An Alternative to a singularity?

[Jeffocal]

Are we 100 percent sure that a singularity exists in the center of a black hole.

 

Einstein in his General Theory of Relativity predicted time is dilated or moves slower when exposed to gravitational field than when it is not. Therefore, according to Einstein's theory a gravitational field, if strong enough could stop time.

 

As a star cools and contacts, the gravitational forces at its surface increase as its circumference decrease. We know this because observations of gravitational forces tell us they are proportional to the square root of the stars mass divided by its circumference.

 

In 1915, Karl Schwarzschild discovered that according to Einstein's General Theory of Relativity the gravitational field associated with the mass of a star greater than approximately 2.0 times a solar mass would stop the movement of time if it collapsed to a one-dimension point in space. He also defined the critical circumference or boundary in space around this one-dimensional point where the strength of a gravitational field will result in time being infinitely dilated or slowing to a stop.

 

In other words as a star contacts and its circumference decreases the time dilation on the surface of the mass associated with that stars gravitational field will increase. At a certain point the contraction of that mass will produce a gravitational field strong enough to stop the movement of time. Therefore, critical circumference defined by Karl Schwarzschild is a boundary in space where time stops relative to the space outside of that boundary.

 

This critical circumference is called the event horizon because an event that occurs on the inside of it cannot have any effect on the environment outside of it.

 

Many physicists believe the existence of black holes is an inevitable outcome of Einstein's General Theory of Relativity.

 

However, it can be shown using the concepts developed by Einstein, this may not be true.

 

In Kip S. Thorne book "Black Holes and Time Warps", he describes how in the winter of 1938-39 Robert Oppenheimer and Hartland Snyder computed the details of a stars collapse into a black hole using the concepts of General Relativity. On page 217 he describes what the collapse of a star would look like, form the viewpoint of an external observer who remains at a fixed circumference instead of riding inward with the collapsing stars matter. They realized the collapse of a star as seen from that reference frame would begin just the way every one would expect. "Like a rock dropped from a rooftop the stars surface falls downward (shrinks inward) slowly at first then more and more rapidly. However, according to the relativistic formulas developed by Oppenheimer and Snyder as the star nears its critical circumference the shrinkage would slow to a crawl to the external observer because of the time dilatation associated with the relative velocity of the star's surface with respect to the external observer. The smaller the star gets the more slowly it appears to collapse because the time dilation predicted by Einstein increases as the speed of the contraction increases until it becomes frozen at the critical circumference.

 

However, the time measured by the observer who is riding on the surface of a collapsing star will not be dilated because he or she is moving at the same velocity as the surface of that star.

 

Therefore, the proponents of black holes say the contraction of a star can continue until it becomes a one-dimensional point in space because time has not stopped on its surface even though it has stopped to an observer who in remains at fixed circumference to that star.

 

But one would have to draw a different conclusion if one viewed time dilation in terms of the gravitational field of a collapsing star instead of in terms of the velocity of the contraction.

 

Einstein showed that time is dilated by a gravitational field. Therefore, the time dilation on the surface of a star will increase relative to an external observer as it collapses because, as mentioned earlier gravitational forces at its surface increase as its circumference decrease.

 

This means, as a star nears its critical circumference its shrinkage slows with respect to an external observer who is outside of the gravitation field because the increasing strength of its gravitational field causes a slowing of time on its surface. The smaller the star gets the more slowly it appears to collapse because the gravitational field at its surface increase until time becomes frozen for the external observer at the critical circumference.

 

Therefore, the observations an external observer would make using conceptual concepts of Einstein's theory regarding time dilation caused by the gravitational field of a collapsing star would be identical to those predicted by Robert Oppenheimer and Hartland Snyder in terms of the velocity of its contraction.

 

However, Einstein developed his Theory of General Relativity based on the equivalence of all inertial reframes which he defined as frames that move freely under their own inertia neither "pushed not pulled by any force and therefore continue to move always onward in the same uniform motion as they began".

 

This means that one can view the contraction of a star with respect to the inertial reference frame that, according to Einstein exists in the exact center of the gravitational field of a collapsing star.

