On the nature of the "infinite" fall toward the EH

Austin0

I was not suggesting any special significance to BH's or the vicinity of the horizon. I understood Mike_Holland's statements regarding g dilation as being general so took Steves post in the same context..
Are you suggesting that with a system accelerating under thrust we just disregard Born rigidity and acceleration if the system is smaller than the Empire State building???
Joke. ;-)

PAllen

I'm just suggesting that to the extent we assume rigidity in practice for relativity large objects is achievable for modest tidal forces and accelerations, then rigidity to a similar extent may be assumed for an appropriately chosen event horizon (for free fall). [Edit: for a static observer near a horizon, proper acceleration approaches infinity, so rigidity is impossible for a static object. But for free fall, it is easy. Remember, it is the static observer that is analogous to the accelerating observer in SR.]

DaleSpam

I can calculate it explicitly if you like, but it is exceedingly well-founded.

See above. It isn't an assumption. It falls out of the math quite naturally.

No, Achilles' proper time is finite as Zeno coordinate time goes to infinity. I thought that it would be obvious, but apparently it isn't.

Achilles' proper velocity is clearly constant.

Yes, I disagree. I think that the math is so unfamiliar to you that you have a whole bunch of mistaken beliefs about how this works out. To me it is pretty obvious that none of the claims you made in your previous post are correct.

Mike Holland

Very nice in theory, but it doesn't work in practice if you believe that physics is universal. How is that free-fall observer, using Rain coordinates, going to explain that contracting mass that is accelerating towards him while he is motionless? Where are the rockets that are making it accelerate?

Gravity and acceleration may give the same answers, but if there is a heavy mass present, then gravity wins over acceleration.as an explanation - or at least as part of the explanation where both are involved.

PAllen

The principal of equivalence is local. It will show, for example, that within a moderately large capsule in free fall, physics is the same as in 'empty space'. Or that from top to bottom of building on a planetary surface, you have the same behavior as an accelerating rocket. If you go global, it doesn't apply in GR because there are neither extended uniform gravitational fields, nor global inertial frames.

However, the coordinate dependence of gravitational time dilation is shown by something like Lemaitre coordinates, which reproduce all measurements of any other coordinates, but display no gravitational time dilation (in the sense of dilation as a function of position). The key physical point is that measured time dilation is always done between two clocks; or between an emitter and receiver. It is thus a feature of two world lines and signals between them. The measurements are coordinate independent, and gravitational redshift or time dilation are not necessary to compute any such measurement.

Note also that the general way to define clock and redshift comparison between two world lines applies perfectly well to the vicinity of co-orbiting neutron stars. Meanwhile, there is no way to even define gravitational time dilation for such an inherently non-static field.

Gravitational time dilation is a useful concept for static spacetime - but it is not a general feature of GR, and it is never necessary to use. J. L. Synge, in his classic book on general relativity, argued against using it at all - because one universal method may be used in all cases (kinematic, cosmologic, and in strong, non-static geometry) instead.

zonde

I was talking about scientific method, not about math.

You see the problem is that our observations are not expressed as invariants but as coordinate dependant physical quantities instead. So if you want to compare predictions with observations you would have to convert your invariants into coordinate dependant physical quantities.

As far as my understanding of GR goes this is out of scope of GR.

DaleSpam

The math is what the theory uses to make testable predictions for the scientific method. If you do not understand the math then you do not understand the theory well enough to address it with the scientific method. Hence the disagreements.

This is simply false. All experimental measurements are invariants. If they were not invariant then you could always construct a paradox of the form "Dr. Evil builds a bomb which is detonated iff device X measures Y, device X measures Y under coordinate system A, but Z under coordinate system B. Therefore the bomb explodes in one coordinate system but not in the other."

Two different coordinate systems may disagree on the meaning of the measurement, e.g. they may disagree whether or not the rod is accurately measuring length, but they must agree on what value is measured.

OK, so considering all other mainstream physics theories as well. What would prevent the formation of a horizon?

zonde

Stellar blackholes are formed in violent explosions - this is quite close to direct observation.
What you are telling is (mainstream) speculation. The closest thing to something like that as I can imagine is galactic collisions.

You can ask of course. But it does not mean you can get testable answer.
I for example can ask what people experience when they die. So what?


As I see speculations about BHs relies on one important thing that GR takes as a postulate. That is that laws of physics are independent of (Newtonian) gravitational potential. If we assume that this assumption holds without bonds then we have no reason to assume that anything will happen with a clock falling into the hypothetical BH.
But I don't buy the idea about assumptions holding without bonds. And that takes it out of domain of GR.

