Black holes and time dilation around the event horizon

[GODISMYSHADOW]

It's quite possible (and, I think, usual) to describe all aspects of black holes without the use of complex numbers.

(By the way, electrical and electronic engineers frequently describe alternating current and electronic circuits using complex numbers. I suppose that casts doubts on the reality of electronics?)

Electronic engineers call it the "j" operator (so as not to confuse it with current). It's trig intensive and they use Euler's formula and De Moivre's theorem along with phasors to make calculations easier. It's a means to an end.

 

[Q-reeus]

So there was no attempt to answer my queries in #24. Sweet - no-one here is under compulsion to always respond; still it would have been a nice courtesy. Just maybe there was an oblique response by way of comments in #26 and #28. In #26, PAllen wrote:

"...in GR have coordinate independent definitions. In this case, the observation is round trip radar interval as a function of proper time of of the 'stationary lab'. The coordinate interpretation issue is the separation of this into position and travel time. The former is the observable, the latter is pure coordinate convention..."

But unless there is some unspecified mixing of light travel time with planetary orbital travel time here, 'round trip radar interval' IS 'travel time'. Should that not then read 'the separation of this into integrated coordinate length (thus length scale enters) and integrated coordinate time (thus time dilation enters)'? The issue is still there, just masked imo.
And in #28:

"I don't see what this has to do with setting up a coordinate system. I pick one worldline to be a 'time axis'. Then, I define both simultaneity and distance by round trip radar time (for position, I supplement my radar defined distance from origin with angular coordinates). I have unambiguous polar coordinates."

I am unclear as to where these unambiguous polar coordinates are centered, and what the mathematical form of such is.

"Now, if desired, I can switch to a Cartesian style system from the same origin (my chosen origin world line).
In this coordinate system, coordinate distances don't match proper distances for the given simultaneity surfaces. So what? That is true of many coordinates systems, including SC, and isometric SC."

This implies there actually is even one coordinate system where everything, over an extended region of curved spacetime, can correspond to proper values?!

"In these coordinates, geodesics are 'near ellipses' with a bump.
Despite more complex mathematical form, such coordinates actually have some virtues:
- a minimum of interpretation; they just codify direct measurements.
- Radial speed of light everywhere is constant by definition..."

The radial light speed is constant in such - but only radial? In such a 'preferred' coordinate system where radial c is a constant, orbital 'bumps' appear rather than Shapiro time delay. But positing a coordinate invariant c goes against what GR is about surely, and as suggested in #24, the 'bumps' are therefore not a perfectly valid alternative explanation at all, but unphysical artifacts.
Would you care to now answer just this bit from #24:
"Or is the best we can do is refer to a 'quotient' of f = c'/(2*d') (f the redshifted clock frequency, c', d' the 'redshifted' values of light speed, mirror displacement, respectively), with neither c' or d' individually definable?"

 

[pervect]

So there was no attempt to answer my queries in #24.

I'm not quite sure what the question was. I am guessing SC = Schwarzschild coordinates and ISC = Isotropic Schwarzschild coordinates.

Pick either one (or any other coordinate system you like), and you'll get the same results for the observed doppler shift at the top of the tower. You won't even need any notion of "distance in the large" to perform such a calculation. You'll just need the coordinates of both ends of the light clock, and the coordinates of the observer of the tower. For a light clock in an arbitrary orientation, you'll have different doppler shifts from each end, of course. I'm assuming the light clock transmits a signal to the tower "ping" or "ping+timestamp" every time the light pulse it holds makes a round trip, if you imagine the light pulse is short you can imagine that the light clock transmits a signal every time the light pulse is "at that end" of the lightclock.

As far as measuring distances goes, the more-or-less standard prescription, according to Wald (and with some input from George Jones- see https://www.physicsforums.com/showpost.php?p=2933053&postcount=9 ) is as follows:

[add. This discussion, as the title of the thread also indicates, assumes one is talking about "proper distance", in the GR sense, by "distance". This is NOT always the case!