 

(Einstein would consider this point an inertial reference frame with respect to the gravitational field of a collapsing star because at that point the gravitational field on one side will be offset by the one on the other side. Therefore, a reference frame that existed at that point would not be pushed or pulled relative to the gravitational field and would move onward with the same motion as that gravitational field.)

 

The surface of collapsing star from the view point of an observer who is at the center of the collapse would look according to the field equations developed by Einstein as if the shrinkage slowed to a crawl as the star near its critical circumference because of the increasing strength of the gravitation field at the surface of the star relative to it's center. The smaller the star gets the more slowly it appears to collapse because the gravitational field at its surface increases until time becomes frozen at the critical circumference.

 

Therefore, because time stops or becomes frozen at the critical circumference for both an observer who is at the center of the clasping mass and one who is at a fixed distance from its surface the contraction cannot continue from either of their perspectives.

 

However, Einstein in his general theory showed that a reference frame that was free falling in a gravitational field could also be considered an inertial reference frame.

 

As mentioned earlier many physicists assume that the mass of a star implodes when it reach the critical circumference. Therefore, the surface of a star and an observer on that surface will be in free fall with respect to the gravitational field of that star when as it passes through its critical circumference.

 

This indicates that point on the surface of an imploding star, according to Einstein's theories could also be considered an inertial reference frame because an observer who is on the riding on the surface of an imploding star will not experience the gravitational forces of the collapsing star.

 

However, according to the principals of Relativity he will observe the differential gravitational forces caused by an imploding mass with respect to someone who remains at a fixed circumference or is at the center of the collapsing mass. But according to the Einstein theory of relativity, as a star nears its critical circumference an observer who is on the stars surface will perceive the differential magnitude of the gravitational field relative to an observer who is in an external reference frame to be increasing. Therefore, he or she will perceive time as slowing to a crawl with respect to those reference frames that are not on its surface as it approaches the critical circumference. The smaller the star gets the more slowly time appears to move with respect to an external reference frame until it becomes frozen at the critical circumference.

 

However, the contraction of a stars surface must be measured with respect to the external reference frames in which it is contracting. But as mentioned earlier Einstein's theories indicate time would become infinitely dilated or stop in the reference frames that were not on the surface of a collapsing star as it nears its critical circumference. Therefore, because motion is not possible in a reference frame or an environment where time has stopped, the collapse of a star's surface cannot continue beyond the critical circumference.

 

This contradicts the assumption made by many that the implosion would continue for an observer who was riding on its surface.

 

Therefore, based on the conceptual principles of Einstein's theories relating to time dilation caused by a gravitational field a collapsing star must maintain a minimum volume which is equal to of greater than the critical circumference defined by Karl Schwarzschild and cannot implode to a one dimensional point or a singularity as many physicists believe.

 

This means either the conceptual ideas developed by Einstein are incorrect or the field equations many physicists used to predict the existence of a singularity are incomplete because the theoretical predications regarding its existence are contradictory.

 

Only observations can determine which one is correct because both are based on the validity of the concepts presented in Einstein's theories and the mathematical equations he developed.

 

Source: The Imagineers chronicals

[modest]

Hi Jeff, welcome to Hypography.

 

You can't tell the difference gravitationally between a point mass (a singularity in this case) and a spherically symmetric ball from a point outside the ball. This is true not only in Schwarzschild's metric, but GR in general. It's called Birkhoff's theorem. So, from outside a black hole (assuming the mass is spherically symmetric) it's impossible to determine if there is a singularity versus some lower density configuration of the matter.

 

Schwarzchild's metric does not predict a singularity. Because the metric is static (it only works for observers who are not moving) it breaks down at the event horizon—the point where things can no longer be static. It's possible to change up the metric a bit and describe the interior of the horizon which is done with Kruskal–Szekeres coordinates which then show that inside the horizon all future paths lead to a true singularity (rather than the coordinate singularity in Schwarzchild's metric).

 

The surface of collapsing star from the view point of an observer who is at the center of the collapse would look according to the field equations developed by Einstein as if the shrinkage slowed to a crawl as the star near its critical circumference because of the increasing strength of the gravitation field at the surface of the star relative to it's center. The smaller the star gets the more slowly it appears to collapse because the gravitational field at its surface increases until time becomes frozen at the critical circumference.

