How much time dilation is there within a contracting remnant

[DC0]

Relative to a remote point, when using the standard GR method, how does the rate of time passage (1 / gravitational time dilation) typically vary with radius within a contracting 5 solar mass supernova remnant, when its’ outer radius crosses a value of about 1.6 times the Schwarzschild radius?

 

[PeterDonis]

Are you familiar with the math that applies to this scenario? (You should be since you marked this thread as "A" level.) Have you tried to calculate the answer?

 

[DC0]

Even though I am looking for an answer, I am not familiar with the GR math that applies to this scenario. I have used relativistic equations derived from general relativity that were factored into the Newtonian model. I have calculated results and I'm trying to make a comparison.

 

[PeterDonis]

I am not familiar with the GR math that applies to this scenario.

Based on that, I have changed the thread level to "I".

 

[PeterDonis]

I have used relativistic equations derived from general relativity that were factored into the Newtonian model. I have calculated results and I'm trying to make a comparison.

Please show your work.

 

[Dale]

I have used relativistic equations derived from general relativity that were factored into the Newtonian model.

This seems like a highly problematic approach. Do you have a professional scientific reference that details this approach, or are you just kind of “winging it”?

I would appreciate that as well as the posted work requested by @PeterDonis

 

[DC0]

To set up the density, the initial values of Δr and r are adjusted for each layer by going through several iterations of the next three equations:

The pressure for each layer is

P_r = Σ_n^N (Δm_n) x (m_br) G /((r²) (4π r²)) ,

where m_n is the mass within layer n and m_br is the mass below that radius. The pressure and density settle into where the density is greatest at the center. The density as a function of pressure is expressed by:

ρ = 10^{(0.4838 Log P + 1.2372)}.

Then the new radius for each shell is calculated by.

r_n = ((m_br - m_br(n-1))/( ρ (4/3)π) + r³ (n-1))^(1/3),

For time contraction, the change in the gravitational potential across any Δr layer is

ΔΦ = F Δr = - m_br G Δr / r²

The gravitational potential at any radius r within the remnant is

Φ = - GM/R + Σ_R^r ΔΦ = - GM/R - m_br G Δr / r²

Here the relative slowing of time, the comparison of the rate of clock ticks, is used rather than dilation, the time between ticks. Time contraction is expressed by

t / T = (1+2 Φ / C2 )^1/2.

Δr is contracted by the same time contraction value (t / T) and these 3 equations are either iterated several more times until it becomes stable or until (t / T) becomes 0 (frozen).

For a contracting remnant greater than 2.2 solar masses, the decreased gravitational potential causes time to relatively freeze at the center, and stop the contraction before the pressure gets high enough to stop it, as it would in a neutron star. I’m trying to find out if a relative time freeze like this occurs during the contraction using the standard GR method.

The program described, is written in Excel and takes about a day to run. It has been in the works for many years.

 

[Dale]

The pressure for each layer is

Pr = ΣnN (Δmn) x (mbr) G /((r2) (4π r2)) ,

This equation assumes Newtonian gravity and flat spacetime.

The density as a function of pressure is expressed by:

ρ = 10(0.4838 Log P + 1.2372).

Where did you get this?

Then the new radius for each shell is calculated by.

rn = ((mbr - mbr(n-1))/( ρ (4/3)π) + r3 (n-1))(1/3),

What are you doing here?

For time contraction, the change in the gravitational potential across any Δr layer is

ΔΦ = F Δr = - mbr G Δr / r2

I think this is also only valid in flat spacetime.

Δr is contracted by the same time contraction value (t / T)

I am not sure that this is valid or even meaningful. I need to see some valid references as I requested earlier.

Here is a full GR approach
http://insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a6c_9.pdf

 

[PeterDonis]

I’m trying to find out if a relative time freeze like this occurs during the contraction using the standard GR method.

First of all, my reaction to your math is similar to @Dale's. It looks to me like you are pulling equations out of the air. I'm not even sure they're all valid Newtonian equations.

That said, the term "relative time freeze" is not a correct description of the phenomenon you are referring to, which does occur in the correct GR models of situations like this. What actually happens is that, if a body is more compact than a certain threshold--which is that its Schwarzschild radial coordinate (which is actually a measure of the surface area of the 2-sphere labeled $r$) is 9/8 of the Schwarzschild radius corresponding to its mass--it is impossible for it to be static, because the worldline of the point at the center of the body would have to be null--lightlike--instead of timelike. That is not possible.

