B Black hole at the beginning of time

[Physics Slayer]

TL;DR Summary

If all the matter was condensed to a single point at the beginning of the universe (at the big bang) why wasn't there a formation of a black hole.

If all the matter was condensed to a single point at the beginning of the universe, then why didn't it all collapse into a black hole? I have heard speculation that the laws of physics change with time, is this the reason why there was no black hole at the beginning or is the reason more technical and complicated?

 

[PeroK]

This question gets asked a lot. E.g.

https://www.physicsforums.com/threads/shouldnt-the-big-bang-be-physically-impossible.996573/

 

[Melbourne Guy]

This question gets asked a lot. E.g.

I'm not surprised, @PeroK, it seems a reasonable naive question! And as a naive answer, @Physics Slayer, the big 'bang' occurred everywhere in space, it was not actually from a single point like the explosive name implies. And energy was uniformly distributed so did not create a gravitational gradient such that a black hole could form. Honestly, the big bang is a poor description of the (very) early universe.

 

[PeroK]

I'm not surprised, @PeroK, it seems a reasonable naive question! And as a naive answer, @Physics Slayer, the big 'bang' occurred everywhere in space, it was not actually from a single point like the explosive name implies. And energy was uniformly distributed so did not create a gravitational gradient such that a black hole could form. Honestly, the big bang is a poor description of the (very) early universe.

A better answer is given by @Ibix in the above thread:

A black hole is a region in space surrounded by vacuum, and there's a sense of "down" towards the centre. The universe at large was always filled more or less isotropically with stuff (matter, radiation, etc), so there can be no down, nor up. In other words, it looked nothing like a black hole.

 

[Ibix]

If all the matter was condensed to a single point at the beginning of the universe, then why didn't it all collapse into a black hole?

It wasn't a single point, and it didn't collapse because it wasn't a more-or-less static region of matter surrounded by vacuum, it was a universe full of expanding matter. You need some very dense region surrounded by a much much less dense region to give a sense of a center for matter to collapse towards, and a nearly uniform matter distribution filling the whole of space doesn't meet that criterion.

 

[Ibix]

Heh - I see I nearly repeated myself.

 

[Ibix]

Honestly, the big bang is a poor description of the (very) early universe.

That's not true. Popular descriptions of the Big Bang are frequently wrong, though.

 

[PeroK]

That's not true. Popular descriptions of the Big Bang are frequently wrong, though.

Perhaps Ubiquitous Bang is better than Big Bang!

 

[Ibix]

Perhaps Ubiquitous Bang is better than Big Bang!

Wasn't "Big Bang" coined by Hoyle, a steady state proponent, as a dismissive nickname for a theory he didn't like? (Edit: yes.) Perhaps a better name is needed. Don't see it happening, mind you - too much inertia behind current terminology.

 

[PeroK]

Wasn't "Big Bang" coined by Hoyle, a steady state proponent, as a dismissive nickname for a theory he didn't like? (Edit: yes.) Perhaps a better name is needed. Don't see it happening, mind you - too much inertia behind current terminology.

Perhaps even better is to coin the term U-big-uitous Bang!

 

[Melbourne Guy]

That's not true. Popular descriptions of the Big Bang are frequently wrong, though.

I agree with one of these statements But I agree the term 'big bang' won't be going away for some time, if ever.

 

[PeterDonis]

I agree with one of these statements

This implies that you disagree with the other. But both of them are true. Which one do you disagree with, and why?

 

[Ibix]

I agree with one of these statements

It would have been helpful to say which one.

Nevertheless, there may be some miscommunication going on here. My initial reading of your post #3 was that you were saying that Big Bang theory is not an accurate description of the early universe. That is not correct - it's a very well tested theory, and we have none more accurate.

If, on the other hand, you were making the same point that PeroK made in #8, that "Big Bang" may not be the best name for the theory in light of popsci tendencies to wildly misinterpret it as a pointlike explosion in space, then I have some sympathy, as in #9.

 

[cianfa72]

A topic puzzling me is about the meaning of the time 'U-big-uitous Bang' (see #10) occurred in the past history of our Universe. From GR we know that time makes sense only along the worldline followed by an observer. So which is let me say the 'underlying observer' wrt. define the history of the Universe ?

 

[PeroK]

A topic puzzling me is about the meaning of the time 'U-big-uitous Bang' occurred in the past history of our Universe. From GR we know that time makes sense only along the worldline followed by an observer. So which is let me say the 'underlying observer' wrt. define the history of the Universe ?

It's generally time in comoving coordinates.

 

[cianfa72]

It's generally time in comoving coordinates.