No, the field is strongest (greatest gravitational potential) at the center of the mass. Gravitational time dilation is proportional to gravitational potential where potential is taken to be positive, greatest at the center of the object, and decreases to zero at infinity. With smaller potential comes faster clocks. You can use the following where U is potential:

 

[math]T = \frac{T_0}{\sqrt{1-U/c^2}}[/math]

 

So clocks will tick faster on the surface of the collapsing star as considered from the center.

 

There's a good page on this topic (the geometry inside a black hole) here:

 

Schwarzschild Geometry

 

~modest

[Jeffocal]

You can't tell the difference gravitationally between a point mass (a singularity in this case) and a spherically symmetric ball from a point outside the ball.

~modest

 

I believe the contradictions mentioned in the origin article would not only apply to the size of mass or "ball" but also on its ability to pass through what physicist call an event horizon.

 

Therefore, the theoretical predications regarding existence of an event horizon would also be contradictory based on the conceptual principles of Einstein's theories relating to time dilation caused by a gravitational field because the arguments presented in the origin article indicate that a collapsing star must maintain a minimum volume which is equal to of greater than the critical circumference or event horizon defined by Karl Schwarzschild.

[modest]

I believe the contradictions mentioned in the origin article would not only apply to the size of mass or "ball" but also on its ability to pass through what physicist call an event horizon.

 

Did you read the link on Birkhoff's theorem? From outside a spherically symmetric mass (like a star) there are no gravitational effects if the mass were to shrink or grow. It doesn't matter gravitationally if the mass is that of a low-density star or an infinitely dense singularity. If a person outside a black hole wanted to consider a black hole as a shell of mass where all the mass is stuck at the event horizon never being able to pass through because of time dilation then that's fine. By Birkhoff's theorem there is no way of distinguishing that from a singularity.

 

So, what is the contradiction? The person outside sees objects falling in a black hole become more and more time dilated and more and more redshifted until the object gets infinity close to the horizon and disappears (reaching infinite redshift). What does that contradict?

 

~modest

[Jeffocal]

Did you read the link on Birkhoff's theorem?

~modest

Yes I did however, I believe that Birkhoff's theorem contradicts the basic concepts of Relativity because as was mentioned in the original article I think relativity allows one to define the time dilation associated with the gravitational forces on the surface of collapsing mass in terms of an inertial reference frame at its center.

 

"Someone who was on that surface would the observe the differential gravitational forces caused by an imploding mass with respect to someone who remains at a fixed circumference or is at the center of the collapsing mass. But according to the Einstein theory of relativity, as a star nears its critical circumference an observer who is on the stars surface will perceive the differential magnitude of the gravitational field relative to an observer who is in an external reference frame to be increasing. Therefore, he or she will perceive time as slowing to a crawl with respect to those reference frames that are not on its surface as it approaches the critical circumference. The smaller the star gets the more slowly time appears to move with respect to an external reference frame until it becomes frozen at the critical circumference.

 

However, the contraction of a stars surface must be measured with respect to the external reference frames in which it is contracting. But as mentioned earlier Einstein's theories indicate time would become infinitely dilated or stop in the reference frames that were not on the surface of a collapsing star as it nears its critical circumference. Therefore, because motion is not possible in a reference frame or an environment where time has stopped, the collapse of a star's surface cannot continue beyond the critical circumference."

 

I believe this contradicts the assumption made by many that the implosion would continue for an observer who was riding on its surface.

[Pluto]

G'day from the land of ozzzzzz

 

With this link I'm not trying to prove a point, I just thought that this paper is interesting.

 

[0904.3520] Time is not the problem

Time is not the problem

 

Authors: Olaf Dreyer

(Submitted on 22 Apr 2009)

 

Abstract: Attempts to quantize general relativity encounter an odd problem. The Hamiltonian that normally generates time evolution vanishes in the case of general relativity as a result of diffeomorphism invariance. The theory seems to be saying that time does not exist. The most obvious feature of our world, namely that time seems to progress and that the world changes accordingly becomes a problem in this presumably fundamental theory. This is called the problem of time. In this essay we argue that this problem is the result of an unphysical idealization. We are caught in this "problem of time" trap because we took a wrong turn in the early days of relativity by permanently including a split of geometry and matter into our physical theories. We show that another possibility exists that circumvents the problem of time and also sheds new light on other problems like the cosmological constant problem and the horizon problem in early universe cosmology.

