Oppenheimer-Snyder model of star collapse

[Mike Holland]

I have trouble imagining the Krauss quantum phenomena in the case of PAllen's trillion star contractring cluster. Surely in this case an event horizon would form long before any quantum radiation is emittted. The stars are still well separated when the black hole forms!

Mike

 

[PeterDonis]

A time code is emitted from Earth that can be received by Voyager. Voyager emits its proper time code s1 that is sent back to Earth together with the then received time stamp t1 from Earth (we'll ignore the technical difficulties).

An observer on Earth with the name Kraus calculates the expected (s1,t1) signal from Voyager as function of expected UTC, for the approximation or assumption that the black hole is completely formed.

Ok, just to make sure I understand:

- Earth emits a signal time stamped with the time t1 of emission according to Earth clocks.

- Voyager receives the signal, and emits a return signal time stamped with the time s1 of emission according to Voyager's clock, plus the Earth emission timestamp t1 of the Earth signal just received.

- Earth wants to predict the (s1, t1) pairs that it will receive in Voyager's return signal, as a function of the time UTC that it receives the return signal.

UTC , (s1 , t1)
--------------
100 , 40.3, 200
1E3 , 41.2, 1.5E3
1E4 , 41.5, 1E5
1E5 , 41.7, 1E7
1E6 , 41.9, 1E10
1E100 42.0, 1E1000

Assuming my understanding above is correct, the first and last columns are wrong as given. The last column is reasonable as a set of "UTC" values; the first column isn't usable at all as given.

A correct set of numbers would look something like this (I haven't calculated these numbers exactly, I've just tried to give a fair approximation of the qualitative behavior):

t1, s1, UTC
-------------
40, 40.3, 200
40.5, 41.2, 1.5E3
40.7, , 41.5, 1E5
40.8. , 41.7, 1E7
40.9, 41.9, 1E10
40.99, 41.99, 1E1000
(...)
41, 42, (Earth never receives any return signal from here on)
41.3, 43
41.6, 44
41.8, 45
42, 46
42.2, 47
42.3, 48
42.300001, (Voyager never receives any Earth signal from here on, it is destroyed in the singularity at tau = 48)

 

[PeterDonis]

To help make sense of the numbers in my last post, attached is a Kruskal-type plot of the scenario. (I made it using fooplot.com, which seems like a neat if simple online tool for generating plots.)

Quick description of the plot:

- The horizontal and vertical axes are the Kruskal U and V coordinate axes.

- The black hyperbola at the top is the singularity at r = 0.

- The crossing 45 degree gray lines are the horizon (up and to the right) and the antihorizon (up and to the left). In a more realistic model where the black hole was formed by the collapse of a massive object, the antihorizon would not be there; instead, there would be the surface of the collapsing object on the left as in the diagram DrGreg posted some time ago.

- The blue hyperbola on the right is the Earth's worldline.

- The dark red curve that leaves Earth at U = 0 (i.e., just as Earth crosses the horizontal axis--this is also t = 0 on Earth's clock) is Voyager's worldline; Voyager leaves Earth and falls into the hole.

- The three progressively darker green lines, running from Earth up and to the left towards Voyager, are three of the light signals emitted from Earth, at Earth times (according to the numbers in my previous post) 40 (more or less--the qualitative behavior is the key here, not the exact numbers), 41, and 42.3. Note what happens to them:

Signal #1 reaches Voyager before it crosses the horizon; Voyager then emits a return signal (the 45 degree line going up and to the right from where #1 reaches Voyager), which reaches Earth further up its worldline, at t = 200 (more or less). You can see that signals emitted in between #1 and #2 from Earth will be received by Voyager closer and closer to the horizon, so Voyager's return signals will reach Earth further and further up its worldline, i.e., at later and later times, increasing without bound.

Signal #2 reaches Voyager just as it crosses the horizon. Voyager's return signal therefore stays at the horizon; it never reaches Earth. Signals emitted from Earth between #2 and #3 will reach Voyager between the horizon and the singularity, so its return signals will stay below the horizon and also never reach Earth (eventually each of these return signals will hit the singularity).

