Calculate Red Shift w/ Shwarzschild Metric: A Hypothetical Black Hole Any Size?

[elevin]

TL;DR Summary

Using the Shwarzschild metric to calculate cosmic inflation

You can unconventionally use the Shwarzschild metric to calculate red shift at given stellar distances with some accuracy right out to the edge of the universe. Take the mass of the known universe ~10^55kg. Try it! A bit ironic.

I understand this is not how to use the Shwarzchild; however, given size/distance are relative, (EG. A handful of sand is the same mass as when thrown in the air) a black hole (if hypothetically broken down "thrown in the air", like stars/galaxies in the sky) could be any size given gravitational falloff is never zero.

With gravity/acceleration being equivalent, why wouldn't extra-galactic red shift (per Hubble) be caused by the total gravitational potential within a given radii from the observer (eg. greater distances = greater red shift due to gravity within a "large" sphere)? Why must these galaxies accelerate away when we have all the fuel we need for red shift in the form of gravity?

I don't even want to bring this up for fear of getting kicked off, but I've been on this for a long time and the more I look, the more interesting it gets with (seemingly) lots of observational support.

Disclaimer: I'm an armchair astrophysicist so go easy on me math-man, but please tell me I'm wrong and why so I can let this go!

 

[PeterDonis]

I understand this is not how to use the Shwarzchild

Then why are you doing it?

please tell me I'm wrong and why so I can let this go!

You already know why you're wrong: look at the quote I gave from your own post above.

I could also point out that your thread title contains a false premise: black holes aren't "impossibly dense". A black hole is vacuum: there is nothing inside. Also a black hole does not have a well-defined volume. So it doesn't have a well-defined density.

 

[PeterDonis]

With gravity/acceleration being equivalent

They aren't.

 

[Nugatory]

With gravity/acceleration being equivalent,

That equivalence is very restricted in its applicability: The effects of gravity are indistinguishable from the effects of constant acceleration, but only within a region of spacetime small enough that gravitational tidal effects are not detectable. In other words, it only works inside Einstein's elevator.

 

[Dale]

Why do black holes have to be impossibly dense?

Actually, they don't. In fact, if you have a spherically symmetric distribution of matter of any finite density then there exists a mass at which the vacuum outside that distribution will have an event horizon. So they don't have to be impossibly dense, but just really big. Then they can collapse later and become dense, but the event horizon can form before that.

 

[elevin]

Then why are you doing it?You already know why you're wrong: look at the quote I gave from your own post above.

I could also point out that your thread title contains a false premise: black holes aren't "impossibly dense". A black hole is vacuum: there is nothing inside. Also a black hole does not have a well-defined volume. So it doesn't have a well-defined density.

I love how certain you are of things so far out of reach Peter :) I think you're missing the point of my query. This is a thought experiment and I have a question about its validity. I may not be using terminology in a scientifically exacting way (as you pointed out) because this is a forum not a physics paper. The premise however is not as "dense" as your tone might imply. Just run the formula and then lets talk in a way that's mutually supportive and helpful.

 

[elevin]

They aren't.

Equivalence principle applied. Get in your elevator on this one please ;)

 

[elevin]

That equivalence is very restricted in its applicability: The effects of gravity are indistinguishable from the effects of constant acceleration, but only within a region of spacetime small enough that gravitational tidal effects are not detectable. In other words, it only works inside Einstein's elevator.

The entire premise exists outside the confines of our planet, the solar system, even our galaxy so I'm not sure how tidal forces are a thing. In empty space would the principles of equivalence not apply? That's what I'm referring to. As I mentioned to Peter, we may be missing the point here if we're chopping up my sloppy terminology. If we could just answer the question posed, that would be outstanding.

 

[elevin]

Actually, they don't. In fact, if you have a spherically symmetric distribution of matter of any finite density then there exists a mass at which the vacuum outside that distribution will have an event horizon. So they don't have to be impossibly dense, but just really big. Then they can collapse later and become dense, but the event horizon can form before that.

YES! This is what I'm talking about Dale! The S Metric does something really interesting. The more mass you throw at it, the larger and less dense said "black hole" becomes. In other words, the more mass you have, the easier it is to arrive at a net total gravitational force that bends light to the point of an event horizon. This "traditionally" is used to describe collapsed stars, however with the discovery of larger and larger black holes (do voids count here?) perhaps that brings about the question of "what's inside" which I think you're indicating here. My thought is: what happens when you blow the scale of this thing wayyyy out of proportion to the tune of the formula mentioned in my initial post? Ironically the Schwarzschild radius of the known matter in the universe approximates the size of the known universe. This would indicate that the sum total gravity within our own personal event horizon is proportional to what we can calculate with our radio telescopes and etc. ...or I could be way off, but please, educate.