So we're actually assuming an FWR model, I think.

 

[PeroK]

So we're actually assuming an FWR model, I think.

That's the standard Big Bang model/theory.

 

[PAllen]

So we're actually assuming an FWR model, I think.

Which basically amounts to the physical assumption of isotropy everywhere, on average. In principle, one could posit a GR consistent model in which deviations from isotropy increase going back in time. In such a case, there would be no 'preferred cosmological time', but it would remain true for any timelike world line, that local total mass/energy density increases as you go back in time along that world line. In fact, this is true for FRW models - an arbitrary time like world line (not just a comoving one) will measure ever increasing local total energy density as you follow it back in proper time.

 

[cianfa72]

In such a case, there would be no 'preferred cosmological time'

Do you mean the time defined by the family of spacelike hypersurfaces orthogonal to the congruence of comoving observers in case of FRW models ?

 

[PAllen]

Do you mean the time defined by the family of spacelike hypersurfaces orthogonal to the congruence of comoving observers in case of FRW models ?

Yes.

 

[cianfa72]

We had a thread some time ago, however I'm not sure if the proper time along each comoving observer worldline in the congruence between any two given spacelike hypersurfaces othogonal to the congruence is the same or not.

 

[PAllen]

We had a thread some time ago, however I'm not sure if the proper time along each comoving observer worldline in the congruence between any two given spacelike hypersurfaces othogonal to the congruence is the same or not.

It is the same. That is what defines a synchronous coordinate system (assuming the congruence is geodesic, which comoving world lines certainly are).

 

[cianfa72]

It is the same. That is what defines a synchronous coordinate system (assuming the congruence is geodesic, which comoving world lines certainly are).

ok however, if I remember correctly, those (geodesic) timelike worldlines in the FRW comoving congruence are not integral curves of a timelike Killing vector field (KVF), though.

 

[Ibix]

those (geodesic) timelike worldlines in the FRW comoving congruence are not integral curves of a timelike Killing vector field (KVF), though.

There isn't a timelike KVF in FLRW spacetime, so yes, you are correct.

 

[Melbourne Guy]

If, on the other hand, you were making the same point that PeroK made in #8, that "Big Bang" may not be the best name for the theory in light of popsci tendencies to wildly misinterpret it as a pointlike explosion in space, then I have some sympathy, as in #9.

That's often the trouble with 'at a distance' communication, @Ibix (and @PeterDonis), what's obvious to me may not be obvious to you. Plus, humour doesn't always translate, sorry about that!

Yes, it's the popsci term, "big bang".

 

[cianfa72]

There isn't a timelike KVF in FLRW spacetime, so yes, you are correct.

Hence along the worldline of each comoving observer in the congruence the local geometry on spacelike hypersurfaces around the intersection point does not stay the same: it changes along the worldline.

 

[Ibix]

Hence along the worldline of each comoving observer in the congruence the local geometry on spacelike hypersurfaces around the intersection point does not stay the same: it changes along the worldline.

Yes - the universe expands. But at any chosen time it changes in the same way for all this family of observers. All see the same $a$, $\dot{a}$ etc.

 

[cianfa72]

It is clear to me what it means to have a given geometry locally around the intersection point of a comoving observer worldline with a spacelike hypersurface. However what does this mean from a global point of view ? We said in FLRW models each comoving observer sees the Universe homogeneous and isotropic. Now the 'space' for each a such observer at a given cosmological time (i.e. its proper time) is defined as the set of points (events) belonging to the 'intersecting' spacelike hypersurface.

Do the two properties above (i.e. homogeneous and isotropic) apply to each of those 'space' slices ?

 

[Ibix]

Do the two properties above (i.e. homogeneous and isotropic) apply to each of those 'space' slices ?

Yes.

The starting point for deriving the FLRW solution is the notion that a surface of constant coordinate time (i.e. "space at one time") is homogeneous and isotropic. When you feed this through the maths you find that a surface of constant coordinate time corresponds is the set of events of equal proper time since $a=0$ for all observers who are stationary in these coordinates. So "space", as used in cosmology, means a spacelike surface which is isotropic and homogeneous, and observers who see the universe as isotropic and homogeneous will all, on this surface, have experienced the same time since the singularity (or whatever is actually back there).

 

[PeterDonis]

The starting point for deriving the FLRW solution is the notion that a surface of constant coordinate time (i.e. "space at one time") is homogeneous and isotropic.

Actually, one can state the assumption in a way that does not depend on any choice of coordinates. The assumption, stated in an invariant way, is that the spacetime can be foliated by spacelike 3-surfaces that are all homogeneous and isotropic. This of course implies that one can choose a coordinate chart in which each such 3-surface is a surface of constant coordinate time.