Signal #3 reaches Voyager just as it hits the singularity. Any signal emitted from Earth after #3 will never reach Voyager, because it is destroyed in the singularity; these signals will hit the singularity instead.

 

[harrylin]

Ok, just to make sure I understand:

- Earth emits a signal time stamped with the time t1 of emission according to Earth clocks.

- Voyager receives the signal, and emits a return signal time stamped with the time s1 of emission according to Voyager's clock, plus the Earth emission timestamp t1 of the Earth signal just received.

- Earth wants to predict the (s1, t1) pairs that it will receive in Voyager's return signal, as a function of the time UTC that it receives the return signal.

Assuming my understanding above is correct, the first and last columns are wrong as given.

Oops yes, sorry for the glitch - indeed I swapped the two Earth times in the table.

The last column is reasonable as a set of "UTC" values; the first column isn't usable at all as given.

A correct set of numbers would look something like this (I haven't calculated these numbers exactly, I've just tried to give a fair approximation of the qualitative behavior):

t1, s1, UTC
-------------
40, 40.3, 200
40.5, 41.2, 1.5E3
40.7, , 41.5, 1E5
40.8. , 41.7, 1E7
40.9, 41.9, 1E10
40.99, 41.99, 1E1000
(...)
41, 42, (Earth never receives any return signal from here on) [..]

I suppose that with "from here on" you mean after UTC > 1E10000000000000000000000000000000000000000.
Correct?

The t1 numbers in the beginning are surprising to me; you seem not to account for the ca. 20 light years in "distant" units in your estimated prediction. And/or you assume that the different time dilation factors largely compensate each other.

[Addendum]: in fact I assumed the Voyager to circle for some years in orbit, thus ticking slower; and I suddenly realize that I added instead of subtracted the 20 years - I was in a hurry! What could be relevant for this discussion (although likely also not) is your (t1,s1) = (40.99, 41.99). I don't know how you get that 1 year difference, is that just a coincidence?

Now I'll study the rest; the issue is really (t1,s1)= (41.3, 43).
I do think that Earth must get a signal back (41.3, 41.9999999999) according to O-S-1939.

 

[PeterDonis]

Oops yes, sorry for the glitch - indeed I swapped the two Earth times in the table.

Ok, good.

I suppose that with "from here on" you mean after UTC > 1E100000000000000 - correct?

No, I mean that signals emitted by Voyager at or after s1 = 42 are never received by Earth (because they remain at or inside the horizon). There is no invariant way to relate that to a "time" on Earth's worldline; it depends on which simultaneity convention you choose. Some conventions (like that of standard SC coordinates) don't allow you to assign a "t" coordinate to events on Voyager's worldline with s1 >= 42 at all; no surface of simultaneity in that convention passes through any event on or inside the horizon. Other conventions (like that of Painleve coordinates or Eddington-Finkelstein coordinates) allow you to assign a finite "time" coordinate in those charts to events on or inside the horizon.

The s1 numbers in the beginning don't make sense to me. I accounted (very roughly) for about a factor 2 time dilation due to the high speed of Voyager on its way towards the black hole, aas measured in Schwartzschild time t. I find that time dilation lacking in your estimation. However, that is perhaps not important for this discussion.

I don't think the exact numbers are important (I wasn't trying to get them exact anyway), but the qualitative behavior is. Your t1 numbers were *larger* than your s1 numbers, and your t1 numbers increased very fast (though not as fast as your UTC numbers) as your s1 numbers approached 42. That's wrong. The t1 numbers should be *less* than the s1 numbers, and the t1 numbers should, if anything, grow more slowly than the s1 numbers as the s1 numbers approach 42, because the t1 timestamps are made before the Earth light signals travel inward towards Voyager; that light-speed travel time delay should more than cancel out the time dilation factor due to Voyager's inward motion (though I'm not quite as sure about that last; I'll have to do the calculation when I get a chance). Looking at the diagram I posted may be helpful.

 

[harrylin]

Oops I was still editing my post, trying to reconstruct what went wrong in not -so-important details.