[Jeffocal]

The problem may not be with time but with the relativistic concept of a space-time dimension.

 

We agree that One of the most persistent observations regarding time is that it is not perceive as matter or space but as an irreversible physical, chemical, and biological change in physical space.

 

Why then do physicist's assume that it can interact with the physical properties of space when there is no observational evidence for that assumption.

 

Defining time only in terms of a measure of the sequential ordering of the causality of an event would provide an unambiguous definition of time that is more consistent with both physical and mathematical observations than defining it in terms of the physical properties of a dimension.

 

Source: the Imagineer's Chronicles

[Pluto]

G'day from the land of ozzzzzzz

 

Time cannot interact, it measures the interaction.

[Jeffocal]

The fact that time is only measure an interaction may be a clue as to why "Attempts to quantize general relativity encounter an odd problem" because a measurement by definition is continuous.

 

Maybe instead of trying to quantize relativity we should try to relativize the quantum properties of mass and energy because a continuous medium can not be made up of quantized parts but can be divided into them.

 

Source: The Imagineer's Chronicles

[modest]

Yes I did however, I believe that Birkhoff's theorem contradicts the basic concepts of Relativity

 

It is derived directly from relativity.

 

because as was mentioned in the original article I think relativity allows one to define the time dilation associated with the gravitational forces on the surface of collapsing mass in terms of an inertial reference frame at its center.

 

Well, you don't define time dilation, you solve for it. Nonetheless, yes you can solve the difference in proper time between the center and the collapsing surface. The equation I gave earlier would work fine for that (although it is a first order approximation). Where the OP goes wrong is thinking that the field is stronger on the surface of the collapsing star than at the center and therefore clocks tick slower on the surface vs. the center. The opposite is true. Clocks at the center of a collapsing star would tick slower than clocks on the surface, and clocks at the surface would again tick slower than clocks further out. There's no contradiction in that.

 

~modest

[Jeffocal]

Clocks at the center of a collapsing star would tick slower than clocks on the surface, and clocks at the surface would again tick slower than clocks further out.

~modest

 

I disagree with your conclusion that the clocks would tick slower on a surface of a collapsing star than the ones further out because the gravitational force and therefore the time dilation would be strongest at it' surface. Both the clocks*at the center and at an infinite distance would slow to a complete stop relative to the surface of a mass as it collapsed through its event horizon because relativity tells us that the differential gravitational forces they would experience at those points in space would be infinite.

 

However, the contraction is not related to the time dilation between the reference frames on its surface and an external one. It is related to the fact that Einstein's theories indicate that the contraction of a stars surface must be measured relative to the external reference frames in which it is contracting. But as mentioned earlier Einstein's theories indicate time would become infinitely dilated or stop in the reference frames that were at its center or at an infinite from a collapsing star as it nears its critical circumference. Therefore, because motion is not possible in a reference frame or an environment where time has stopped, the collapse of a star's surface cannot continue beyond the critical circumference.

 

This contradicts the assumption that a star can continue to collapse beyond its critical circumference to form an event horizon.

 

A more serious and what I believe to be a fatal flaw in the argument for the existence of an event horizon is the fact that the laws of physics for those on a surface of a collapsing star as it passed through an event horizon would be different than for those that are not. The laws of motion would not apply to either of thoes reference fames relative to the other because each would view time in the others to have stopped.

[modest]

I disagree with your conclusion that the clocks would tick slower on a surface of a collapsing star than the ones further out because the gravitational force and therefore the time dilation would be strongest at it' surface.

The field strength is stronger on the surface than further out. A stronger field means more dilation (as in: slower clock). Clocks on the surface (where the field is stronger) are then slower than clocks further out (where the field is weaker).

 

EDIT: Sorry, it's not the strength of the field that determines gravitational time dilation, but the value of gravitational potential. Clocks on the surface (where potential is less [considering it as negative]) are then slower than clocks further out (where potential is greater).

