A 4D Black Hole?

One feature of a universe undergoing Flat Universal Granular Expansion is that the density of the universe is critical and that density is equal to the density of a Schwarzschild black hole with the radius equal to the age of the universe multiplied by the speed of light.

An obscured assumption there is of a 3D black hole, but the universe has at least one more dimension, since time and space are interchangeable under general relativity (see On Time for an example of this interchangeability).  So … the question arises, what happens with a black hole in more than 3 dimensions?

There’s some disagreement as to whether black holes are already 4D, because they exist in a 4D universe (with three dimensions of space and one of time) and they persist over time.  While most say yes, of course, I have seen the occasional person say no, of course not – basically because there is an inherent assumption that we only talk about spatial dimensions in relation to objects.  We can’t imagine a .  That’s not really the question I was pondering though.  I was thinking about how mass would be distributed in a spatially 4D black hole, working from the intuition that because there is more volume in the 3D surface of a 4D glome than in a sphere, then there would need to be more mass.

This is going about things the wrong way though.

Consider a standard Schwarzschild black hole, a mass (M_S) that is neither charged nor rotating, in a volume defined by its radius (r_S=2GM/c²).  This is the point at which the escape velocity is the speed of light, meaning that nothing, not even light can escape.

We can show this easily – to escape, a body must have more kinetic energy than its gravitational potential energy, or

[K+U(r)=½mv²-GMm/r]>0

 

So, at the limit where the kinetic energy is not quite enough,

½v²=GM/r

 

Since we are talking about where the limit is the speed of light at r_S, and rearranging we get:

r_S=2GM/c²

 

Note that we are effectively talking about vectors here, a body must have a velocity, not just a speed, in order to escape.  In other words, the body must have a velocity greater than v=c in the direction of the separation from the mass M.  Every point on the sphere defined by the radius r_S around the centre of mass M_S defines a vector from the locus of the sphere.  The sphere, however, is not really relevant to each body that is able or is not able to escape.  All that matters is the vector from the centre of the mass and the body’s own centre of mass.

The same applies to a 4D black hole and also a hypothetical 2D black hole.  The limit will always be given by r_S=2GM/c², meaning that mass does not change.  A 4D (along with a hypothetical 2D) black hole has the same mass as a 3D black hole.

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Note that if we talk in terms of natural Planck units, the equation resolves to r_S=2M, because both G and c are effectively just exchange rates between units, and it can be seen that the radius is two units of Planck length for every unit of Planck mass.  This could have been intuited from the Mathematics for Imagining a Universe, where the conclusion that “mass … is being added to the universe at the rate of half a Planck mass … per Planck time” is reached, noting that in the scenario, that universe is expanding at the speed of light.

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Some may note that there is a problem.  In On Time, I noted that the equation for kinetic energy is an approximation.  It is noted elsewhere that E_k=½mv² is a first (or second) order approximation.

As v->c, the approximation becomes less valid, as E_k=mc².(1/√(1-v²/c²)-1) tends towards infinity.  The derivation above explicitly uses v=c, so … it doesn’t work.  (Other than the fact that, obviously, it does.)

So, perhaps it’s better come at it from the other side.  From outside of the black hole, using the relativistic effects of gravitation.

Time is dilated and space is contracted (according to normal parlance) in proportion to the force of gravity.  The relevant equation is this:

t₀=t_f.√(1-(2GM/c²)/r)

 

Note that t₀ is “proper time” between two events observed from a distance of r from the centre of the mass M, and t_f is the “coordinate time” between the same two events observed at distance of r_f such that the gravitation acceleration experienced (a_f=GM/r_f²)≈0 (and assuming that no other effects are in play).  This can be made even more complicated by considering that the events must be either collocated or equidistant from both observation locations, but we could just be a little less stringent, and say that t_f is “normal time” and that t₀ is “affected time” (that is, affected by gravitation due to proximity to the mass M).

As r decreases with increasing proximity to the mass M, affected time between standard events (in the normal frame) gets shorter until, when 1-(2GM/c²)/r=0, all normal events have no separation at all.  This represents a limit as you can’t get less than no separation (and what you get instead if something were to approach more closely to the mass M is the square root of a negative number).

The value of r at this limit is, of course, r=r_S=2GM/c².  Note that while we have used the value r, we were not specifically talking about a radius, merely a separation.  Therefore the same principles are mentioned above apply.

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But, of course, that equation is predicated on “the Schwarzschild metric, which describes spacetime in the vicinity of a non-rotating massive spherically symmetric object”, so it already assumes r_S=2GM/c².