Some common alternatives (besides proper distance in the SR sense, which is confusingly not the same) include Fermi-normal distance (soometiems called fermi distance, but this is an oversimplification) and various generalziations therof. See for instance http://arxiv.org/PS_cache/arxiv/pdf/0901/0901.4465v2.pdf, for a discussion including some references about why sometiemes Fermi-normal coordinates are sometimes considered to have "a physical meaning coming from the principle of equivalence.".

SOrry - continuing on with how you define proper distance:

You slice space-time up into spatial slices of constant time, then use the 4-metric of space-time to induce a 3-metric on your spatial subslice. This is pretty easy if your surfaces of constant time are defined by dt=0, in which case you simply eliminate dt from your line element, turning your 4-element line element (dt, dr, dtheta, dphi) into a 3-element line element (dr, dtheta, dphi).

If your surfaces of constant time don't have the property that dt=0 along them, you have to do more work.

Then you define the distance as, informally, the "length of the shortest path between two points", or more formally as "the greatest lower bound of the lengths of all curves connecting the two points".

So your notion of distance will depend on the details of how you slice up space-time into surfaces of constant time. Fortunately, in the case of a static observer there's a fairly obvious choice of how to do the split, and it corresponds to the easy choice of making dt=0 in Schwarzschild coordinates. The spatial surface defined in this way (dt=0) will be orthogonal to the timelike worldlines of all static observers.

But you don't really need to know any distances-in-the-large to compute the doppler shifts. In fact, it's probably an easier calculation without worrying about distance.

 

[Q-reeus]

I'm not quite sure what the question was. I am guessing SC = Schwarzschild coordinates and ISC = Isotropic Schwarzschild coordinates.

Yes - sorry I probably should not have assumed everyone would know what that shorthand represented.

Some common alternatives (besides proper distance in the SR sense, which is confusingly not the same) include Fermi-normal distance (soometiems called fermi distance, but this is an oversimplification) and various generalziations therof. See for instance http://arxiv.org/PS_cache/arxiv/pdf/...901.4465v2.pdf , for a discussion including some references about why sometiemes Fermi-normal coordinates are sometimes considered to have "a physical meaning coming from the principle of equivalence.".

A look at VI. EXAMPLES OF EXTENDED FERMI COORDINATES - g. The Schwarzschild metric on p11 there, eqn's 50-52 indicates to my layman mind just how complicated even a 'simple' example can become, thanks to an infinity of derivatives in GR, and then there's the parallel transport issue - where I can appreciate ambiguities will arise in general. Despite all that, the simplest case given should not be that difficult an exercise surely - certainly not to a first order calculation in weak gravity anyway.

...Then you define the distance as, informally, the "length of the shortest path between two points", or more formally as "the greatest lower bound of the lengths of all curves connecting the two points".
So your notion of distance will depend on the details of how you slice up space-time into surfaces of constant time. Fortunately, in the case of a static observer there's a fairly obvious choice of how to do the split, and it corresponds to the easy choice of making dt=0 in Schwarzschild coordinates. The spatial surface defined in this way (dt=0) will be orthogonal to the timelike worldlines of all static observers.

Right - which is surely true in the case given. The coordinate prescription for finding just that in that case is what is wanted. But as soon as one attempts that using SC's there is the rebuff "r is just a coordinate, not a length". Which leaves one where to go?

But you don't really need to know any distances-in-the-large to compute the doppler shifts. In fact, it's probably an easier calculation without worrying about distance.

Of course I agree there is no need to think in terms of distance scale - we have a redshift prescription in terms of GM/r. The point was though one should surely be able to derive it by an alternate means that does involve distance scale and light speed. That after all was implied in the bouncing light and mirrors example given by PAllen to illustrate slowing light. My point was without also a handle on distance scale, coordinate light velocity has no properly defined meaning. To reiterate; there is enough problems with non-linearity in GR to make almost any scenario infinitely complex, but if we just take a first-order simplest case, things should not be all that troublesome. However as the OP was looking at EH scenario and thus strong gravity I will not persist here but will look at reviving another thread where certain issues remain unresolved.