 

Both the clocks*at the center and at an infinite distance would slow to a complete stop relative to the surface of a mass as it collapsed through its event horizon because relativity tells us that the differential gravitational forces they would experience at those points in space would be infinite.

 

Your previous quote disagreed that clocks on the surface are slower than clocks further out and your quote here says clocks on the surface do tick slower than clocks further out. :confused:

 

Some specific numbers are calculated by 2 different methods in this post:

Some numberic examples of gravitational time dilation in and arround Sol

And this one:

Re: Some numberic examples of gravitational time dilation in and arround Sol

 

You can also think of time dilation as the inverse of redshift (wavelength being the inverse of frequency where photons make great clocks). As a photon travels from the center of a collapsing mass to the surface it climbs out of a gravity well and is redshifted. This means the frequency of the photon is greater at the center and less at the surface... time ticks slower (as revealed by the photon) at the center vs. the surface.

 

Any time you climb uphill fighting the force of gravity your proper time gets larger relative to a clock which you leave behind. This is a fundamental aspect of general relativity by the equivalence principle. If you consider a rocket accelerating then clocks on the nose of the rocket tick faster than on the floor (by the engine). For the person on the floor of the rocket to reach the nose he must climb uphill against the force of acceleration.

 

To think of this graphically (and rather simply) you can plot the gravitational potential of a homogeneous ball:

-

source

Points on the curve that have a steeper slope have clocks that run slower than points on the curve which have less slope. The "strength of the field" *is* the slope of the curve of potential. Edit: Again, sorry, Time dilation is a function of potential not field strength! So, the further down on that plot the slower clocks run. The slope of the curve has nothing to do with it but rather how far up and down the curve it is. Sorry :eek:

 

So, yes, clocks at the center of a mass where the field is strongest run slower than clocks on the surface where the field is weaker. EDIT: and, again, it's not the strength of the field but the value of gravitational potential. This is most easily shown mathematically with the equation for time dilation as a function of gravitational potential (U):

[math]T = \frac{T_0}{\sqrt{1-U/c^2}}[/math]

where U is:

[math]U = \frac{GM}{2a^3}(3a^2-r^2)[/math]

-

source

You might also notice the first sentence in the wiki for gravitational time dilation:

Gravitational time dilation is the effect of time passing at different rates in regions of different gravitational potential; the lower the gravitational potential [where it is taken to be negative] (closer to the center of a massive object), the more slowly clocks run.

I don't know what else to do to convince you of this. Do you have a link explaining or agreeing with what you're thinking?

 

~modest

[Jeffocal]

 

 

Your previous quote disagreed that clocks on the surface are slower than clocks further out and your quote here says clocks on the surface do tick slower than clocks further out. :)

 

~modest

Ya I guess I am a bit :eek: but hope we can work through it :)

 

However if you agree with my other statement that

 

Both the clocks*at the center and at an infinite distance would slow to a complete stop relative to the surface of a mass as it collapsed through its event horizon because relativity tells us that the differential gravitational forces they would experience at those points in space would be infinite.

 

Then the arguments I have presented are still valid. Could you please elaborate on any logical inconsistency which may invalidate them. Because if they are logically consistent within the confines of relativity then the assumption made by physicist that an event horizon is form when a star collapse is invalid.

[modest]

Ya I guess I am a bit :) but hope we can work through it :eek:

 

That's a good approach :)

 

However if you agree with my other statement that

 

Both the clocks*at the center and at an infinite distance would slow to a complete stop relative to the surface of a mass as it collapsed through its event horizon because relativity tells us that the differential gravitational forces they would experience at those points in space would be infinite.

 

No, I would disagree. The person on the surface of the collapsing star should see clocks at the center as time dilated and running slower than his own. Looking at clocks further out away from the star he would find them running faster than his own. The further a clock is from the center of the collapsing star, the faster it ticks.

 

~modest

[modest]

Jeff, I'm sorry, I've been moving through this thread on autopilot. I've been saying that gravitational time dilation is greater or less depending on the "strength of the field". That's not true at all. It is directly proportional to gravitational potential regardless of field strength.