[..] No, I mean that signals emitted by Voyager at or after s1 = 42 are never received by Earth (because they remain at or inside the horizon). There is no invariant way to relate that to a "time" on Earth's worldline; it depends on which simultaneity convention you choose.

I specified that the black hole and solar system are in rest wrt to each other, and that that time convention is used for t. t>∞ is in number simulation indicated as t>1E100000000000000. As a reminder, the O-S model:
"we see that for a fixed value of R as t tends toward infinity, τ tends to a finite limit".
That is also what online simulators find (in fact I now found a nice one in Java. )

your t1 numbers increased very fast (though not as fast as your UTC numbers) as your s1 numbers approached 42. That's wrong. The t1 numbers should be *less* than the s1 numbers, and the t1 numbers should, if anything, grow more slowly than the s1 numbers as the s1 numbers approach 42, because the t1 timestamps are made before the Earth light signals travel inward towards Voyager; that light-speed travel time delay should more than cancel out the time dilation factor due to Voyager's inward motion (though I'm not quite as sure about that last; I'll have to do the calculation when I get a chance). Looking at the diagram I posted may be helpful.

I'm too tired now, it was a long day and I squeezed this example in-between. But yes, you are certainly right about that point (except that I did not assume Voyager to free-fall straight towards the black hole).
The real issue is the last point in my addendum, which was also the intended point of the illustration. To be discussed tomorrow!

 

[PeterDonis]

I specified that the black hole and solar system are in rest wrt to each other, and that that time convention is used for t.

Which is fine for events outside the horizon; but you can't just declare by fiat that those are the only events that exist. If you want to say that, for purposes of your scenario, those are the only events we can consider, then some of the questions you are trying to ask simply do not have answers at all.

(except that I did not assume Voyager to free-fall straight towards the black hole).

That's the simplest assumption from a mathematical standpoint, so it's the one I used. A more complicated assumption would not change the central conclusions, it would just make the calculations more complicated.

I'll comment on your addendum in a separate post.

 

[PeterDonis]

[Addendum]: in fact I assumed the Voyager to circle for some years in orbit, thus ticking slower

Doing that just adds a long period of time where Voyager can exchange light signals with Earth before it falls in. There are no stable orbits inside r = 6M (three times the horizon radius), and no orbits at all, even unstable ones that have to constantly be maintained by rocket thrust, inside r = 3M (1.5 times the horizon radius). Time dilation at those altitudes is not very great by relativisitic standards, and anyway, as I said, the period of orbiting is irrelevant to the central question we're addressing.

What could be relevant for this discussion (although likely also not) is your (t1,s1) = (40.99, 41.99). I don't know how you get that 1 year difference, is that just a coincidence?

As I said, I wasn't calculating exact numbers, just trying to qualitatively describe the general pattern; so if any numbers happen to match something else, it's just a coincidence. I won't have time to do any detailed calculations until after this weekend.

Now I'll study the rest; the issue is really (t1,s1)= (41.3, 43).
I do think that Earth must get a signal back (41.3, 41.9999999999) according to O-S-1939.

O-S 1939 is consistent with everything I said up to (t1, s1, UTC) -> (41, 42, infinity) (qualitatively speaking--as I said, I haven't done detailed calculations of the exact numbers). After that point O-S 1939 doesn't cover the scenario at all; they don't say it's possible and they don't say it's impossible. They simply leave their analysis incomplete. (Their analysis has been completed since--for example, it's in MTW and other GR textbooks--and the completion of the analysis is what I've used to generate the qualitative behavior I illustrated.)

O-S do say, however, that when the surface of the infalling matter reaches the horizon radius (what they call r_0)--this corresponds to Voyager's clock reaching tau = 42--outgoing light can no longer escape (hence the infinity as the limit of the UTC times above as t1, s1 -> 41, 42). This seems like a pretty clear indication that *if* O-S had continued their analysis and discovered that points on Voyager's worldline with tau > 42 could exist, they would find (as modern analyses have found) that those points would not be able to send light signals back to Earth; since if outgoing light can't escape from the event where tau = 42, at r = r_0, any event with tau > 42 must have r < r_0 (since r > r_0 would require Voyager to move faster than light from the tau = 42 event, and even r = r_0 would require Voyager to move at the speed of light from the tau = 42 event), and would also not be able to send signals back to Earth (since those signals would also have to move faster than light).

