Large diffeomorphisms in general relativity

In summary, the discussion focused on "large diffeomorphisms" in general relativity, specifically in the context of a 2-torus. These large diffeomorphisms, known as "Dehn twists," involve cutting the torus, rotating one of the generated circles by an angle theta, and reattaching the two circles. There are two cases of interest - one where theta is arbitrary and the other where it is a multiple of 360°. In the former case, the Dehn twist is not a diffeomorphism, while in the latter case it is. This raises questions about the role of these large diffeomorphisms in general relativity and their relation to topology and topology change. The discussion
  • #1
tom.stoer
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"Large" diffeomorphisms in general relativity

We had a discussion regarding "large diffeomorphisms" in a different thread but it think we should ask this question here.

For a 2-torus there are the so-called "Dehn twists"; a Dehn twist is generated via cutting the 2-torus, rotating one of the two generated circles by some angle theta and gluing the two circles together again.

Two things are interesting:

A) using an arbitrary angle theta this is not a homeomorphism (and therefore not a diffeomorphisms either) as neigboured points are not mapped to neighboured points. Nevertheless the torus is mapped to a torus.

B) using an angle theta which is amultiple of 360° this is a diffeomorphism, but it seems that it should be called a "large" diffeomorphism as the two ccordinate systems are not transformed into each other via l"local" deformations.

Now in GR we expect everything to be invariant regarding diffeomorphisms. The Dehn twist is a rather simple example but one can easily construct similar transformations in higher dimensional spaces.

Questions:

In the case A) the twist is not a diffeomorphism, therefore we need not expect invariance; but can this case A) be generated via dynamics in GR? Or are there "diffeomorphic superselection sectors"?

In the case B) we have a diffeomorphism, but nevertheless it seems that there is a discrete structure regarding the different N*360° rotations labelling "different" (but diffeomorphic) tori. Again: are such "different" but diffeomorphic manifolds of any relevance.

General question: is there some topology of the diffeomorphism group in n dimensions which is related to these Dehn twists and other "lagre diffeomorphisms" in higher dimensions?

Thanks
Tom
 
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  • #2


Tom, could you provide a link to the thread where the notion of large diffeomorphisms was developed?

Suppose I take the unit square in the (x,y) plane and glue opposite edges together so that it makes a torus. Then anything of the form (x,y)->(f(x,y),y) is a diffeomorphism if f is smooth, 1-1, and onto.

It seems to me that you are arbitrarily singling out some specific y0 and talking about the properties of f at y0, but why is that interesting?

In other words, when you describe a diffeomorphism in terms of gluing, you're implicitly assuming that in areas away from the cut-and-glue line, we distort the torus in some smooth way -- but if I only provided you with information on the distortions (the diffeomorphism itself), I don't see how you would even know where the cut-and-glue line was.

-Ben
 
  • #3


We didn't develop that anywhere else. I am discussing this just here. Perhaps this will help: http://en.wikipedia.org/wiki/Dehn_twist

Looking at the torus in the (x,y) plane I take one y0 where I (Dehn :-) define the cut and the twist. But the y0 is arbitrary and after the gluing it is no longer visible. Any other y1 would do as well, it wouldn't affect the result of the gluing, neither the torus nor the curves on the torus.

And please keep in mind that this is just one simple example in low dimensions. I think you can do something like that for arbitrary n-tori and possibly other compact manifolds.
 
  • #4


But at the cut and glue line, you are making points into neighbors that weren't before? I don't really know the rules of diffeomorphisms versus coordinate transforms, but is topology change really a diffeomorphism (e.g. Ben's example is a topology change)? Certainly, changing how you glue coordinate patches together produces different manifolds. Is diffeomorphism meant to include substantively different manifolds?
 
  • #5


PAllen said:
But at the cut and glue line, you are making points into neighbors that weren't before?
...
changing how you glue coordinate patches together produces different manifolds.
It depends; as I said in case A) where I allow for an arbitrary angle you are right, but in case B) where the angle of the twist is restricted to 360°*n the transformation is topology-preserving and is definately a diffeomorphism. Le's focus on that case B) Can we say something regarding this "large" diffeomorphism?
 
