Singularities

[FindailsCrispyPancakes]

Following the recent thread about black holes, I decided to go all in and try to put something together about singularities. So what is a black hole singularity?

Put simply, it's a patch of spacetime that can't be described by the equations that are used to describe spacetime. If they are used to describe spacetime above a certain density, the equations fail and start spitting out infinities. This leads to singularities being described as points of infinite density.

Is this realistic, or is it just a limitation of the equations? If it's a limitation of the equations, is there another way for us to at least work out just how much stuff can be crammed into a patch of spacetime before the equations fail?

Q) Just how much does one cubic millimetre of singularity weigh? Will it be an infinite amount?

A) A guy called Max Planck came up with the answer about a century ago. I'm going to do my best to reproduce it here with as simple an interpretation as I can provide.

A BUNCH OF BORING MATHS YOU CAN SKIP IF YOU HATE THAT SORT OF STUFF:

I'll start by describing the parts Planck already knew before he figured it out.

1st formula: The speed of light

Let's begin with Maxwell's speed of light in a vacuum (commonly referred to as c) which is 299,729,458 metres per second.

That's a horrible looking number, so I'm delighted to start things off by throwing the number away and concentrating on the units of measurement instead. In the case of the speed of light there is no reference to a mass, but it does refer to units of length (metres) and units of time (seconds).

Using the letters M to refer to mass, L for length and T for time it is possible to write a dimensional formula for the speed of light.



2nd formula: The gravitational constant

The other thing Planck knew about was Newton's gravitational constant G, which describes the strength of the fundamental force of nature we refer to as gravity.

There's an equation of Newton's I remember from the physics lessons of my teenage years that gives me deja vu every time I see it. It describes the gravitational force (F) between two masses.



I'm going to rewrite that dimensionally by using the same trick again, only this time it will require a few more steps. I'll start by using L to denote the distance between the two masses instead of r and a capital M for the mass instead of a lowercase m.



Cheerfully ignoring the mass/length quantities and just concentrating on the units of measurement, we can just say there are two masses and two lengths. Moving it all around the equals sign to get G on its own gives us a good start on the dimensional formula:



So far so good, so what's the dimensional formula for force (F)?

As our science teachers of yesteryear kept telling us, force is equal to mass times acceleration. Ok, that's M=1 multiplied by acceleration.

Great, so what's acceleration? It's a changing velocity divided by a period of time. So the dimensional tally is now M=1 T=-1 multiplied by velocity.

So what's velocity? It's distance divided by time, which adds L=1 and T=-1 to our tally. So in terms of a dimensional formula, our tally for force equates to M=1 L=1 T=-2.

Let's substitute that for F in our previous equation



This is why the value for G is given in the following units:



We now have dimensional formulae for two constants of nature



3rd formula: Planck's constant

Planck comes into the story by virtue of his discovering another constant of nature.

What Planck's constant describes is a quantity of energy that is the minimal amount of change possible in the energy of a system between one event and the next. All energy occurs only in multiple integers of Planck's constant, which is written as h.

The angular momentum of any system only occurs in units of h divided by twice the value of pi, a symbol referred to as h bar. This is what we're interested in here.



Let's try and get a dimensional formula for h bar.

What's a joule? It's the work done by a force of one Newton through a space of one metre. That takes the dimensional tally to L=1 multiplied by the dimensional formula for Newtons.

What's a Newton? It's the amount of force (M=1 L=1 T=-2) necessary over a period of a second, to increase the velocity of a mass of a kilogram by one meter per second.

That means we've got M=1 L=1 T=-2 to add to the L=1, giving us M=1 L=2 T=-2 as the dimensional formula for energy, which is measured in Joules.

The dimensional formula for h bar should therefore be energy multiplied by time.



So far, so good, so what now?

So approximately a century ago, Max Planck became the first person in history to be armed with these three dimensional formulae:



At some point, Herr Planck found himself wondering if the formulae could be used to derive fundamental units of mass, length and time. Here's what he discovered.

The dimensional formula for c has the following property which will come in useful later:



So, what happens when we fool around with h bar and G? Dividing h bar by G gives:



From this it is fairly obvious that multiplying by c will make the length and time terms disappear.