 

[pervect]

Right - which is surely true in the case given. The coordinate prescription for finding just that in that case is what is wanted. But as soon as one attempts that using SC's there is the rebuff "r is just a coordinate, not a length". Which leaves one where to go?

The entire purpose of the metric is to convert changes in coordinates (r, theta, phi) into changes in length.

dr isn't a length, until you multiply it by the appropriate metric coefficient.

Once you have the spatial metric (the induced 3-metric I mentioned earlier), you find the length of any curve by integrating dL along the curve, where dL is given by

dL^2 = g_{rr} dr^2 + g_{{\theta}{\theta}} d\theta^2 + g_{{\phi}{\phi}} d\phi^2

Just remember we're using the induced three-metric here, not the space-time metric. Fortunately, for the simple case in question, it's hard to mess this part up because the two are identical, except for the omission of the dt terms. This is true for this specific case, not generally true.

And the distance is just the shortest curve connecting two points (or the greatest lower bound). Since you probably don't want to calculate an infinite number of curves to find the greatest lower bound :-), you use either the calculus of variations, the geodesic equations, or a Lagranian approach to make sure you've chosen the curve that minimizes distance.

 

[G. E. Hunter]

is black hole evaporation in the real universe, or is this just theory? Any actual observations of this phenomenon? Actual observed decreases in size or mass over time?

 

[Drakkith]

is black hole evaporation in the real universe, or is this just theory? Any actual observations of this phenomenon? Actual observed decreases in size or mass over time?

It has not been observed, and unless a black hole is very very small it will actually absorb more energy from the background radiation than it gives off itself, thus gaining mass.

 

[Zula110100100]

Regardless of the validity of any of the statements, assuming they are all true the OP's outcome should not be what is expected, if we take all the supposed mechanisms to work as stated then the mass making up the black hole would not suddenly drop to 0 once the swartzchild radius did, at that point we would be able to see the mass, and the person who was stuck entering the radius we would see hitting that mass.

 

[Q-reeus]

The entire purpose of the metric is to convert changes in coordinates (r, theta, phi) into changes in length.

I made a slip-up using the term 'coordinate' rather than 'parameter' in that quote, but the gist is the same.

dr isn't a length, until you multiply it by the appropriate metric coefficient.
Once you have the spatial metric (the induced 3-metric I mentioned earlier), you find the length of any curve by integrating dL along the curve, where dL is given by
dL^2 = g_{rr} dr^2 + g_{{\theta}{\theta}} d\theta^2 + g_{{\phi}{\phi}} d\phi^2

right, that's how I basically understood it. But in another thread, the g_rr factor, even in that context, was explained as something quite different in meaning to a straight metric operator on dr - i.e. one cannot infer a coordinate dr = dL*(g_rr)^-1/2, but rather a relation of differential volume to differential area. While the latter is apparently a standard interpretation, couldn't see where it came from.

Just remember we're using the induced three-metric here, not the space-time metric.

Not being familiar with a lot of GR jargon, loked up http://en.wikipedia.org/wiki/Induced_metric , which gave me just enough clues that 'induced metric' is somewhat akin to the idea behind 'partial derivatives' - in this case holding t constant (mapping on to a spatial hypersurface or somesuch?).

And the distance is just the shortest curve connecting two points (or the greatest lower bound). Since you probably don't want to calculate an infinite number of curves to find the greatest lower bound :-), you use either the calculus of variations, the geodesic equations, or a Lagranian approach to make sure you've chosen the curve that minimizes distance.

Phew -right about not getting into calculating an infinite number of curves! Can we simplify that even more if one restricts to just measuring along a short tangent surface geodesic r*sinθ*dϕ, or radial displacement dr - assuming exterior Schwarzschild metric where I take it g_θθ = g_ϕϕ = 1 applies?