# Exploring Black Hole Formation: Schwarzschild Radius and Relativity

• scott haig
In summary, the conversation discusses the concept of black holes and their similarities to the Big Bang. The density of the universe at the time of the Big Bang is proportional to the critical density, and various forms of energy scale differently during expansion. The dark energy, in the form of a cosmological constant, remains constant throughout expansion. There is also a discussion about whether the universe can be considered a black hole, and whether it is closed. The redshift z is defined as a function of the scale factor, and the Hubble parameter is dependent on the matter density, radiation, and dark energy parameters.
scott haig
. They understand the classical Schwarzschild radius argument for black hole formation. They have bare rudiments of special and general rel. I sound like an idiot saying "there's this anti-gravity" force that kicks in when the collapsing system gets really big. Any help?

Conceptually, we look at a black hole from the outside, but we are inside the universe. All of spacetime was collapsed at the Big Bang.

However, there are useful analogies between a black hole and the Big Bang. Like the event horizon beyond which we cannot see.

If "big bang" is an event happened in the past, then how can it be "black hole"? Black hole is an object, Not an event

You may find this website helpful.

thanks-but I still need some help with the minutes after big bang--
Is there a "proper time" argument that leads to the tensor that somehow keeps space flat while its density is so high? Is there a negative energy term from a pressure? And what about Einstein energy/dark matter during expansion?

Not too sure what you mean by proper time argument, but for the flat space part, even though the density was certainly higher in the moments after the big bang, the parameter that leads to the geometry (i.e. flat, open, or closed) is the ratio of the density to the critical density

$$\Omega=\frac{\rho}{\rho_c}$$

The critical density $$\rho_c=\frac{3H^2}{8\pi G}$$ as you can see is proportional to H^2 so and as such is not a constant ( H=H(t) ). Different forms of energy scale differently during expansion, radiation increases at earlier times as

$$\rho_r=\rho_r^0(1+z)^4$$, and the matter like protons and neutrons as $$\rho_m=\rho_m^0(1+z)^3$$, so in the distant past the radiation was dominant, and matter was negligible. The Dark energy in the form of a cosmological constant does not change during expansion, it's by definition a constant. It was also therefore negligible in the early universe. The dark matter also scales as $$(1+z)^3$$ like normal (baryonic) matter, so was also not important in the early times. Good luck, I wish I had learned cosmology in high school, how great :)

Cheers

THANKS MUCH ASTROROYALE-
We're chewing this over...
s

ok but--

I thought the reason Hubble's constant (the simple ratio of recessional velocity to distance) changes with time is because of the time dilatation associated with the increased density (rho) as you go backward in time and the universe is more dense. And similarly there's that scaling factor z that came from the metric tensor--it's time dependent by virtue of size, therefore it changes with density, yes?

By proper time I mean time for an observer in the frame under consideration. Now I don't know if its legitimate to consider an arbitrary point in the universe at some short (by our clock) time post big bang a rest frame (has anyone addressed the first few minutes from a Newtonian point of view?--would expansion be associated with a measurable acceleration?) but I know that down that big gravitational well a clock would seem slowed relative to one now.

And so,

All the size terms are already time dependent because we've assumed size changes with time. There seems to be something wron with the logic underlying all of this. And to fix it I am left with dark energy-- this time-invariant, deus-ex-machina number that I'm shoving into equations-- which were based on measurements-- to explain away an apparent but never-measured anti-gravity observation (expansion). I know it's an artifice, but when I try to explain it it seems so "artificial". Is there any way to simplify the explanation?
S

If you think a while about the equation for the Schwartzschild radius:

r=2GM/c^2

and about the mass of the universe (various calculations can be found, one as "accurate" as 1.59486 × 10^55 kg), you could possibly come to the conclusion that the universe in its entirety still is a black hole. This would give radius of 1.183 x 10^26 m. Most estimates for the radius of the universe lie around 10^26 m.

So, we could lie comfortably within the Schwartzschild radius of the universe's mass.