So what this phenomenon is actually telling us is that, once a body gets to 9/8 of the Schwarzschild radius corresponding to its mass, there is absolutely no way for it to avoid collapsing into a black hole. In any real scenario, of course, the threshold of unavoidable collapse will come sooner than that.

There is another thing here which is worth pointing out: thinking of "rate of time flow" at all doesn't really work for a collapsing object; it only works for a static object. For a static object, you can imagine standing on its surface and being at rest relative to an observer at infinity, so the two of you can exchange light signals and compare your relative clock rates. That's not possible if you are riding on the surface of a collapsing object, because you aren't at rest relative to the observer at infinity.

 

[DC0]

ρ = 10^{(0.4838 Log P + 1.2372)}

was derived from a graph found at
http://iopscience.iop.org/article/10.1088/0004-637X/773/1/11/meta figure 6, which describes neutron matter.

r _n = ((m_br - m_br(n-1))/( ρ (4/3)π) + r³_(n-1))^(1/3)
is used to feed r into the equation for Φ. It is derived from
ρ = (m_br - m _br(n-1))/(4/3π ( r³_n -r³ _(n-1) ))

My reference point was not any where near the surface of a collapsing object but at a very distant point. It seemed to me that if the rate of time flow relative to this distant point was 0 then the collapsing object would be relatively stopped. In this model, time gets very close to freezing first at the center, and as the remnant contracts, this freeze condition moves almost to the surface. The surface would be just above the Schwarzschild radius but below the 9/8 R_s you mentioned. Here t/T would be about .01, red shifted to where it would be hard to see. This is all relative to the remote point. All calculations are done in a real (not imaginary) condition. t/T approaches 0 but does not cross it. I'm not sure what to do about getting out of flat space time. I'll check the full GR approach reference.

 

[PeterDonis]

r n = ((mbr - mbr(n-1))/( ρ (4/3)π) + r3(n-1))(1/3)
is used to feed r into the equation for Φ. It is derived from
ρ = (mbr - m br(n-1))/(4/3π ( r3n -r3 (n-1) ))

And where does that equation come from?

My reference point was not any where near the surface of a collapsing object but at a very distant point.

Your "reference point" for "zero time dilation", yes. I'm not disputing that. But the point whose time dilation you are calculating with respect to that distant reference point is somewhere on or inside the object. I used the surface because it's a convenient choice, but nothing I said changes if you pick a point deep inside the object instead.

It seemed to me that if the rate of time flow relative to this distant point was 0 then the collapsing object would be relatively stopped.

It might seem that way to you, but it's not correct. What this is actually telling you is what I said in my previous post.

Also, there is another problem which I briefly mentioned at the end of my last post; let me elaborate on it. In flat spacetime, i.e., in the absence of gravity, we can give a well-defined meaning to time dilation due to relative motion, because the concept of "relative velocity" is well-defined for spatially separated objects. That is no longer true in curved spacetime, i.e., in the presence of gravity. So the only time dilation that is well-defined in curved spacetime is time dilation between two observers at rest at different points in a gravitational field. (And even that is only well-defined in a certain limited class of curved spacetimes.) There is no well-defined "time dilation" between a distant reference and a point inside a collapsing massive object.

 

[DC0]

r _n = ((m_br - m_br(n-1))/(ρ (4/3)π) + r³_(n-1))^(1/3)
is used to feed r into the equation for Φ. It is derived from
ρ = (m_br - m_br(n-1))/(4/3π ( r³_n -r³_(n-1) ))

And where does that equation come from?