PAllen

No, galaxies are believed to contain supermassive central black holes, 10 billion or more sun's worth in some cases.
True, but this is not the the only case of physical theories including untestable predictions. To better understand a theory (and its limits), it is useful to understand what a theory predicts for such things. GR + known theories of matter (classically) predict continued collapse. GR must be modified in some way to avoid this.
Fine - you agree that GR must be modified to get the result you want. What you call laws being affected by something like Newtonian potential is a fundamental violation of the principle of equivalence, which is built in (as a local feature) to the math and conceptual foundations of GR. Note, for gravity to be locally equivalent to acceleration, a direct consequence is that free fall must have locally the same physics everywhere. (Otherwise, observing what happens inside a (small) free falling system would locally distinguish gravity from corresponding acceleration.)

pervect

There's growing experimental evidence for the existence of event horizons. Basically, black hole candidates are very black, and don't appear to surface features.

WHen matter falls onto a neutron star, the surface heats up and re-radiates. The spectra signature is rather distinctive, also there are "type 1 x ray bursts".

Black hole candidates do not appear to have any such "surface" features, and it's already very difficult to explain by any means other than an event horizon how they can suck in matter without , apparently re-radiating anything detectable.

For the details, see

See for instance http://arxiv.org/pdf/0903.1105v1.pdf

and check for other papers by Naryan in particular.

Austin0

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Well this whole post of yours is nothing more than a repetitive bald assertion that you are right and I am wrong without content or justification so yes some hint as to the math you are referring to would be appropriate.

Where in the stated parameters :

is the mathematical basis for the derivation of time dilation . I.e. justification of its insertion into a classical scenario???



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DaleSpam

I will work it out in full and post it either later tonight or early tomorrow. I am sorry that it isn't obvious to you from pervect's description, but I think when you are unfamiliar with the math that you would be better served to simply ask for a detailed derivation instead of asserting that well qualified individuals like pervect are wrong or implying that they are acting deceptively.

Nugatory

It sounds as if we still haven't clearly explained the Zeno time analogy... Let's try a different tack...

When we say that there is time dilation between two observers, what are we saying in coordinate-independent terms? We are saying that:
1) There are two points on A's worldline; call them A1 and A2. Call the proper time between them ΔA.
2) There are two points on B's worldline; call them B1 and B2. call the proper time between them ΔB.
3) A claims, using some more or less reasonable definition of "simultaneous", that A1 and B1 are simultaneous and that A2 and B2 are simultaneous.
4) Now A calculates the ratio ΔA/ΔB. If that ratio comes out to be greater than unity, then A says that B's clock is running slow because of time dilation. Obviously this result depends on the simultaneity convention used to choose the endpoints B1 and B2 as well as the metric distance between them on B's worldline.

The standard SR definition of time dilation is contained in this more general definition; you just use the obvious and only sensible simultaneity convention, namely that all events on a line of constant t in a given frame are simultaneous in that frame, and you'll get the SR time dilation formula.

This is the only definition of time dilation that can be made to work in GR, although this is somewhat obscured by the need to do the calculations in SOME coordinate. Note that in GR the choice of simultaneity convention is arbitrary, and that if you cannot draw null geodesics from B1 to A1 and from B2 to A2 there's no reason to even prefer one convention over another.

The same method even applies in a classical scenario (although it is trivial and uninteresting). There's only one possible simultaneity convention, that defined by the Newtonian absolute time, and the ratio ΔA/ΔB always comes out to one, so there's no reason to mess with any of this coordinate-independent description.

But that is the point of the Zeno time analogy. We pick a deliberately absurd time coordinate instead of the obvious Newtonian one; it's so absurd that we cannot assign any time coordinate to event B2 ("the arrow hits the wall"), and then we calculate in this coordinate system that the arrow cannot hit the wall. Of course we know that the arrow does in fact hit the wall, so we know that something is wrong with the coordinate system and that the ratio of zeno time to arrow time is not telling us anything.

And it's the same way with the Schwarzchild time coodinate. The ratio of A's Schwarzchild time coordinate to proper time on B's worldline serves only to mislead. The interesting quantity is the ratio of proper time between any two points on B's world line and any two points on A's worldline.

Austin0

So according to what you are saying here , Pervects stated conditions are to be taken as outside of Newtonian uniform time
so then have an implicit assumption of time dilation. That Achilles clock is running at a different rate and his velocity is constant.
Well of course given these conditions everything else is obvious. But then you have simply rewritten Zeno's paradox completely. Simply stuck Zeno's and Achilles names on the conditions of free fall in Sc coordinates.

And those conditions are not derivable from the stated Zeno time as above ,alone.