Whether this means the universe actually is a black hole is open to interpretation. http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BlackHoles/BlackHoles.html" from the University of Toronto says

For a mass of 2.5 x 10^53 kg, i.e. a 2 and a 5 followed by 52 zeroes kg, the Schwarzschild radius is about 17 billion light years. This huge mass is an estimate for the total mass of the universe. Also, given that the age of the universe is 15 billion years or so, 17 billion light years is awfully close to the size of the universe. Does this mean that the universe itself is a black hole?

It turns out that this question is the same as asking: is the universe closed. If the universe is closed, then it is fairly accurate to say that it is a black hole.

I tend to agree.

cheers,

neopolitan

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Hi again,

Sorry for the delay, here's a few points.

And similarly there's that scaling factor z that came from the metric tensor--it's time dependent by virtue of size, therefore it changes with density, yes?

The redshift z is defined as the following;

$$(1+z)=\frac{a_0}{a(t)}$$

where a is the scale factor. So when we say redshift 1, what we really mean is that the universe is currently (1+z)=2 times the size now then when the light was emitted by the source. So yes, the redshift z does change with time, it's actually just like time really. The Hubble paramter H(z) is defined as follows,

$$\frac{da/dt}{a(t)}=H(z)=H_0[(1-\Omega_{tot})(1+z)^2 + \Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_\Lambda]^{\frac{1}{2}}$$

Where $$\Omega_m$$ is the matter density parameter, $$\Omega_r$$ is the radiation parameter, $$\Omega_\Lambda$$ is the dark energy in the form of a cosmological constant, and $$\Omega_{tot}=\Omega_m+\Omega_r+\Omega_\Lambda$$
You can show that for any given $$\Omega_{tot}$$ that during the expansion it can never change the sign of the curvature parameter, ie if $$\Omega_{tot} < 1$$ at some time, it will always be < 1, or if $$\Omega_{tot} > 1$$ then it will always be >1, and if $$\Omega_{tot} = 1$$ it will always =1. In fact, for any $$\Omega_{tot}$$, it will tend to $$\Omega_{tot} = 1$$ at very high redshift.

Now I don't know if its legitimate to consider an arbitrary point in the universe at some short (by our clock) time . . .

It is absolutely legitimate to do so, every point in the Friedmann-Robertson-Walker universe is completely equivalent, it has to be for the requirements of isotropy and homogeneity. And it is possible to derive the friedmann equations for the expansion via Newtonian mechanics. I think Liddle does so in his book, might be worth checking out.

to explain away an apparent but never-measured anti-gravity observation (expansion).

Hmmm, not too sure about this one, but the dark energy is not necessary to explain an expanding Universe. It was actually introduced by Einstein to create a static Universe, which was the popular model before the observations that the Universe was expanding. After the observations by Hubble/Slipher of expansion, it was no longer needed. That's why he called it his "biggest blunder". It's become fashionable again lately b/c it is needed to describe an accelerating expansion.

This probably just confused things more :) Just curious, are you teaching in the states?

Cheers

neopolitan said:
If you think a while about the equation for the Schwartzschild radius: r=2GM/c^2

and about the mass of the universe (various calculations can be found, one as "accurate" as 1.59486 × 10^55 kg), you could possibly come to the conclusion that the universe in its entirety still is a black hole. This would give radius of 1.183 x 10^26 m. Most estimates for the radius of the universe lie around 10^26 m

>> the universe in its entirety still is a black hole?
Nonsense!

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Xeinstein said:
>> the universe in its entirety still is a black hole?
Nonsense!

Context is important, you can argue with what you snipped out. Can you argue with the larger context which you quoted (noting there was more to it as below)?

neopolitan said:
If you think a while about the equation for the Schwartzschild radius r=2GM/c^2 and about the mass of the universe (various calculations can be found, one as "accurate" as 1.59486 × 10^55 kg), you could possibly come to the conclusion that the universe in its entirety still is a black hole. (Given that t)his would give radius of 1.183 x 10^26 m (and m)ost estimates for the radius of the universe lie around 10^26 m

Note that I said in the same post

Whether this means the universe actually is a black hole is open to interpretation.