  • #7


tom.stoer said:
It depends; as I said in case A) where I allow for an arbitrary angle you are right, but in case B) where the angle of the twist is restricted to 360°*n the transformation is topology-preserving and is definately a diffeomorphism. Le's focus on that case B) Can we say something regarding this "large" diffeomorphism?

In this case, it occurs to me that the 'large' aspect is an artifact. The same result could be achieved with no cut and glue. Just, a smooth, in place twist will have an identical result. Hypothesis (no proof): if we ban topology change and non-smooth transform, then every 'large' diffeormorphism (that qualifies as a diffeomorphism) can be recast as an ordinary diffeomorphism.
 
  • #8


PAllen said:
In this case, it occurs to me that the 'large' aspect is an artifact. The same result could be achieved with no cut and glue.
If you look at a drawing of a torus and a closed curve C with winding numbers (m,n) = (1,0) according to the fundamental group Z² of the torus T², then you see that a Dehn twist with 360° changes this to (m,n') = (m,n+1) = (1,1), right?

Therefore this is not an artifact.
 
  • #9


tom.stoer said:
If you look at a drawing of a torus and a closed curve C with winding numbers (m,n) = (1,0) according to the fundamental group Z² of the torus T², then you see that a Dehn twist with 360° changes this to (m,n') = (m,n+1) = (1,1), right?

Therefore this is not an artifact.

Ok, I see, very interesting. No matter how I define a smooth twisting without cut and re-attach, it won't change the winding number. On the other hand, a smooth point mapping function can achieve this. So the idea is to distinguish this type of diffeomorphism.
 
  • #10


PAllen said:
So the idea is to distinguish this type of diffeomorphism.
Yes, exactly. And not only that - the idea is to understand the physical meaning :-)

Origininally Dehn twists were studied in string theory (world sheet transformations); so my idea was that if these large diffs. exist in dim=2 they may also exist in dim=n (dim=4 especially) and they may play a role for the large scale structure / topology of spacetime. The T² case was only a warm-up.
 
  • #11


Thanks for the further information, Tom.

Something similar to this happens with discrete symmetries. For example, a closed FRW solution has a discrete symmetry under time-reversal. It seems to me that discrete symmetries don't integrate as cleanly into the structure of GR as they do into the structure of a subject like Newtonian mechanics or QFT, because there is not even a general way to define them. For example, a manifold describing a spacetime in GR may not even be time-orientable, so there may be no way to define a time-reversal operator.

This whole topic of how to physically interpret general covariance has a muddy and inconclusive history. There are arguments that general covariance is trivial. There are arguments that it's not. There are arguments that it's not the interesting notion to study, and the interesting notion is really something more like background independence.
 
  • #12


bcrowell said:
This whole topic of how to physically interpret general covariance has a muddy and inconclusive history. ... There are arguments that it's not the interesting notion to study, and the interesting notion is really something more like background independence.
For "small" diffeomorphisms it's rather clear (as long as you do try to quantize them :-) they are afaik local coordinate transformations. For "large" diffeomorphisms it becomes more complicated as one mixes the topology of the manifold with the topology of the diffeomorphism group. I studied similar aspects of quantum gauge theories where the topological structure of the gauge group (or bundle) plays a central role. There are rather well-known effects related to this topology (Aharonov-Bohm, instantons, ..., Gribov copies, ...)

But I haven't found similar studies (topological structure of the diffeomorphism group in n dimensions), neither in GR nor in QG. String theorists have studied the 2-dim. diff. invariance of the string world sheet, but this is very different from the target space (in addition the diff. group in 2 dim. is a very special case).

So basically it boils down to a better undersanding of the topological structure of the diffeomorphism group in n dimensions. Any further ideas?
 
  • #13


Which metrics can one twist? Just guessing, Minkowski seems ok, but how about if the manifold is geodesically incomplete like in the Schwarzschild or FRW solutions?
 
  • #14


bcrowell said:
This whole topic of how to physically interpret general covariance has a muddy and inconclusive history. There are arguments that general covariance is trivial. There are arguments that it's not. There are arguments that it's not the interesting notion to study, and the interesting notion is really something more like background independence.