During the contraction as I was working layer by layer to the surface, I needed an equation, that described the new radius as a function of the new density caused by the increased pressure. The density for this next layer is the mass of that layer (m_brn -m_br(n-1)) divided by the volume of the shell (4/3π (r³_n -r³_(n-1))). Or the mass below that layer minus the mass below the previous layer, divided by the difference of the volume of the two spheres. Then solving for r_n we get the wanted radius which was used in the equations that followed. Concerning my statement “Δr is contracted by the same time contraction value (t/T)”, when watching something fall into a black hole, as it gets close to the event horizon, it flattens out as time freezes. Observing a black hole earlier during its contraction, starting at the center the gravitational potential will get down to -c²/2 where time freezes. At this point, during the contraction, the radius R of the remnant is about 1.75 times the Schwarzschild radius. As the contraction continues, the radius that freezes works its way toward the surface. It seems that this flattening out or relative dimensional contraction would also be true as an object approaches any radius where the time was frozen. Since the gravitational potential at any point is the sum of all the gravitational potentials above, all points below the uppermost frozen point would meet the condition of Φ being less than -c²/2. This would make t/T imaginary if it wasn’t for the freeze of information flow. As the freeze point progresses outward from the center, the freezing of the speed of light and the flow of information on gravitational potential prevent t/T from becoming imaginary. From our remote point of view, the inner frozen points do not receive the information about the outer matter contracting in and changing the gravitational potential. Relative to any frozen point r, the r and Δr dimensions above are greatly increased. This would relatively increase the gravitational potential and prevent (t/T)² from becoming less than 0. Points within the frozen region are relatively protected from becoming imaginary. Would it be possible for you to help me adjust the equations so that they would represent the required curved space?

 

[PeterDonis]

I needed an equation, that described the new radius as a function of the new density caused by the increased pressure. The density for this next layer is the mass of that layer (mbrn -mbr(n-1)) divided by the volume of the shell (4/3π (r3n -r3(n-1))).

All of this is wrong in GR, because you can't assume that the geometry of space is Euclidean.

The way this is usually done in GR, for a static object, is to assume spherical symmetry and define the radial coordinate $r$ such that the surface area of a 2-sphere labeled $r$ is $4 \pi r^2$. Then adding on a thin shell with coordinate thickness $dr$ adds mass $dm = 4 \pi r^2 \rho dr$, where $\rho$ is the density. But $r$ is not the physical radius of the shell; it's just a coordinate, a label. To do this you have to know the density $\rho(r)$ as a function of $r$; in other words, that's an initial input to the model. If you have that, you can figure out everything else by solving the Einstein Field Equation.

For a collapsing object, one way this can be done in GR is to first start with the static initial state of the object, and label shells with a radial coordinate $r$ as above. Then you keep the same label for each shell as it collapses. That means, of course, that once the collapse starts, the surface area of a shell labeled $r$ will no longer be $4 \pi r^2$; it will be smaller.

Another way to handle the collapse case is to keep the meaning of $r$ the same, and find a function $r(\tau)$ for each shell that gives its radial coordinate (i.e., its surface area) as a function of its proper time $\tau$.

when watching something fall into a black hole, as it gets close to the event horizon, it flattens out as time freezes. Observing a black hole earlier during its contraction, starting at the center the gravitational potential will get down to -c2/2 where time freezes. At this point, during the contraction, the radius R of the remnant is about 1.75 times the Schwarzschild radius. As the contraction continues, the radius that freezes works its way toward the surface. It seems that this flattening out or relative dimensional contraction would also be true as an object approaches any radius where the time was frozen.

All of this is wrong. I explained why in an earlier post. The "freeze" does not mean what you think it means.

Since the gravitational potential at any point is the sum of all the gravitational potentials above

Where are you getting that from?

Would it be possible for you to help me adjust the equations so that they would represent the required curved space?

No. This problem is much too complex for a PF thread, and it also appears to be beyond your current level of understanding. I recommend taking some time to work through a GR textbook treatment of this type of problem; MTW, for example, has a good discussion of the 1939 Oppenheimer-Snyder model of gravitational collapse, which is the simplest model of this type.

 

[DC0]

Φ = - GM/R + Σ_R^r ΔΦ = - GM/R - m_br G Δr / r²

Since the gravitational potential at any point is the sum of all the gravitational potentials above

Where are you getting that from?

The equation was miss stated. It should have been
Φ = - GM/R + Σ_R^r ΔΦ = - GM/R - Σ_R^r m_br G Δr / r²
Simply put, the gravitational potential at any radius r within the star remnant is the gravitational potential at the surface plus all the incremental gravitational potential decreases down to that radius.

In the model, each shell always maintained the same mass through out the process and the shell thickness Δr was adjusted to establish the density which was a function of pressure. The shell volumes were calculated using (4/3π ( r³_n -r³_(n-1) )) rather than (4πr²_nΔr ) to improve accuracy.