It seems the concept is also open for dismissal, with no reason given. I would like to hear more about why is it so clearly "nonsense".

cheers,

neopolitan

neopolitan; said:
If you think a while about the equation for the Schwartzschild radius: r=2GM/c^2 and about the mass of the universe (various calculations can be found, one as "accurate" as 1.59486 × 10^55 kg), you could possibly come to the conclusion that the universe in its entirety still is a black hole. This would give radius of 1.183 x 10^26 m. Most estimates for the radius of the universe lie around 10^26 m.

So, we could lie comfortably within the Schwartzschild radius of the universe's mass.

Whether this means the universe actually is a black hole is open to interpretation.

Can you tell me if the universe is flat or curved and open or closed?
How could the universe be a black hole?

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If the universe is a black hole then from an outside observer would time appear to be stopped? If you could look into a black hole?

scott haig said:
.

Help with explaining why BigBang was not a Black Hole...

They understand the classical Schwarzschild radius argument for black hole formation. They have bare rudiments of special and general rel. I sound like an idiot saying "there's this anti-gravity" force that kicks in when the collapsing system gets really big. Any help?

the Schw. radius formula is meaningless when applied to the rapidly expanding early universe. the Schw. BH radius formula was meant to apply to non-expanding situations.

Your question can be discussed either on a classical or a quantum cosmology level.
on a classical level, the model breaks right at the start of expansion so it doesn't say anything about the very start. So you have to postulate initial conditions. Classically you postulate both extremely high density and extremely high rate of expansion!

the Hubble parameter, instead of being like 70, is millions of time larger. Classically, you just postulate that because if you start the model running right it will fit the data.

All gravity can do is SLOW THE EXPANSION DOWN, and in the classic model that is what it immediately proceeds to do, but that is a gradual process and we still have (greatly diminished) expansion left over. it takes time to slow it down and by the time it has slowed some, then the universe less dense----matter has thinned out, so it doesn't have such a strong effect.

YOU CAN ALSO DISCUSS the same question using fledgling QUANTUM cosmology models.
In that case you don't have to postulate initial conditions at the start of expansion because the models run back in time to BEFORE the start of expansion and they go some way towards explaining why there was this rapid expansion in the early days.

scott haig said:
ok but--

I thought the reason Hubble's constant (the simple ratio of recessional velocity to distance) changes with time is because of the time dilatation associated with the increased density (rho) as you go backward in time and the universe is more dense. And similarly there's that scaling factor z that came from the metric tensor--it's time dependent by virtue of size, therefore it changes with density, yes?...

I think the red statement may indicate a misconception. My suggestion would be to look up Friedmann equation. It is a simple differential equation, derived from the main GR equation, which shows how density governs the gradual reduction of H.
Actually there are two Friedmann equations which together describe the evolution of a(t) the scalefactor in the usual metric that cosmologists use. You sound as if you are familiar with a lot of this. I wouldn't say that the Friedmann equations describe a "time dilatation" effect. It doesn't seem to me to be a helpful way to look at it. But this is just my opinion an maybe someone else will step in and clarify.

## 1. What is a Schwarzschild radius?

A Schwarzschild radius is the distance from the center of a black hole at which the escape velocity exceeds the speed of light. It is named after the German physicist Karl Schwarzschild who first calculated it in 1916.

## 2. How is the Schwarzschild radius related to black hole formation?

The Schwarzschild radius is the defining characteristic of a black hole. Once a massive object collapses to a size smaller than its Schwarzschild radius, it becomes a black hole and its gravity becomes so strong that even light cannot escape from it.

## 3. What is the significance of the speed of light in relation to black holes?

The speed of light is the maximum speed at which anything can travel in the universe. When an object falls within the Schwarzschild radius of a black hole, the escape velocity exceeds the speed of light, making it impossible for anything, including light, to escape from the black hole's gravitational pull.

## 4. How does relativity play a role in black hole formation?

Relativity, specifically Einstein's theory of general relativity, is essential in understanding the formation and behavior of black holes. It describes the curvature of spacetime caused by massive objects, and this curvature becomes infinitely strong at the center of a black hole, creating a singularity.

## 5. Can we observe black hole formation directly?

No, we cannot observe black hole formation directly as it occurs within the black hole's event horizon, which is the boundary beyond which nothing can be seen or detected. However, we can observe the effects of black hole formation, such as gravitational lensing and the emission of high-energy radiation, on the surrounding matter and light.

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