Einstein, and strong equivalence principles). In GR the equality of gravitational and inertial mass (more precisely of passive gravitational mass and inertial mass) is enforced by the fact that the worldline of an object--its trajectory through spacetime--depends only on its initial 4-velocity and the geometry of the spacetime, not on the mass of the object. Depending on how you want to interpret "gravitational", that could be linked to either the weak or the Einstein equivalence principle.

Tertius said:
I assumed this meant that an observer would always see the same 'physics'.
Only if you exclude the effects of the spacetime geometry (curvature) from "physics". But that doesn't seem reasonable.

Tertius said:
local measurements could not differentiate between an accelerated frame, and a frame at rest in a gravitational field
More precisely, that locally (i.e., restricting to a region of spacetime that is small enough that the effects of curvature are negligib le) one cannot differentiate between having a given proper acceleration due to, say, being accelerated by a rocket in otherwise empty space, or sitting at rest in a gravitational field. This is one way of stating the equivalence principle, but it's limited because it only talks about the case of nonzero proper acceleration. But the EP also applies to the case of zero proper acceleration, i.e., free-fall motion.

Tertius said:
if an observer is on the surface of a planet, they will see an upward burst of light at a particular frequency, but their friend in space will see that same burst of light at a lower frequency from the gravitational red-shift. So, obviously their measurements show different physical quantities
Yes, but that's because even in flat spacetime, observers at different heights in an accelerating rocket will observe a redshift/blueshift when they exchange light signals. So this effect counts as a "local" effect that can't tell you anything about spacetime curvature. (If you have three or more observers at different heights and they are separated enough that you can measure the specific rate of change of redshift with height, then you can detect effects of spacetime curvature; that would no longer count as a "local" effect.)

Tertius said:
is this what you meant that in general spacetime, the 'physics' is not always the same?
No. See above.

Tertius said:
Perhaps I thought incorrectly that a frame bundle was the same thing as the tangent space of a manifold.
Indeed this is incorrect. A frame bundle is useful to model an "observer" that has a finite size, so a single worldline is not sufficient to model the observer; you need a bundle of worldlines. The "frame" part is what I was describing in post #13 when I talked about the 3 mutually orthogonal spacelike vectors that an "observer" would carry along their worldline; for an observer modeled by a bundle of worldlines, such a thing is carried along each of the worldlines, and the whole thing is a frame bundle.

The Bill and vanhees71
Tertius said:
Here is something I am not understanding -> if we rearrange the FRW metric for a null path, it is quick to see that ##\frac{dr}{dt} = \frac{c}{a(t)}##.
More precisely for a null radial path in a spatially flat FRW universe, for a particular choice of FRW coordinates. There is more than one form for those coordinates.

Tertius said:
Doesn't this mean the coordinate speed of a null path is decreasing with the expansion of the universe?
Yes, sorry, I was too hasty before. The coordinate speed of a null path in conformal coordinates is constant. But the coordinates you were using are not conformal. In non-conformal coordinates, yes, the coordinate speed of a null path will change as the scale factor changes.

Tertius said:
Wouldn't this change in speed cause a red-shift?
No, because coordinates can't cause anything. The redshift is caused by the change in scale factor and the effect it has on the geometric relationship between comoving worldlines.

Tertius said:
Better wording -> how would you make a system of coordinates where the null paths change speed, if you cannot do this for FRW metrics?
You can. See my comment above about conformal coordinates.

The Bill and vanhees71

One question: "coordinates can't cause anything" makes a lot of sense, and clears up a few things for me. Does this mean it is not physically possible for the cosmological red-shift to be caused by a coordinate change in the speed of light? And does suggesting the redshift is due to a change in the speed of light require that change to be a real, measurable (local) change? This I assume would violate Lorentz invariance and cause a host of other issues

Tertius said:
Does this mean it is not physically possible for the cosmological red-shift to be caused by a coordinate change in the speed of light?
The question doesn't even make sense. "A coordinate change in the speed of light" isn't even a physical thing to begin with.