 

[PeterDonis]

I love how certain you are of things so far out of reach Peter :)

A black hole is a definite solution of the Einstein Field Equation--the one you named, the Schwarzschild metric--whose properties are well known. Those are what I was describing. Those properties are certain. They're math.

Whether or not particular things in our actual universe are described by this solution is a separate question.

This is a thought experiment

I don't see any thought experiment in your OP. I see a bunch of assertions and a question. The question is pointless because it is based on a false premise. I pointed out the place in your own post where you admitted the false premise.

I may not be using terminology in a scientifically exacting way

There is no issue with your terminology. The term "Schwarzschild metric" is quite clear. So are the following facts: the Schwarzschild metric is a vacuum solution (which I stated); and the Schwarzschild metric is not applicable to the universe as a whole (which you stated, when you said "this is not how to use" it).

Just run the formula

What formula? The only one you have referred to is the Schwarzschild metric, and you admit this metric is not applicable to what you're asking about. So there is no formula to run.

If you want to use formulas to ask questions or investigate thought experiments about the universe as a whole, you need to use the right formulas. So it seems like the first thing you should do is to figure out what those are.

 

[PeterDonis]

if you have a spherically symmetric distribution of matter of any finite density then there exists a mass at which the vacuum outside that distribution will have an event horizon.

This only applies if there is a vacuum outside. That is not the case for our universe; the EFE solution that describes our universe is spherically symmetric, but has no vacuum outside. So the Schwarzschild metric is simply not applicable to our universe as a whole, and its properties cannot be used to analyze the behavior of the universe as a whole.

This would indicate that the sum total gravity within our own personal event horizon is proportional to what we can calculate with our radio telescopes and etc. ...or I could be way off, but please, educate.

You are way off, for the same reason you gave in your OP, that I pointed out in post #2. I've expanded on it a bit in response to @Dale above.

 

[PeterDonis]

In empty space would the principles of equivalence not apply?

The equivalence principle always applies in GR. It just doesn't say what you think it says. It doesn't say "gravity is equivalent to acceleration". What it does say is that, if you confine your attention to a small enough patch of spacetime that the effects of spacetime curvature are not observable, the local laws of physics are the same as they would be in flat spacetime. That is true whether you are in "empty space" or not. But it doesn't help at all in addressing your question, because your question is not confined to a small enough patch of spacetime that the effects of spacetime curvature are not observable. So the equivalence principle is irrelevant.

 

[elevin]

A black hole is a definite solution of the Einstein Field Equation--the one you named, the Schwarzschild metric--whose properties are well known. Those are what I was describing. Those properties are certain. They're math.

No argument here, just not the point of my query.

Whether or not particular things in our actual universe are described by this solution is a separate question.

That's pretty much the question. Take a look at Dale's response.

I don't see any thought experiment in your OP. I see a bunch of assertions and a question. The question is pointless because it is based on a false premise. I pointed out the place in your own post where you admitted the false premise.

Fair enough, I was writing in short hand. I'd be happy to expand on the thought experiment if you would entertain it. I'm really just looking to learn here, but in order to get the the learning part we need to get off chopping up my query and just get to the point. I really do appreciate and respect adherence to the scientific process. I imagine you're probably pretty amazing with whatever course you follow in your field.

There is no issue with your terminology. The term "Schwarzschild metric" is quite clear. So are the following facts: the Schwarzschild metric is a vacuum solution (which I stated); and the Schwarzschild metric is not applicable to the universe as a whole (which you stated, when you said "this is not how to use" it).

As Dale's description seemed to imply, black holes may be a variety of different sizes / densities. This is the area I'm curious about.

What formula? The only one you have referred to is the Schwarzschild metric, and you admit this metric is not applicable to what you're asking about. So there is no formula to run.

If you want to use formulas to ask questions or investigate thought experiments about the universe as a whole, you need to use the right formulas. So it seems like the first thing you should do is to figure out what those are.

Just entertain me and input the known mass of the universe into the schwarzschild metric and see what you get. I'm happy to do it, but I'm assuming someone would likely shred my presentation. Do you not find it ironic that the results approximate the known size of our universe?

 

[elevin]

This only applies if there is a vacuum outside. That is not the case for our universe; the EFE solution that describes our universe is spherically symmetric, but has no vacuum outside. So the Schwarzschild metric is simply not applicable to our universe as a whole, and its properties cannot be used to analyze the behavior of the universe as a whole.You are way off, for the same reason you gave in your OP, that I pointed out in post #2. I've expanded on it a bit in response to @Dale above.