Yes, there has been some confusion from the beguinning, starting with Einstein that mistakenly thought that General covariance was related to the General principle of relativity, but as experts in the field (like Michel Janssen or Norton,etc) have shown, the physical meaning of General covariane is actually the Equivalence principle, it is just the mathematical way to implement it. Interpreted this way is certainly not trivial.
There has been further confuson in posterior years because people with ulterior motives have tried to interpret general covariance as absolute freedom to change coordinates.
 
  • #15


Tom, connecting with your question about large diffeomorphism's physical meaning in GR, we could say that general covariance in the last sense I mentioned allow us to make all kinds of those large diffeomorphisms, gluing, twisting and patching without any concern for the topological structure implications of those changes that are certainly topologically non-trivial.
 
  • #16


Let's make an example. Assume R*T³ solves Einstein equations in vacuum (it does not, but that doesn't matter here). Assume we have a closed geodesic curve of a test object with winding numbers (1,0); this should be OK as T³ is flat and therefore a straight line with (0,1) should work.

No let's do the cut-twist-glue procedure. What we get back is a different closed curve with winding number (1,1). Questions:
a) does this generate a new, physically different spacetime?
b) does this generate a different path of a test object on the same spacetime?
c) did I miss something, e.g. did I miss to check whether this new curve can still be a geodesic?
 
  • #17


tom.stoer said:
Let's make an example. Assume R*T³ solves Einstein equations in vacuum (it does not, but that doesn't matter here). Assume we have a closed geodesic curve of a test object with winding numbers (1,0); this should be OK as T³ is flat and therefore a straight line with (0,1) should work.

No let's do the cut-twist-glue procedure. What we get back is a different closed curve with winding number (1,1). Questions:
a) does this generate a new, physically different spacetime?
b) does this generate a different path of a test object on the same spacetime?
c) did I miss something, e.g. did I miss to check whether this new curve can still be a geodesic?

I'll throw out a few thoughts, but based back on the torus example, which I can picture better.

1) Modeling the physical action of cutting, twisting, mending the torus. Here, lengths, adjacency, geodesics, and obviously winding number change. One way of describing mathematical operations would be: imagine the torus is embedded in 3-space. Before, we have a set coordinates for the torus, and also the torus is described in Euclidean 3-space coordinates. Distances, etc. computed in toroidal coordinates match those of the relevant curve embedded in the 3-space. Now we do the surgery. The embedding space remains the same, with the same metric. Pre-surgery, the torus was described by some x(a,b), y(a,b), z(a,b). Post surgery, it is defined by some x(a',b'), y(a',b),z(a',b'), such that a'(a,b), b'(a,b) causes (x,y,z) points to move as specified by the surgery . However, the way we impute a metric onto the torus remains to use the unchanged Euclidean metric applied directly to x(a',b'), y(a',b),z(a',b'). As a result, distances, geodesics, etc. all change. That is, a curve that was geodesic no longer is. In effect, the metric has not been transformed as if this was a coordinate transform.

2) As above, but starting by constructing a metric for (a,b) such as to capture the toroidal geometry , while distances also come out the same as using Euclidean metric on x(a,b),y(a,b),z(a,b). Now treat the transform as coordinate transform, mapping (a,b) to (a',b'), transforming the imputed metric by standard rules. Now, I think all intrinsic geometric calculations come out the same, despite the 'large' change. In x,y,z coordinates, you see all the changes, but using (a',b') with the properly transformed metric, you don't. Note that using Euclidean calculations on x(a',b'), etc. will now *not* agree with computations using (a',b') with transformed metric.
 
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  • #18


I think 1) is irrelevant as we know that the embedding of T² in R³ does itself change the metric; T² admits a flat metric, but the embedding in R³ does not. So 1) is an artefact of the embedding.
 
  • #19


tom.stoer said:
Let's make an example. Assume R*T³ solves Einstein equations in vacuum (it does not, but that doesn't matter here). Assume we have a closed geodesic curve of a test object with winding numbers (1,0); this should be OK as T³ is flat and therefore a straight line with (0,1) should work.