Since everyone basically agrees there is no merit in the logic in the classic Zeno argument, then by association and implication anyone considering the possible validity of the Sc case is obviously silly, right???
What other point was there as you simply made the scenarios identical (I.e. completely different from the classic argument).???

If those assumptions had been explicitly stated by Pervect then it would have been quite obvious that Zeno time was explicitly dilated and outside any classical context .

Nugatory

Aaargh... I'm still not being clear enough.... Zeno time is not "explicitly dilated" because it's a coordinate time so doesn't "dilate" - dilation is a statement about the ratio between two amounts of proper time, not coordinate time.

Is there any reason to take the Schwarzchild time coordinate in spacetime more or less seriously than the Zeno time coordinate in classical space?
(IMO the answer is "yes", but for a rather unsatisfying and unfundamental reason - there are some problems that are computationally easier if you choose to work them using the SC time coordinate, while AFAIK there are no interesting problems that are more easily solved by transforming into Zeno coordnates).

pervect

Austin0 was asking for some more detailed math. I'd suggest looking at Caroll's GR lecture notes online.

I'll also add that while Caroll's online notes are perfectly fine (one can't trust everything online, Caroll's online notes are drafts of a book by a physics profesor that was later published. From my POV the main advantage of them is that they're free).

I could also quote similar statements from some of my other GR textbooks, (i.e. Caroll is not an isolated occurrence among textbooks). However, I think it would be better from a pedagogical point of view if interested people went out and found their own textbooks if they don't like Caroll (though I can't think of any valid reason for not liking Caroll).


But - onto Caroll:

These just the words - the actual calculation consists of solving for the trajectory of the worldline. If one does this in the usual format, one doesn't even have to integrate the length of the worldline to get proper time, instead one solves the geodesic equations to find t(tau) and r(tau).

One can then observe directly that the event horizion is reached at a finite tau, even though t(tau) is infinite.

If one is willing to just take the limit as r approaches the event horizon one can do this all in Schwarzschild coordinates. This is even observable. One can say that's it's possible to observe the limiting sequence of proper time as one approaches the event horizon from outside, and observe that the limit is finite.

To go futher and carry the trajectory smoothly through the event horizon, one needs coordinates that are better behaved, which is what Caroll does next.

The point of the Zeno analogy is to demonstrate a simple example of how a coordinate time can be infinite while the time actually measured on a clock is finite.

Specifically, zeno time is infinite, while as far as Achilles is concrned, there's a finite time at which he passes the tortise.

I'm afraid I don't understand the difficulties people are having in understanding the analogy. It could be my fault, sometimes I "leap ahead' too far when I write.

The way you demonstrate that the proper time on an infalling clock is actually finite rigorously is that you calculate it.

Post #12 in this thread
http://www.physicsforums.com/showpos...4&postcount=12

(and a later post after it, #13)

One can see that at tau = -8/3 , which is finite, r=4 so one is at the event horizon. Furthermore, t(tau) is infinite because of one of the ln(...) terms.

To verify this is a solution one needs to demonstrate that said trajectory satisfies the geodesic equations. You'll find them in my post #12, Caroll's GR lecture notes, for starters.

The idea behind the Zeno analogy isn't to "prove" anything - that's what textbooks are for. The idea behind the Zeno analogy is to illustrate how t can be infinite and tau can be finite in a simple, easy-to-understand example.

WEll, the Zeno analogy does prove one thing. It demonstrates that just because you have a time coordinate t going to infinity doesn't prove that something doesn't happen. It's an example of how t going to infinity can be the result of a poor choice of coordinates. It's a counterexample to the argument "t goes to infinity, therefore it can't happen".

Historically, I do believe that the "tortise coordinate" was named after the tortise in Zeno's paradox, but I haven't seen anything really detailed on this in textbooks. There was something in Scientific American about it a long time ago as well, I think.

Austin0

Actually I misspoke. It is Achilles' time which is dilated within the context of Pervects conditions if we add the condition that Achilles' velocity is constant.
I really just meant that time dilation was in effect within the stated conditions and coordinates

And yes I am quite aware of the meaning of dilation and it is the ratio of rates or intervals of two different clocks. In a recent post I made the simple statement that time dilation was inherently relative. Self evidently true for exactly this reason. It is meaningless applied to a single clock. Like the term length contraction or the word faster. It intrinsically requires and implies a comparison.
But somehow I got a bunch of flack from several people telling me I was wrong.
???????????

I am not convinced that Sc coordinates are necessarily preferred or correct. I am still just learning their subtleties and details and trying to synthesize a logically coherent structure up to the horizon. My exception to this analogy was purely logical. You all may be ultimately right about Sc coords and the horizon but this use of Zeno added nothing of logical probative value to the debate and was actually misleading in it's subtle reframing of Zeno.