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So … we have to go deeper.  Looking at relativistic kinetic energy, we can see that it’s not quite so simple.  At relativistic speeds, kinetic energy (K, to remain consistent with above) is given by:

K=√(p²c²+m²c⁴)-mc²

 

Where p is the linear momentum, p=mγv, where γ=1/√(1-v²/c²) which approaches infinity as v->c.  Substituting and rearranging the above then, we get:

K=√((mγv)²c²+m²c⁴)-mc²=√(m²γ²v²c²+m²c⁴)-mc²

K=mc²(√(γ²v²/c²+1)-1)

But since γ=1/√(1-v²/c²), and just focussing on the term within the square root for the sake of clarity:

γ²v²/c²+1=v²/c²/(1-v²/c²)+1=v²/(c²-v²)+1

=v²/(c²-v²)+(c²-v²)/(c²-v²)=c²/(c²-v²)=1/(1-v²/c²)

And so:

K=mc²(√(1/(1-v²/c²))-1)= mc².(γ-1)=mγc²-mc²

 

Which is the same equation as shown in On Time (just the terminology is different), which puts us back to where we were before.

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So, while the intuitive solution has some procedural difficulties, we really need to go via the derivation of the Schwarzschild metric to see that r_S is the radius at which the metric is singular (in this case meaning that there is a division by zero).

It is interesting that, in this case, using an approximation that is based on the assumption that v<<c works perfectly in a situation where v=c.  Why that is the case is currently unclear to me.

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Update: Having thought about it a little more and looking at some other views on why the 2 factor is in the Schwarzschild radius, my suspicion is that the curvature of space in the region around a black hole is such that the escape velocity at any given distance from the black hole’s centre of mass (where r>=r_S) is given by v=√(2GM/r), even if that is the result you might reach simplistically (and by making incorrect assumptions about the value of kinetic energy at relativistic speeds).

Here's my thinking – the escape velocity at the Schwarzschild radius is, due to completely different factors, of a value such that the following equation is true: ½mv²=GMm/r, where m is irrelevant due to being on both sides of the equation, M is the mass of a black hole, v is the escape velocity (in this case c) and r is the distance between the body in question and the centre of the mass M (in this case the Schwarzschild radius).  At a nominal distance r very far from the centre of the mass M, the kinetic energy of a mass m at escape velocity v is indistinguishable from ½mv² and is equal in value to the gravitational potential energy GMm/r, so the equation ½mv²=GMm/r is true (noting that, again, m is irrelevant).

Since we have picked a nominal distance, we could conceivably pick another distance which is slightly closer to the centre of the black hole and have no reason to think that ½mv²=GMm/r is no longer true.  As we inch closer and closer to the black hole, there is no point at which we should expect the equation to suddenly no longer hold, particularly since we know that it holds at the very distance from the black hole at which escape is simply no longer possible because we bump up against the universal speed limit of c.

I accept that it’s not impossible that the escape velocity might gradually depart from the classical equation (becoming lower in magnitude due to the fact that we must consider relativity), only to pop back up to agreement with classical notions right at the last moment.  I don’t have a theory to account for that, and have no intention to create one, but my not having a theory for something doesn’t make it untrue.  It just seems unlikely.  To me.

Half a Problem Solved?

I have intended to write something for quite a while about something that I wrote in Mathematics to Address an Apparent Problem with Imagining a Universe has been bugging me.  That post was “a very brief explanation as to how mass/energy enters to the (…) universe at a rate of one unit of mass/energy per unit of time”.

 

The problem here is that the mathematics tells us that, to remain flat in the presence of expansion, a universe with a volume equal to the Hubble volume (where the Hubble “constant” is the inverse of the age of the universe) must accrue an additional unit of (Planck) mass/energy every two units of (Planck) time.  If the universe got one unit of mass/energy per unit of time, then it would never have done more than collapse in on itself because it would have had a density twice that of a black hole.

 

In Mathematics to Address an Apparent Problem with Imagining a Universe, I wrote:

 

We can get circles from a sphere by sectioning it.  The greatest circle we can create has the same radius as the sphere itself.  The sectioning effectively creates two hemispheres.  Note that I remain aware that the surface area of the curved section of the hemisphere is not equal to the surface area of the circle created by the section.  By analogy, the universe could be notionally sectioned by a spherical section, creating two halves, meaning two (three-dimensional) Hubble volumes, meaning that the nice one “unit” of energy for each “unit” of time during which the radius increases by one “unit” of length is obtained for the universe as a whole.

 

 Usch.

 

Recently, PBS Space Time published the video Could the Universe be Inside a Black Hole? which ended with a question about what exists outside of the black hole.  I slapped together a quick (and somewhat imperfect) comment to the effect that outside our universe – if it were to be a black hole – is not a where-question but rather a when-question.  There were universes before ours and there will be universes after ours, but they would be orthogonal to our universe.

 

I also riffed, in that comment, about the problem above that has been bugging me, writing:

 

My working hypothesis at the moment is that if you project four space (spacetime) onto three space (space), there are two equal solutions.  To understand this imagine looking at a monochrome sphere - ignoring any shading - and it would look like a circle.  That circle incorporates precisely half of the sphere with a complete other circle on the other side that you can't see.