If you think otherwise, please give specific references from the paper. I've read it through now and what I've said about the model in that paper and its limitations is based on what I've read.

A final note about the 20 light-year distance: that would just add an irrelevant constant to every s1 value and every UTC value. Instead of triples like (40, 40.3, 200), you would get, for example, (40, 40.3 + 20 years, 200 + 20 years); and instead of triples like (40.99, 41.99, 1E1000), you would get, for example, (40.99, 41.99 + 20 years, 1E1000 + 20 years), which works out to a very good approximation to (40.99, 41.99 + 20 years, 1E1000). So the 20 years quickly becomes negligible compared to the huge increase in UTC values compared to the other two.

Rather than add 20 years to the s1 and UTC values as above, I chose to ignore the 20 light year distance and assume that Earth was much closer to the hole. But I can put back in the 20 light year distance when I do the detailed calculations if you think it's really important (I don't think it is, since it doesn't change the qualitative behavior).

 

[harrylin]

[..] As I said, I wasn't calculating exact numbers, just trying to qualitatively describe the general pattern; so if any numbers happen to match something else, it's just a coincidence. I won't have time to do any detailed calculations until after this weekend.

Surely that won't be needed. For general interest for this kind of discussions, the following simulation program that I found yesterday may be handy:

http://www.compadre.org/osp/items/detail.cfm?ID=7232
Put r=7.414 and τ gets to nearly 42 as in my original illustration.

O-S 1939 is consistent with everything I said up to (t1, s1, UTC) -> (41, 42, infinity) (qualitatively speaking--as I said, I haven't done detailed calculations of the exact numbers). After that point O-S 1939 doesn't cover the scenario at all; they don't say it's possible and they don't say it's impossible. They simply leave their analysis incomplete. [..]

Sure. To me their model looks straightforward enough to discuss qualitatively (for high numerical precision we should write a little program). Their model is based on standard stationary space of Einstein's GR that is also used in Schwartzschild's model, right?

[..] *if* O-S had continued their analysis and discovered that points on Voyager's worldline with tau > 42 could exist, they would find (as modern analyses have found) that those points would not be able to send light signals back to Earth

in fact, I cited them as saying just that - see my post #50.

However there was an essential point that I overlooked: in the model of a fully formed black hole Voyager remains in free-fall towards the centre, so that it may be expected to outrun certain radio waves (thanks for pointing that out Atyy!).

Consequently I will almost certainly agree with your calculation about by us observable events - thank you too.

 

[PeterDonis]

For general interest for this kind of discussions, the following simulation program that I found yesterday may be handy:

http://www.compadre.org/osp/items/detail.cfm?ID=7232

This looks cool, thanks for the link!

Their model is based on standard stationary space of Einstein's GR that is also used in Schwartzschild's model, right?

For the portion of the spacetime that is vacuum (i.e., outside the collapsing matter), yes. For the portion of the spacetime that is not vacuum (i.e., inside the collapsing matter), no: that portion of the spacetime is not vacuum (of course), it's stationary (it's collapsing), and the boundary between it and the vacuum region is not stationary either (it's shrinking).

However there was an essential point that I overlooked: in the model of a fully formed black hole Voyager remains in free-fall towards the centre, so that it may be expected to outrun certain radio waves (thanks for pointing that out Atyy!).

Yes, that's reflected in my numbers: in my numbers, Voyager will "outrun" any radio wave emitted by Earth after t1 = 42.3, in the sense that Voyager will hit the singularity before the radio wave reaches it.

 

[harrylin]

This looks cool, thanks for the link!

Yes I also think that it's cool, The orbiter can be repositioned and double-clicking on it gives the energy. Seeing such nice programs encourages me to get back to doing some programming . Regretfully I don't know Java.