How about multiplying h bar by G?



It is fairly obvious that dividing by c cubed will remove the time term



Dividing by c to the power of five will remove the length term



At this point Planck simply plugged the values for h bar, c and G back into the equations and found himself presented with fundamental units of mass, space and time, now known as Planck units.



HOW MUCH DOES A CUBIC MILLIMETRE OF SINGULARITY WEIGH?

A cubic metre of space contains an incredibly large number of Planck volumes.



Multiplying this number by the Planck mass will give the maximum possible mass that can be contained within a cubic metre.



NEVER MIND ALL THESE SILLY NUMBERS, GET TO THE RESULT!

So, a quick conversion and the question can be answered.



A very large number indeed. Over 100,000,000,000,000,000,000,000,000,000,000 times the mass of the observable universe per cubic millimetre. That's a lot, but it's not infinite.

[Skyweir]

Well its impressive.. but entirely beyond my intellectual reach. Yet methodically and logically approached .. from a lay scientific perspective.

And no SDT didnt prepare me sufficiently for this giant leap

[Hashi Lebwohl]

Physics 101 students everywhere thank you.

[FindailsCrispyPancakes]

Physics 101 students everywhere thank you.

I only did it so I could make the following terrible pun:

When I started trying to work out the density of a singularity I had no idea things could get so heavy.

(Silence, tumbleweed drifting past, etc.)

[Skyweir]

Omg grrrooooaaaannnn

[Zarathustra]

Multiplying this number by the Planck mass will give the maximum possible mass that can be contained within a cubic metre.

Where in all these derivations does it say that one Planck volume can only hold one Planck mass? I don't see density addressed at all here, much less in terms of a limit. A Planck mass isn't a fundamental limit, like Planck time or Planck length. The only thing I can see as significant about one Planck mass confined to one Planck volume is that it will form a black hole--one with the smallest possible (or meaningful) Schwarzschild radius. But just because this is the Schwarzschild radius for one Planck mass doesn't necessarily mean that this is the maximum amount of mass that can fit within that radius, does it? Or am I missing something? Black holes can have various densities.

We routinely deal with less mass than a Planck mass. However, smaller masses couldn't form black holes, because their Schwarzschild radii would be smaller than a Planck length. So this is a lower limit for mass only in terms of forming black holes. I don't see how it's an upper limit for defining density. But again, maybe I'm missing something.

[FindailsCrispyPancakes]

Omg grrrooooaaaannnn

My jokes are only going to get worse from here on in I'm afraid Sky lol

[Skyweir]

[FindailsCrispyPancakes]

PLANCK PARTICLES, THE PLANCK MASS AND SINGULARITIES:

The two different types of theorised Planck particle I am aware of are black hole models on a scale close to the Planck length.

Planck particle model 1: A Schwarzschild black hole with a Schwarzschild radius equal to its Compton wavelength.

This theorised black hole/particle would have a mass of approximately 1.7 Planck masses, with a radius/Compton wavelength of approximately three and a half Planck lengths and a volume of just over 186 Planck volumes.

Planck particle model 2: A Schwarzschild black hole with a mass equivalent to one Planck mass.

This theorised black hole/particle would have a radius of two Planck lengths, a Compton wavelength of two multiplied by pi Planck lengths and a volume of approximately 33 Planck volumes.

Model number 3: A singularity

This is the model previously described, with a mass equivalent to one Planck mass and a volume equivalent to one Planck volume.

This is the mass/energy density at which it is necessary to switch to an Einstein-Cartan gravity framework, which employs torsion in order to resolve singularities. To fully model the properties of spacetime in excess of this density would require a quantum theory of gravity.

The Planck density is not an absolute upper bound for mass density. It's more accurate to describe the Planck density as being the mass density at which the phase transition from singular to non-singular spacetime takes place in our universe.

In other words, it's the density of the universe one Planck second after the big bang.

[FindailsCrispyPancakes]

It's interesting to take the reciprocal of the Planck mass. To do so requires switching to units where mass and energy are interchangeable via E=mc squared. In this case that's electronvolts (eV).



In theories of massive gravity (where gravitons are not massless) this is a feasible figure for graviton mass if one does not resort to invoking new fundamental forces of nature.