No let's do the cut-twist-glue procedure. What we get back is a different closed curve with winding number (1,1). Questions:
a) does this generate a new, physically different spacetime?
b) does this generate a different path of a test object on the same spacetime?
c) did I miss something, e.g. did I miss to check whether this new curve can still be a geodesic?
Mainstream says none of the above, I'd say a)
Normally you would have b) if you have a)
 
  • #20


I tend to agree with a) But that means that GR is NOT invariant w.r.t. all diffeomorphisms but only w.r.t. "restricted" diffeomorphisms (like small and large gauge transformations, where large gauge trf's DO generate physical effects)
 
  • #21


tom.stoer said:
I tend to agree with a) But that means that GR is NOT invariant w.r.t. all diffeomorphisms but only w.r.t. "restricted" diffeomorphisms (like small and large gauge transformations, where large gauge trf's DO generate physical effects)

I think this would have to show up somehow in the process of transforming the metric, e.g. unavoidable singularities. Otherwise it is just arithmetic that two curves of some length, and orthogonal to each other, and with one intersection, preserve all those feature in new coordinates with properly transformed metric.
 
  • #22


PAllen said:
I think this would have to show up somehow in the process of transforming the metric, e.g. unavoidable singularities.
Why? It is a diffeomorphism and does not create a singularity
 
  • #23


tom.stoer said:
Why? It is a diffeomorphism and does not create a singularity

Well, then it can't change the geometry, at least as defined by anything you can compute using the metric. This really has nothing to do with GR, it is differential geometry.

My understanding is that topology of a differentiable manifold is encoded in how coordinate patches overlap. So, if we don't change this (and we don't need to for the Dehn twist), and we don't change anything computable from the metric, what can change?

In my (1) and (2) I was trying to get at the idea of making the operation 'real' so it does change geometry, versus treating as a pure coordinate transform, such that the corresponding metric transform preserves all geometric facts. I've heard the terms active versus passive difffeomorphism. I don't fully understand this, but I wonder if it is relevant to this distinction.
 
  • #24


PAllen said:
Well, then it can't change the geometry, at least as defined by anything you can compute using the metric. This really has nothing to do with GR, it is differential geometry.
Du you agree that it changes the winding number of a closed curve?
 
  • #25


tom.stoer said:
Du you agree that it changes the winding number of a closed curve?

Certainly, it is changed in my case (1) of my post #17. I'm not sure about as described in case(2) of that post. If you can compute winding number from the metric and topology as encoded in coordinate patch relationships, then it would seem mathematically impossible. If this is an example of geometrical fact independent of the metric and patch relationships, then we would need some definition how to compute it intrinsically, and it would seem to necessitate adding some additional structure to the manifold. In this case, it may well be possible, having specifically introduced non-metrical geometric properties not preserved by coordinate transforms.

Then, the physics question becomes that conventionally formulated GR would attach no meaning to this additional structure, it would become physically meaningful only in the context of an extension to GR that gave it meaning. This is what some of the classical unified field theory approaches did.
 
  • #26


PAllen said:
Certainly, it is changed in my case (1) of my post #17. I'm not sure about as described in case(2) of that post. If you can compute winding number from the metric and topology as encoded in coordinate patch relationships ...
It does even in case (2)

I found an explanation on Baez "this week's finds", week 28:

http://math.ucr.edu/home/baez/week28.html

Baez said:
Now, some diffeomorphisms are "connected to the identity" and some aren't. We say a diffeomorphism f is connected to the identity if there is a smooth 1-parameter family of diffeomorphisms starting at f and ending at the identity diffeomorphism. In other words, a diffeomorphism is connected to the identity if you can do it "gradually" without ever having to cut the surface. To really understand this you need to know some diffeomorphisms that aren't connected to the identity. Here's how to get one: start with your surface of genus g > 0, cut apart one of the handles along a circle, give one handle a 360-degree twist, and glue the handles back together! This is called a Dehn twist.

...

In other words, given any diffeomorphism of a surface, you can get it by first doing a bunch of Dehn twists and then doing a diffeomorphism connected to the identity.

So we can now concentrate on the physical role of these "large" diffeomorphisms.
 
  • #27


tom.stoer said:
It does even in case (2)

I found an explanation on Baez "this week's finds", week 28:

http://math.ucr.edu/home/baez/week28.html



So we can now concentrate on the physical role of these "large" diffeomorphisms.

This was very interesting, but I didn't find any answer to my question my key question: how is it winding number of closed curve computed / defined against the definition of a differentiable manifold?