 

The force of gravity is change in potential. If you're in an area of space where potential changes quickly then a person will feel a large gravitational force. As a person gets farther away from a mass the gravitational potential and the field strength both change, but it is most certainly the value of potential which determines time dilation.

 

It was probably very confusing that I mixed those up. I see now what you were saying that a person in the center of the mass would feel no gravitational force—the field strength would be zero while the field strength would be very large on the surface. If time dilation were a function of field strength then you'd most certainly be correct in your conclusions.

 

But, time dilation is a function of gravitational potential and the person at the center of the star has greater potential [where potential (U) is considered positive] than the person on the surface. The farther from the center of the star, the less the potential and the slower clocks run. EDIT: "...the faster they run." :) I can't keep my clocks straight!

 

So, I think that was the problem. You were thinking that time dilation was a function of field strength (the force of gravity) and I was saying the same without thinking about it—it's most certainly not the case.

 

~modest

[Jeffocal]

 

But, time dilation is a function of gravitational potential and the person at the center of the star has greater potential [where potential (U) is considered positive] than the person on the surface. The further from the center of the star, the less the potential and the slower clocks run.

 

~modest

 

Thanks :)

 

However you will have to explain to me using only the concepts of relativity why the center of a star is consider to have greater gravitational potential than its surface because until then I will stick to the conclusion both the clocks*at the center and at an infinite distance would slow to a complete stop relative to the surface of a mass as it collapsed through its event horizon because relativity tells us that the differential gravitational forces they would experience at those points in space would be infinite.

 

Please try to explain it to me in terms of the relativistic properties the star internal gravitational potential not relative to anything outside of it.

Thanks Jeff

 

Source: The Imagineer’s Chronicles

[modest]

Thanks ;)

 

However you will have to explain to me using only the concepts of relativity why the center of a star is consider to have greater gravitational potential than its surface because until then I will stick to the conclusion both the clocks*at the center and at an infinite distance would slow to a complete stop relative to the surface of a mass as it collapsed through its event horizon because relativity tells us that the differential gravitational forces they would experience at those points in space would be infinite.

 

Please try to explain it to me in terms of the relativistic properties the star internal gravitational potential not relative to anything outside of it.

 

No problem. First, I should add that I edited that post right before you replied to it. I meant to say:

But, time dilation is a function of gravitational potential and the person at the center of the star has greater potential [where potential (U) is considered positive] than the person on the surface. The further from the center of the star, the less the potential and the

faster

clocks run.

rather than slower.

 

But, yeah, gravitational potential energy can either be negative or positive. In Newtonian mechanics it is considered negative and called phi ([math]\Phi[/math]), or V. In general relativity it is considered positive and called U. It doesn't really matter which way you do it, it's just important to keep straight which sign you're using.

 

Probably the easiest way for me to convince you that potential is greatest at the center of the object and decreases to zero at infinity is to quote a good source:

 

[math]V = -\frac{GM}{2a^3}(3a^2-r^2)[/math]

 

This is the expression of gravitational potential for a point inside solid sphere. The potential at the center of sphere is obtained by putting r = 0,

 

[math]V = -\frac{3GM}{2a}[/math]

 

This may be an unexpected result. The gravitational field strength is zero at the center of a solid sphere, but not the gravitational potential. However, it is entirely possible because gravitational field strength is rate of change in potential, which may be zero as in this case.

 

The plot of gravitational potential for uniform solid sphere is shown here as we move away from the center.

 

 

Gravitational potential due to rigid body

 

So, if you look at that blue line it is most negative (in general relativity we would say it is "most positive") at r=0, the center of the solid sphere. That's the point where time dilation would be greatest and clocks would run slowest. The higher up the blue line goes on the graph the less time dilation we have and the faster clocks run.

 

The field strength which is the slope of that blue line at any given point is not the same as the potential. The value of potential at a point on the line is given by how far up and down on the graph the point is. In particular, at r=0 (the center of the sphere) the slope is zero meaning the field strength is zero, but the potential is not zero.

 

You'll find that result is fully derived on that web page: Gravitational potential due to rigid body. We would say, then, that gravitational potential is zero at infinity and increases as one gets closer to a massive body. It continues to increase inside the body and is largest at the center of the mass. The larger the potential, the slower clocks tick.

 

~modest