Tertius said:
does suggesting the redshift is due to a change in the speed of light require that change to be a real, measurable (local) change?
Now you appear to have shifted the meaning of "a change in the speed of light". If by that you mean something observers could actually measure, if such a thing were measured it would be a physical thing that could be the cause of something else--but of course we never have measured any such change in the actually measured speed of light.

Tertius said:
This I assume would violate Lorentz invariance and cause a host of other issues
Not necessarily. If photons were not massless but had an extremely tiny mass, their speed could change without violating Lorentz invariance. But, as above, no experiment has indicated any such thing.

The Bill, vanhees71 and Tertius
PeterDonis said:
I'm not sure what you mean by this. There is no need to invoke any such thing to show why the expanding FRW geometry that is used to describe our universe is time asymmetric. The time asymmetry is obvious from the fact that one can define a family of geodesic worldlines (those of the comoving observers) with an expansion scalar that is always positive (which corresponds to the fact that our universe, in our best current model, will expand forever). In a time symmetric spacetime that would be impossible (the expansion would have to stop at some instant and then reverse).
This criterion is not complete. Consider the Milne congruence in Minkowski spacetime. The latter is time symmetric, yet you have a congruence with positive expansion scalar forever. I wonder if you have to add some statement about complete coverage by the congruence. That is, it is the full Minkowski spacetime that is time symmetric, but the Milne congruence only covers the region within some light cone. Or, maybe you can actually say the region covered by the Milne congruence, treated as it’s own manifold, is not time symmetric.

The Bill
PAllen said:
I wonder if you have to add some statement about complete coverage by the congruence.
That would be one way of dealing with the issue you raise, yes. I think the simplest way would be to specify that every worldline of the congruence must intersect every Cauchy surface of the spacetime.

PAllen said:
Or, maybe you can actually say the region covered by the Milne congruence, treated as it’s own manifold, is not time symmetric.
Yes, that would be another way of dealing with the issue.

The Bill

1. What is the concept of CPT symmetry in general relativity?

CPT symmetry is a fundamental principle in physics that states that the laws of physics should be invariant under the combined operations of charge conjugation (C), parity (P), and time reversal (T). In general relativity, this means that the laws of physics should be the same for particles and antiparticles, for objects in their original orientation and reflected orientation, and for events moving forward and backward in time.

2. How does CPT symmetry relate to discrete symmetries in general relativity?

CPT symmetry is a combination of three discrete symmetries - charge conjugation, parity, and time reversal. This means that if CPT symmetry is violated, it would also mean that at least one of these discrete symmetries is violated in the laws of physics. Therefore, studying CPT symmetry in general relativity can provide insights into the possible violation of discrete symmetries.

3. Is there any experimental evidence for the violation of CPT symmetry in general relativity?

So far, there is no experimental evidence for the violation of CPT symmetry in general relativity. However, there are ongoing experiments and studies that are exploring the possibility of CPT symmetry violation in different physical systems and scenarios. The results of these experiments could provide evidence for or against the violation of CPT symmetry in general relativity.

4. How do scientists study the possibility of CPT symmetry violation in general relativity?

There are various methods and approaches that scientists use to explore the possibility of CPT symmetry violation in general relativity. These include theoretical calculations, numerical simulations, and experimental measurements. Scientists also study the effects of CPT symmetry violation on different physical systems, such as particles, fields, and spacetime, to understand the potential consequences of such a violation.

5. What are the implications of CPT symmetry violation in general relativity?

If CPT symmetry is found to be violated in general relativity, it would have significant implications for our understanding of the laws of physics. It could potentially lead to the discovery of new fundamental particles or interactions, challenge our current theories and models, and open up new avenues for scientific research. However, more studies and evidence are needed before any conclusions can be drawn about the possibility of CPT symmetry violation in general relativity.

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