If it can change while the manifold is considered identical, then it must be computed in a way that is not invariant. Is it some form of coordinate dependent torsion?
 
  • #28


PAllen said:
This was very interesting, but I didn't find any answer to my question my key question: how is it winding number of closed curve computed / defined against the definition of a differentiable manifold?
Yes, not a single word.

PAllen said:
If it can change while the manifold is considered identical, then it must be computed in a way that is not invariant.
I don't agree. If the Dehn twist is a global diffeomorphism (and if we agree that in 2 dimensions homeomorphic manifolds are diffeomorphic and vice versa - which does not hold in higher dimensions) then the two manifolds before and after the twist are identical - there is no way to distinguish them. Now suppose we cannot compute the winding numbers (m,n) but only their change under twists. Then this change is not a property of the manifold but of the diffeomorphism. So we don't need a way to compute the winding numbers from the manifold but a way to compute their change from the diffeomorphism (this is similar to large gauge transformations where the structure is encoded in the gauge group, not in the base manifold). I think for large diffeomorphisms there is some similar concept.
 
  • #29


tom.stoer said:
Yes, not a single word.


I don't agree. If the Dehn twist is a global diffeomorphism (and if we agree that in 2 dimensions homeomorphic manifolds are diffeomorphic and vice versa - which does not hold in higher dimensions) then the two manifolds before and after the twist are identical - there is no way to distinguish them. Now suppose we cannot compute the winding numbers (m,n) but only their change under twists. Then this change is not a property of the manifold but of the diffeomorphism. So we don't need a way to compute the winding numbers from the manifold but a way to compute their change from the diffeomorphism (this is similar to large gauge transformations where the structure is encoded in the gauge group, not in the base manifold). I think for large diffeomorphisms there is some similar concept.

Thankyou! Very interesting. Then I spout my opinion of the physics issue (assuming something like this is what is going on). Conventional GR only gives meaning to metrical quantities, so this aspect of the diffeomorphism would have no physical significance, and anything metrically defined would be preserved. And I come back to the idea that this sort of thing provides an opportunity to extend conventional GR- without changing any of its predictions, you can add new content.
 
  • #30


I found two possibly relevant papers, both focusing on 2+1 dimensions:

http://relativity.livingreviews.org/Articles/lrr-2005-1/ [Broken]

http://matwbn.icm.edu.pl/ksiazki/bcp/bcp39/bcp3928.pdf

If I am reading section 2.6 of the Carlip paper (first above) correctly, it suggests that GR is invariant under large diffeomorphisms, as I guessed above.
 
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  • #31


Thanks a lot for checking that and providing the two links. Looks very interesting.
 
  • #32


This material has always confused me, and its hard to find good references. Over the years I've asked a few specialists but it hasn't helped me much.

In 2+1 dimensions, the whole mapping class group sort of makes good intuitive sense, but then I rarely see it generalized in 4d. Further, the real subleties, at least to me, arise when the diffeomorphisms change the asymptotic structure of spacetime.

Its not clear whether bonafide 'observables', are invariant under these 'gauge' transformations (incidentally, to avoid confusion, the notion of a large gauge transformation is afaik typically done where you fix a spacetime point, fix a vielbein and treat the diffeomorphism group acting on these elements in an analogous way to intuition from gauge theory)
 
  • #33


Haelfix said:
This material has always confused me, ...

In 2+1 dimensions, the whole mapping class group sort of makes good intuitive sense, but then I rarely see it generalized in 4d. Further, the real subleties, at least to me, arise when the diffeomorphisms change the asymptotic structure of spacetime.

Its not clear whether bonafide 'observables', are invariant under these 'gauge' transformations
I agree with you. I can't remember why we started to discuss these issues in GR, but thinking about it, it became clear to me that these large diffeomorphisms are not well-understood (at least not by me).
 
  • #34


tom.stoer said:
I agree with you. I can't remember why we started to discuss these issues in GR, but thinking about it, it became clear to me that these large diffeomorphisms are not well-understood (at least not by me).

It seems clear to me that large difeomorphisms in the GR context are not well understood, and yet all the mainstream experts have decided that spacetimes are invariant under these large diffeomorphisms without offering any real reason.
And yet I'd say this is a vital point, the theoretical base of many yet unobserved physics, such as that of black holes(see the Carlip cite in section 2.6) depends on whether spacetimes are considered as invariant or not for these large diffeomorphisms.
Of course some people that are not very fond of thinking for themselves would rather just obey the conventional opinion on this and let it be like that, so your question Tom, touches a very sensitive spot.
 
