Maximum theoretical curvature of space time

[petrushka.googol]

If the Universe space time is curved, then what is the theoretical maximum ? If we consider an open loop system undergoing progressive curvature then this results at some point in a closed loop system.

Curvature beyond this point is actually not feasible. The topology of space time is then limited in it's manifestations. (I visualize a parabola transmuting to a circle.). However a closed loop system is like a system with zero entropy (thermodynamics) or zero displacement (mechanics). Is this realistic and if so, what does this entail ?

 

Please advise.

 

P.S. I trust you could call this an open-ended question in more ways than one.....

[Strange]

The space-time curvature becomes infinite at the singularity of a black hole. Although whether this can happen in reality or not is an open question.

[ajb]

As strange says, (mod maths and exactly what we mean here) that the classical limit of curvature of a space-time is infinite.

 

However, it is generally believed that quantum effects of gravity would regulate this. But without a full theory we cannot say what the maximum curvature could be. Also we would need to define curvature carefully.

[petrushka.googol]

As strange says, (mod maths and exactly what we mean here) that the classical limit of curvature of a space-time is infinite.

 

However, it is generally believed that quantum effects of gravity would regulate this. But without a full theory we cannot say what the maximum curvature could be. Also we would need to define curvature carefully.

 

 

The space-time curvature becomes infinite at the singularity of a black hole. Although whether this can happen in reality or not is an open question.

 

In the light of the above :

Would a singularity be

 

- dimensionless

- infinite number of dimensions curled up into the infinitesimal (much like the 7 extra dimensions in string theory)

 

Please advise.

[ajb]

In the light of the above :

Would a singularity be

 

- dimensionless

- infinite number of dimensions curled up into the infinitesimal (much like the 7 extra dimensions in string theory)

 

Please advise.

A singularity is a point or a region in a space-time where the space-time fails to be a smooth Riemannian manifold. The usual thing to do is to cut these regions out when doing the mathematics. Singularities are then loosely seen as when paths 'fall off' the space-time.

 

This is classical. Changing space-time to a quantum version should get rid of the regions where the curvature heads towards infinity. Exactly how, no-one is sure. Moreover, we would no longer be dealing with true smooth manifolds, only objects that approach smooth manifolds in the 'large volume' limit, what ever that means exactly.

Edited March 6, 2015 by ajb

[studiot]

 

 

ajb

Also we would need to define curvature carefully.

 

I am not familiar with the differential geometry of more than 3 dimensions so perhaps you would care to comment on the following thought.

 

I agree about the definition of curvature, as in 3D there are two separate (independent) curvatures.

Spacetime is, of course, a minimum of 4D.

[ajb]

I am not familiar with the differential geometry of more than 3 dimensions so perhaps you would care to comment on the following thought.

 

I agree about the definition of curvature, as in 3D there are two separate (independent) curvatures.

Spacetime is, of course, a minimum of 4D.

The usual thing is to consider the components of various curvature tensors associated with the Levi-Civita connection (or some other connection on the tangent bundle); Riemann curvature tensor and the Ricci curvature.

There are other tensors you can consider, all of which classify to different extents what you mean by curvature; i.e. how much the metric is not locally the same as the Euclidean one.

 

The Riemann tensor tells you something about the lack of commutativity of the covariant derivative.

 

The Ricci tensor measures the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

 

Now, both of these depend on the coordinates used, they are tensors and so have nice transformation properties but still. When one talks about the curvature being infinite, one is usually thinking in terms of a scalar invariant of the curvature. In GR this is often the Kretschmann scalar which is a full contraction of the Riemann tensor. This gives you a true coordinate invariant measure of the curvature. By 'infinite curvature' it is this scalar that people are usually talking about.

 

I don't know what petrushka.googol had in mind by 'curvature' but I was thinking of the Kretschmann scalar.

 

EDIT: I should say that the Kretschmann scalar does not fully encode the presence of singularites, but it does for a large class of space-times. Very loosely, as long as we don't have gravitational radiation to worry about it is okay. Without some further details of the kinds of space-times we are considering here, it maybe difficult to fully understand 'infinite curvature = singularity'. But roughly this is okay

Edited March 6, 2015 by ajb

[studiot]

Thank you , ajb, I will think about this.

[petrushka.googol]

The usual thing is to consider the components of various curvature tensors associated with the Levi-Civita connection (or some other connection on the tangent bundle); Riemann curvature tensor and the Ricci curvature.

There are other tensors you can consider, all of which classify to different extents what you mean by curvature; i.e. how much the metric is not locally the same as the Euclidean one.

 

The Riemann tensor tells you something about the lack of commutativity of the covariant derivative.