I believe that the existence of a family of timelike worldlines that is orthogonal to the above spacelike 3-surfaces can also be derived from the above assumption. This then implies that the metric in above coordinate chart has the general FRW form.

 

[PAllen]

Actually, one can state the assumption in a way that does not depend on any choice of coordinates. The assumption, stated in an invariant way, is that the spacetime can be foliated by spacelike 3-surfaces that are all homogeneous and isotropic. This of course implies that one can choose a coordinate chart in which each such 3-surface is a surface of constant coordinate time.

I believe that the existence of a family of timelike worldlines that is orthogonal to the above spacelike 3-surfaces can also be derived from the above assumption. This then implies that the metric in above coordinate chart has the general FRW form.

The most common derivations go about this in different order (which does not imply the above is incorrect). You first note or derive that any metric can be put in synchronous form (for some finite region). Then, argue that isotropy everywhere (which implied homogeneity) puts the space-space metric components in one of 3 forms, each diagonal only (for hyperbolic, flat, or positive curvature spatial slices). Then, argue that the coordinate coverage is global (also due to isotropy/homogeneity). This is basically how Weinberg goes about it, for example.

However, you are trying to get away from coordinates, as much as possible. Since the only assumption above that is unique to FRW is the isotropy/homogeneity, it suggests it should be possible to turn the arguments around as you suggest; I've just never seen it done 'rigorously' in that order.

 

[PeterDonis]

You first note or derive that any metric can be put in synchronous form (for some finite region).

This is equivalent to saying that there is a timelike congruence that is hypersurface orthogonal, correct? Then you can argue that that must be true for the entire spacetime, not just some finite region, because, as you say, that is implied by isotropy everywhere.

it suggests it should be possible to turn the arguments around as you suggest; I've just never seen it done 'rigorously' in that order.

My understanding has been that that is how the argument was originally framed by Friedmann et al., although I have rephrased it in more modern terminology.

 

[PAllen]

This is equivalent to saying that there is a timelike congruence that is hypersurface orthogonal, correct?

Yes. Actually, a timelike geodesic congruence that is hypersurface orthogonal.

Then you can argue that that must be true for the entire spacetime, not just some finite region, because, as you say, that is implied by isotropy everywhere.

Agree.

My understanding has been that that is how the argument was originally framed by Friedmann et al., although I have rephrased it in more modern terminology.

I didn't know that, interesting.

 

[cianfa72]

Then, argue that isotropy everywhere (which implied homogeneity)

So the point is that from isotropy everywhere follows homogeneity everywhere. Is the reverse also true ?

puts the space-space metric components in one of 3 forms, each diagonal only (for hyperbolic, flat, or positive curvature spatial slices).

As space-space metric components I believe you mean the metric tensor components $g_{\mu \nu}$ with both spatial indices. Are these 3 diagonal forms of spatial metric basically the 3 possibile homogeneous and isotropic FLRW models ?

 

[PAllen]

So the point is that from isotropy everywhere follows homogeneity everywhere. Is the reverse also true ?

I meant to type implies. Isotropy everywhere implies homogeneity. The converse is false. Consider the simple example of a two cylinder - no point or small region is distinguishable from any other, so you have homogeneity. However, it is nowhere isotropic, as there is everywhere a distinguishable direction.

As space-space metric components I believe you mean the metric tensor components $g_{\mu \nu}$ with both spatial indices. Are these 3 diagonal forms of spatial metric basically the 3 possibile homogeneous and isotropic FLRW models ?

Yes.

 

[PeterDonis]

Is the reverse also true ?

No. One could have a preferred direction (i.e., lack of isotropy) in a homogeneous spacetime, as long as the preferred direction was the same everywhere.

 

[cianfa72]

The converse is false. Consider the simple example of a two cylinder - no point or small region is distinguishable from any other, so you have homogeneity. However, it is nowhere isotropic, as there is everywhere a distinguishable direction.

In this example the 'preferred/distinguishable' direction is that in which the curvature is null, I believe.

 

[PeroK]

In this example the 'preferred/distinguishable' direction is that in which the curvature is null, I believe.

A simple example is the idealised uniform gravitational field. You have a homogeneous gravitational field (the same everyone), but a common direction of gravitational force.

A geometric example is the surface of an infinite cylinder. Every point is the same, but there's always a circumferencial direction and a longitudinal direction. So, homogeneous but not isotropic.