 

[PeterDonis]

the gravitational potential at any radius r within the star remnant is the gravitational potential at the surface plus all the incremental gravitational potential decreases down to that radius.

Ok, that general rule is fine for a static object. (I'm not sure your specific expression for "all the incremental gravitational potential decreases down to that radius" is correct, but I don't have time to dig into that in detail right now.) I'm not sure it works for a collapsing object, though, for the reason I gave at the end of post #11: the concept of "gravitational time dilation" between two observers or objects assumes that they are at rest relative to each other, and if an object is collapsing, it isn't at rest relative to an observer/object at infinity, which is the "standard" against which the time dilation, or gravitational potential, is defined.

The shell volumes were calculated using (4/3π ( r3n -r3(n-1) )) rather than (4πr2nΔr ) to improve accuracy.

This assumes that the geometry of space is Euclidean. It isn't. I've already mentioned that.

 

[Dale]

This assumes that the geometry of space is Euclidean. It isn't.

This is also my primary objection to the calculations. They are mostly Euclidean, with a strange hodgepodge of GR concepts thrown in or discarded for no apparent reason.

@DC0 the best approach is to simply discard this Frankenstein approach and just do the calculations the right way.

 

[DC0]

the term "relative time freeze" is not a correct description of the phenomenon you are referring to, which does occur in the correct GR models of situations like this.

in #9

Can you describe this? "relative time freeze" does occur in the correct GR models of situations like this.

Thanks

 

[PeterDonis]

"relative time freeze" does occur in the correct GR models of situations like this.

No, it doesn't. What occurs is that a curve of constant "position" becomes null instead of timelike. (I put "position" in quotes because that term implies that the curve is timelike.)

 

[DC0]

What occurs is that a curve of constant "position" becomes null instead of timelike. (I put "position" in quotes because that term implies that the curve is timelike.)

I assume that this is because “timelike” and “position” reverse roles as you cross the event horizon to where things are imaginary. I may be repeating something you have already corrected but I need to make sure. In my model while the radius R of the remnant is about 1.75 times the Schwarzschild radius, the gravitational potential at the center begins to meet the condition to cause time to freeze. This freeze happens just on the outside of this freeze radius where conditions are still real. This is the start of a boundary between the real and imagery. As the remnant contracts and the shell surface that freezes moves out, the freezing surface remains on the real side of this expanding boundary. If this expanding freeze radius were to make it to the surface, it would then meet the mathematical conditions of being the Schwarzschild radius.

You are right about the radius maybe being used as a label because my model shows that at any point after being frozen, the sum of the shell thicknesses is much less than the radius required to cause the freeze. They do not agree.

 

[Dale]

as you cross the event horizon to where things are imaginary

What “things” are imaginary? Please be precise.

the gravitational potential at the center begins to meet the condition to cause time to freeze

What @PeterDonis is rather patiently and repeatedly pointing out is that “time freezing” is not a correct description of the physical situation.

 

[PeterDonis]

I assume that this is because “timelike” and “position” reverse roles as you cross the event horizon to where things are imaginary.

No.

I may be repeating something you have already corrected

Yes, I have. Several times now.

In my model while the radius R of the remnant is about 1.75 times the Schwarzschild radius, the gravitational potential at the center begins to meet the condition to cause time to freeze.

No, it doesn't. In the standard treatment of a case like this, what happens at a certain radius `#R## for a static object is that the metric coefficient $g_{tt}$ at the center of the object goes to zero. That does not mean "time freezes". It means what I said before, that a curve of constant "position" at the center of the object becomes null, not timelike. That means a piece of matter at that "position" would have to move at the speed of light to stay there. That's not possible. So what this is really telling you is that an object with this radius cannot be static; it must be collapsing.

If the object is collapsing, the worldlines of its pieces of matter are not curves of constant "position" in the coordinates you are using, so you can't deduce anything just from the metric coefficient $g_{tt}$. But that appears to be what you are doing. So whatever you are doing, you don't appear to be doing it correctly.

You are right about the radius maybe being used as a label because my model

Yes, but not for the reason you give. The radius is a coordinate; all coordinates are labels. You can't read off any physics just from coordinate labels. You have to look at invariants. You don't appear to grasp that.

 

[PeterDonis]

At this point I am closing the thread since we are just going in circles. The OP is using incorrect math and incorrect concepts.