I said "unconventional use". Seems like you're stuck in particularly granular set of rules prescribed by an extensive knowledge of physics. Let's back out for a sec, and just run the formula.

 

[Dale]

This only applies if there is a vacuum outside. That is not the case for our universe; the EFE solution that describes our universe is spherically symmetric, but has no vacuum outside. So the Schwarzschild metric is simply not applicable to our universe as a whole, and its properties cannot be used to analyze the behavior of the universe as a whole.

Yes, I agree completely.

The question in the title seemed unrelated to the question in the body of the OP for exactly that reason. I was just answering the question in the title since you had already covered the question in the body.

 

[Dale]

what happens when you blow the scale of this thing wayyyy out of proportion to the tune of the formula mentioned in my initial post?

This doesn't work for the reasons already explained by @PeterDonis . I was just answering the unrelated question in the title, not contradicting the correct information he was giving you.

It isn't a question of the scale, it is the fact that the Schwarzschild metric only applies in the vacuum region outside the spherical distribution of matter. There is no such region on cosmological scales.

 

[Vanadium 50]

I love how certain you are of things so far out of reach Peter

Go ahead, poke the bear.

Equivalence principle applied. Get in your elevator on this one please ;)

Um, before indulging in obnoxious snottery, you might want to get your facts straight. The equivalence principle does NOT say gravity and acceleration are equivalent. If anything, the opposite: it explains how to tell gravity from other accelerations.

This thread starts off with a question based on an incorrect assumption, and seems to be wandering from there. Not a good spot to start from and not a good spot to go. Maybe it would be a good idea to rethink this path.

 

[Ibix]

Just entertain me and input the known mass of the universe into the schwarzschild metric and see what you get.

Nobody needs to calculate the Schwarzschild radius of the mass of the observable universe. At this time, plus or minus a few hundred million years or so, you get approximately the age of the universe multiplied by $c$.

If memory serves, this relationship is exactly true in a matter-dominated flat space, but just because the only length scale you'll ever get out of relativity is $GM/c^2$ times some fairly simple numerical factor that happens to be 2 in this case. In any other type of cosmological solution the result is not exact and in fact can vary with time. Given the parameters of our universe it happens to be true (give or take a few percentage points) around now, but was less accurate in the past and gets wildly inaccurate in the future.

I'm also an amateur, but there are plenty of solid learning resources out there. That the Schwarzschild solution looks absolutely nothing like an FLRW solution is easy to verify, and that it looks nothing like the universe out there is also easy to verify. What is the CMB in this model, for example? How are you proposing to calculate gravitational redshift in a non-stationary spacetime?

 

[elevin]

Go ahead, poke the bear.

Um, before indulging in obnoxious snottery, you might want to get your facts straight. The equivalence principle does NOT say gravity and acceleration are equivalent. If anything, the opposite: it explains how to tell gravity from other accelerations.

This thread starts off with a question based on an incorrect assumption, and seems to be wandering from there. Not a good spot to start from and not a good spot to go. Maybe it would be a good idea to rethink this path.

I'm sorry Vanadium, not trying to make enemies here. I'd like to get to the point and get an answer, but not sure of the best way to get my answer without getting shredded or misunderstood.

 

[PeterDonis]

Just entertain me and input the known mass of the universe into the schwarzschild metric and see what you get.

No. I've already explained to you why not: because, as you yourself admitted in the OP, the Schwarzschild metric does not apply to the universe as a whole. It just doesn't. No amount of repetition of the question will change that.

I said "unconventional use".

What you're trying to do isn't an "unconventional" use. It's a wrong use. The reason why has been explained.

Seems like you're stuck in particularly granular set of rules prescribed by an extensive knowledge of physics. Let's back out for a sec, and just run the formula.

No, let's not "run the formula" when we already know it is the wrong formula. And let's not dismiss "rules prescribed by an extensive knowledge of physics" just because you want to use a wrong formula to calculate a meaningless number. That's not a good recipe for getting useful answers.

I'd like to get to the point and get an answer

You've been given the answer. Several times now.

not sure of the best way to get my answer without getting shredded or misunderstood.

If you want to avoid being "shredded or misunderstood", you would do well to actually think about the answer you've been given and the reasons for it that you've been given. And not, as a self-described "armchair astrophysicist", to indulge in snark against experts who are doing their best to help you by explaining the answer you've been given.

This thread is closed.