 

[cianfa72]

A geometric example is the surface of an infinite cylinder. Every point is the same, but there's always a circumferential direction and a longitudinal direction. So, homogeneous but not isotropic.

ok, so the condition of isotropy everywhere (that implies homogeneity everywhere) does mean the spatial curvature everywhere on each spacelike hypersurface point has to be constant. Hence the 3 allowed geometries with constant positive, null or negative curvature, namely spherical, flat or hyperbolic spatial geometry in each spatial slices.

 

[PeroK]

ok, so the condition of isotropy everywhere (that implies homogeneity everywhere)

A non-rigorous argument of the contrapositive (that inhomogenity implies anisotropy) is:

If points $A$ and $B$ are different, then we have anisotropy for a point midway between $A$ and $B$. That should at least help remember which way round the implication goes!

 

[PeterDonis]

In this example the 'preferred/distinguishable' direction is that in which the curvature is null, I believe.

No. Curvature is zero everywhere on a cylinder, and curvature doesn't have a "direction".

The "preferred" direction on the cylinder is the one in which the geodesics are closed circles.

 

[cianfa72]

No. Curvature is zero everywhere on a cylinder, and curvature doesn't have a "direction".

Ah yes, the 'extrinsic' curvature of a cylinder does not matter.

 

[cianfa72]

The "preferred" direction on the cylinder is the one in which the geodesics are closed circles.

So the 'preferred' direction on the cylinder is actually matter of topology.

 

[PeterDonis]

So the 'preferred' direction on the cylinder is actually matter of topology.

No. Geodesics are not a matter of topology; you need to have a metric to know which curves are geodesics.

 

[cianfa72]

No. Geodesics are not a matter of topology; you need to have a metric to know which curves are geodesics.

Sure, however on the cylinder the metric is flat then from a metric point of view all geodesics are indistinguibile. It is the topology that picks a preferred direction.

 

[PeterDonis]

on the cylinder the metric is flat then from a metric point of view all geodesics are indistinguibile.

No, they aren't. Geodesics go in different directions and can be distinguished that way. In fact, without a metric you don't even have a well-defined concept of "direction" to begin with (since you need a metric for angles as well as lengths), so it's not clear that you can define what isotropy means.

It is the topology that picks a preferred direction.

The topology certainly is one factor, since it is what makes one particular set of geodesics closed curves. But you also need the metric to know that they are geodesics (as well as to define the concept of "direction", as above).

 

[cianfa72]

So we can say both (metric and topology) are actually needed in order to define isotropy.

 

[cianfa72]

The assumption, stated in an invariant way, is that the spacetime can be foliated by spacelike 3-surfaces that are all homogeneous and isotropic. This of course implies that one can choose a coordinate chart in which each such 3-surface is a surface of constant coordinate time.

ok, it is simple as defining a function of spacetime (i.e. the coordinate time function $t$) such that its level sets are the given spacelike hypersurfaces (3-surfaces) foliating the spacetime.

I believe that the existence of a family of timelike worldlines that is orthogonal to the above spacelike 3-surfaces can also be derived from the above assumption. This then implies that the metric in above coordinate chart

So, as far as I can understand, the above chart is the one that assigns constant spatial coordinates to each worldline in the family of timelike worldlines orthogonal to the given family of spacelike hypersurfaces and as timelike coordinate the coordinate time function $t$.

 

[PeterDonis]

it is simple as defining a function of spacetime (i.e. the coordinate time function ) such that its level sets are the given spacelike hypersurfaces (3-surfaces) foliating the spacetime.

Yes.

the above chart is the one that assigns constant spatial coordinates to each worldline in the family of timelike worldlines orthogonal to the given family of spacelike hypersurfaces and as timelike coordinate the coordinate time function $t$.

A chart whose coordinate time $t$ behaves as above does not have to assign constant spatial coordinates to each worldline that is orthogonal to the spacelike hypersurfaces of constant $t$. There are an infinite number of possible charts with the same coordinate time $t$. But there is a unique such chart that does assign constant spatial coordinates to each worldline that is orthogonal to the spacelike hypersurfaces of constant $t$; that is the standard FRW chart used in cosmology.

 

[cianfa72]

There are an infinite number of possible charts with the same coordinate time $t$. But there is a unique such chart that does assign constant spatial coordinates to each worldline that is orthogonal to the spacelike hypersurfaces of constant $t$; that is the standard FRW chart used in cosmology.

Actually, strictly speaking, I believe there is a complete family of such charts since we can just continuously remap the values of spatial coordinates assigned to each of the worldlines orthogonal to the spacelike hypersurfaces of constant $t$ by adding the same constant to the old values assigned to them.