  • #35


This is similar to "QCD is invariant w.r.t. SU(3) gauge transformations". As long as one studies infinitesimal ones everything is fine, but as soon as you try to study large gauge transformations on different topologies it becomes interesting (in the axial gauge "A³(x)=0" you cannot gauge away a dynamical zero mode a³=const., Gribov ambiguities, winding numbers, instantons and merons, center symmetry, ...).

There is an important question: gauge transformations arise due to unphysical degrees of freedom (in contradistinction to other global symmetries like flavor) which have to be gauge-fixed (e.g. via Dirac's constraint quantization in the canonical formulation). But it seems that in gauge theories unphysical local and physical global aspects are entangled).

I guess that soemthing similar will happen in GR as well. Many aspects in gauge theory become visible during quantization. So as long as we do not fully understand QG, some aspects may be hidden or irrelevant; example: what is the physical meaning of Kruskal coordinates? we don't care classically - but we would have to as soon as during BH evaporation the whole Kruskal spacetime has to be taken into account in a PI or whatever. If spacetime will be replaced by some discrete structure many problems may vanish, but if spacetime as a smooth manifold will survive quantization than these issues become pressing (diffeomorphisms in 4-dim. are rather complicated - see Donaldson's results etc. )
 
<h2>1. What are large diffeomorphisms in general relativity?</h2><p>Large diffeomorphisms in general relativity refer to transformations of the spacetime coordinates that significantly alter the geometry of the spacetime. These transformations are typically non-linear and can result in changes to the curvature of spacetime, which can have significant implications for our understanding of gravity.</p><h2>2. How are large diffeomorphisms different from small diffeomorphisms?</h2><p>Small diffeomorphisms are local transformations of the spacetime coordinates that do not significantly alter the geometry of the spacetime. In contrast, large diffeomorphisms are global transformations that can result in significant changes to the curvature of spacetime.</p><h2>3. What is the significance of large diffeomorphisms in general relativity?</h2><p>Large diffeomorphisms play a crucial role in understanding the full dynamics of general relativity. They allow for the exploration of non-linear effects in the theory and can lead to new insights into the nature of gravity.</p><h2>4. How do large diffeomorphisms affect the solutions of the Einstein field equations?</h2><p>Large diffeomorphisms can lead to new solutions of the Einstein field equations that were not previously considered. They can also change the behavior of known solutions, leading to a better understanding of the underlying physics.</p><h2>5. Are large diffeomorphisms observable in the real world?</h2><p>While large diffeomorphisms are a fundamental aspect of general relativity, they are not directly observable in the real world. However, their effects can be indirectly observed through their influence on the curvature of spacetime and the behavior of massive objects in the universe.</p>

1. What are large diffeomorphisms in general relativity?

Large diffeomorphisms in general relativity refer to transformations of the spacetime coordinates that significantly alter the geometry of the spacetime. These transformations are typically non-linear and can result in changes to the curvature of spacetime, which can have significant implications for our understanding of gravity.

2. How are large diffeomorphisms different from small diffeomorphisms?

Small diffeomorphisms are local transformations of the spacetime coordinates that do not significantly alter the geometry of the spacetime. In contrast, large diffeomorphisms are global transformations that can result in significant changes to the curvature of spacetime.

3. What is the significance of large diffeomorphisms in general relativity?

Large diffeomorphisms play a crucial role in understanding the full dynamics of general relativity. They allow for the exploration of non-linear effects in the theory and can lead to new insights into the nature of gravity.

4. How do large diffeomorphisms affect the solutions of the Einstein field equations?

Large diffeomorphisms can lead to new solutions of the Einstein field equations that were not previously considered. They can also change the behavior of known solutions, leading to a better understanding of the underlying physics.

5. Are large diffeomorphisms observable in the real world?

While large diffeomorphisms are a fundamental aspect of general relativity, they are not directly observable in the real world. However, their effects can be indirectly observed through their influence on the curvature of spacetime and the behavior of massive objects in the universe.

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