 

This still doesn’t get me to where I want to be, because there is a suggestion of contiguous spacetime, with a surface volume that would have a density that is equivalent to that required (for flatness) for half that volume and I am back to having a similar problem – too much space now rather than too much mass/energy.

 

However, it did get me thinking and – coincidentally – there was something else that has been bugging me for a while.  I hinted at it in The Messiness of Layered Spheres, the notion I had in my head, was that for each instant (Planck time) that the universe expanded, there would be a cascade of Planck volumes filling in the gaps created by the expansion.  But it’s messy and the mechanism was utterly obscure.

 

Recently, although little less recently than the PBS video, I also saw an episode of QI in which Alan Davies argues with Sandi Toksvig about the shape that of the cells that bees make.  Hexagonal, obviously.  But no.  The cells are circular, they just make hexagons because that’s the shape that arises when you pack circles (and are minimising the use of resources such as wax).  Those facts together got me thinking about things a differently.

 

I provided some visualisations about the lifecycle of a universe back in Apparent Problems with Imagining a Universe, with the last image looking like this:

The idea is that time in the universe to the left is upwards, and time in the universe to the right is rightwards (thus orthogonal).  Implied is that the spatial dimensions are also orthogonal with no intention to imply that any spatial dimension in the second universe would be aligned with the temporal dimension in the first.

 

The equations that support this notion relate to the approach of the event horizon of black hole:

 

and

 

These become zero at the event horizon (where r = 2GM/c²) and then, inside the event horizon, become the square root of a negative number.  Now, when we are first introduced to the idea of the square root, we are channelled into the thinking that, for example, the square root of 4 = 2.  However, it is more accurate to think that the two solutions to the square root of 4 are +2 and -2 – because (+2)² = 4 and (-2)² = 4.  That is, the absolute value of the square root of 4 is 2, but it could be positive or negative.

 

I was thinking that what goes on, on the other side of the event horizon, was a new universe shooting off in the i direction (that is, orthogonally, so really off in the i, j, k, τ direction).  However, it makes more sense mathematically if there are two directions – the i, j, k, τ direction and the negative i, j, k, τ direction*.

 

That would imply that our universe is one of a matched pair – linked in a way, without any way to communicate or interact – which are being “fed” at a rate of one unit of mass/energy per unit of mass time, equally split between the two of us, resulting in both universes being eternally flat. 

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Why were the bees relevant?  Because I realised that I had been trying to solve the wrong question - How do bees make hexagons?  They don't, they make circles.

It puts me in mind of one of my favourite jokes.  How do you get down from an elephant?  You don't, you get down from a duck.

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* This is deliberately vague.  I'm still thinking about it.  One issue that remains is that neither +τ nor -τ would have precedence, which opens up the possibility that neither would have precedence over any other potentially orthogonal temporal dimension.  A possible solution to that is that at the event horizon all time and space is effectively compressed to zero - meaning that everything on the outside would have been fed through a singularity simultaneously and the orientation of i, j, k, τ and their opposites selected - and set - instantaneously.  As far as the universe(s) on the inside of the event horizon is (are) concerned, the orientation of i, j, k, τ (and their opposites) are and have always been what they are.

 

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Update: 3 April 2022

 

I just listened to the Skeptic's Guide to the Universe episode from 2 April 2022 (audio) - which was live and recorded some days earlier, somewhere in northeastern US.  Bob Novella speaks of a new theory in published in the Annals of Physics (reported at Live Science), in which there is a mirror (or anti-)universe running "backwards in time".  The physicists involved have come to this conclusion a different way, because they were looking for CPT (charge, parity, time) symmetry as an explanation for dark matter.

 

There are some interesting (and parallel) ramifications of the theory, including "a CPT-respecting universe naturally expands and fills itself with particles, without the need for a long-theorized period of rapid expansion known as inflation".  Compare this with my model (Universe (in) a Black Hole - published 7 June 2019) in which our universe is "fed" with one unit of Planck mass/energy per two Planck units of time (see Is the Universe Getting More Massive? (Flatness, not Fatness) – published 30 July 2019) and the expansion follows the Hubble radius = Schwarzschild radius metric, where the Hubble radius is the speed of light multiplied by the age of the universe - which implies that there is no inflation (see My Universal (and Expanding) Struggles – published 9 July 2019).

 

Fascinating stuff.  Note that the model that these physicists are suggesting (at least according to Live Science) is one in which there is a pair of universes in which one consists (largely) of matter and the other (largely) of antimatter, which would explain the dearth of antimatter in our universe.  I don't think that my model necessarily implies that - I think that both universes would have mass/energy of some sort, as necessary to obtain one Planck unit of mass/energy per unit of Planck time.  It should be noted that antimatter has positive mass and that equal parts of matter and antimatter do not cancel each other out, when they collide each is annihilated but they release energy.  You could say that their matter (and antimatter) nature is destroyed, but not their energy.