 

The Ricci tensor measures the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

 

Now, both of these depend on the coordinates used, they are tensors and so have nice transformation properties but still. When one talks about the curvature being infinite, one is usually thinking in terms of a scalar invariant of the curvature. In GR this is often the Kretschmann scalar which is a full contraction of the Riemann tensor. This gives you a true coordinate invariant measure of the curvature. By 'infinite curvature' it is this scalar that people are usually talking about.

 

I don't know what petrushka.googol had in mind by 'curvature' but I was thinking of the Kretschmann scalar.

 

EDIT: I should say that the Kretschmann scalar does not fully encode the presence of singularites, but it does for a large class of space-times. Very loosely, as long as we don't have gravitational radiation to worry about it is okay. Without some further details of the kinds of space-times we are considering here, it maybe difficult to fully understand 'infinite curvature = singularity'. But roughly this is okay

 

Does this not go against the basic tenet that a singularity is indeterminate ?

[studiot]

 

Does this not go against the basic tenet that a singularity is indeterminate ?

 

 

That depends upon what you understand a singularity to be and also what you intend to do with it.

[petrushka.googol]

 

 

That depends upon what you understand a singularity to be and also what you intend to do with it.

 

Could you please elaborate....

[MWresearch]

So kappa can go from 0 to infinity, but what about like...the angle? I feel like when there's curvature, somehow the angle matters because when I look at the Lorentz transformation the rotation doesn't happen past a certain 45 degree line.

[ajb]

Does this not go against the basic tenet that a singularity is indeterminate ?

In what way?

 

You need some way of describing what you mean by a singularity. The method using the Kretschmann scalar works for some classes of space-times, for example the Schwarzschild solution. Using this scalar tells you you have a singularity at r=0 and not at the event horizon.

 

You cannot really use this scalar as a definition of a singularity. The universally accepted definition is lack of geodesic completeness (being non-technical here), basically your geodesics 'fall off' the space-time in some finite 'time' and this signifies a singularity.

[petrushka.googol]

In what way?

 

You need some way of describing what you mean by a singularity. The method using the Kretschmann scalar works for some classes of space-times, for example the Schwarzschild solution. Using this scalar tells you you have a singularity at r=0 and not at the event horizon.

 

You cannot really use this scalar as a definition of a singularity. The universally accepted definition is lack of geodesic completeness (being non-technical here), basically your geodesics 'fall off' the space-time in some finite 'time' and this signifies a singularity.

 

Correct me if I am wrong, Could we visualize this as a vortex in the gravitational field analogous to the vortices in the magnetic field ?

[ajb]

Correct me if I am wrong, Could we visualize this as a vortex in the gravitational field analogous to the vortices in the magnetic field ?

I don't know, you would need to be more specific with this analogy. I am not aware of any works really linking black holes to vortices; other than analogies with turbulence 2d systems.

[Brad Watson_Miami FL]

petrushka.googol,

Spacetime is written as one word. Einstein was the first to introduce time as the 4th dimension and inseparably connected it to space. Therefore, he wrote the term as "spacetime". It's symbolically important!

[petrushka.googol]

petrushka.googol,

 

Spacetime is written as one word. Einstein was the first to introduce time as the 4th dimension and inseparably connected it to space. Therefore, he wrote the term as "spacetime". It's symbolically important!

Forgive my semantics...

[Strange]

I would hyphenate it: space-time. But it's not a big deal: space time, space-time, spacetime ... all the same thing.

[kyle465]

Wouldn't it make more sense imagining a black hole's mass gravity simply tearing spacetime and basically acting as a portal to a higher dimension? When you look at gravity displayed virtually, it's usually as a bend in spacetime, and spacetime is represented as a 2d plane. So naturally, a dip in that plane would require another dimension. Couldn't the mass gravity of a black hole simply curve spacetime so much that it tears the fabric of our dimensions and opens up a way into a higher dimension? I find the idea of an infinite curvature to be harder to imagine. Also, wouldn't a singularity put a limit on how much matter a black hole could consume? If the hole converges to a point, where is all the matter going that it is consuming?

Edited May 28, 2015 by kyle465

[Strange]

When you look at gravity displayed virtually, it's usually as a bend in spacetime, and spacetime is represented as a 2d plane. So naturally, a dip in that plane would require another dimension.

 

The 4D curvature of space-time is "intrinsic" curvature; which means that it does not need to exist in a higher dimension. (This is a hard concept to get your head round, and I haven't seen a good non-mathematical analogy.)

 

Also, wouldn't a singularity put a limit on how much matter a black hole could consume? If the hole converges to a point, where is all the matter going that it is consuming?

 

If the singularity exists then the density is infinite because the mass is compressed to zero volume. However, most people don;t think the singularity is "real"; it is just a mathematical consequence of the fact our theories are incomplete - for example, not taking quantum effects into account.