Now that I finally got an understanding of the "inside region" arguments, I can zoom in on the real issues - which did not go away. But before continuing I want to make sure of one thing:

[..] For the portion of the spacetime that is vacuum (i.e., outside the collapsing matter), yes. For the portion of the spacetime that is not vacuum (i.e., inside the collapsing matter), no: that portion of the spacetime is not vacuum (of course), it's [not]stationary (it's collapsing), and the boundary between it and the vacuum region is not stationary either (it's shrinking). [..] .

I think that you misunderstood. What I meant is that O-S are developing further Schwartzschild's model, which uses stationary space coordinates. That is consistent with Einstein's 1905 purpose ("the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place").
Like me, you seem to relate the motion of matter with respect to such a reference system in which space does not have a velocity vector; and my impression is that the O-S model that they presented is consistent with that.

 

[PeterDonis]

I think that you misunderstood. What I meant is that O-S are developing further Schwartzschild's model, which uses stationary space coordinates.

They use these coordinates in the first part of the paper; but in the second part of the paper they use different coordinates, ones which are comoving with the collapsing matter.

However, I wasn't making a statement about coordinates; I was making a statement about physics. The original Schwarzschild model was of a spacetime that is entirely static--nothing changes with time. The O-S model is of a spacetime that is only partially static; the region containing the collapsing matter is not static, it changes with time, and so does the radius of its boundary with the vacuum region. So if I am at a certain radius that is greater than the radius r_0 (what we would now call the horizon radius), the metric in my vicinity only becomes static once the collapsing matter falls past me to a smaller radius. That's true regardless of what coordinates I use.

That is consistent with Einstein's 1905 purpose ("the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place").

I don't have any particular problem with this, but I don't see how it's relevant to what we're discussing here. A coordinate system that is comoving with the collapsing matter doesn't have to "assign a velocity-vector to a point of the empty space", any more than a stationary coordinate system does.

 

[harrylin]

Atyy gave a for me useful reference about a nearly equivalent system with accelerating rockets, http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html#FREEFALL

The interesting phrase for me is:
"Eve could claim that Adam never reaches the horizon as far as she's concerned. However, not only is it clear that Adam really does cross the horizon".

I agree with that, but it appears for different reasons than some others.

In fact, according to 1916 GR, Eve's point of view is equally valid as that of Adam; according to that, acceleration and gravitation are just as "relative" as velocity, and their coordinate systems are valid GR systems.
However, the interpretation of what "really" happens is very different, even qualitatively; and in modern GR many people reject "induced gravitation" and agree that we can discern the difference between gravitation and acceleration.

We thus distinguish in that example that Eve's acceleration is real, and that her gravitational field is only apparent because the effect is not caused by the nearby presence of matter. For that reason I think that we should prefer Adam's interpretation. Similarly, in case of a real gravitational field that we ascribe to the presence of matter, it is Eve's interpretation that we should prefer.

Now, it is still not clear to me if O-S used what Einstein called a Gaussian coordinate system, or if they fitted two such systems together that correspond to the same interpretation, or with conflicting interpretations. So, I want to make sure that their model is self-consistent. I guess that it is; the only difference between their inner and outer region modelling is the presence of matter - correct?

 

[Dale]

However, the interpretation of what "really" happens is very different, even qualitatively; and in modern GR many people reject "induced gravitation" and agree that we can discern the difference between gravitation and acceleration.

We thus distinguish in that example that Eve's acceleration is real, and that her gravitational field is only apparent because the effect is not caused by the nearby presence of matter. For that reason I think that we should prefer Adam's interpretation. Similarly, in case of a real gravitational field that we ascribe to the presence of matter, it is Eve's interpretation that we should prefer.

The problem with this whole statement is not the different perspectives or interpretations, but the ambiguous term "real" which makes both paragraphs rather handwavy. What definition of "real" are you using, and can you demonstrate that Eve's acceleration indeed qualifies as "real" under that definition?

 

[PeterDonis]

In fact, according to 1916 GR, Eve's point of view is equally valid as that of Adam; according to that, acceleration and gravitation are just as "relative" as velocity, and their coordinate systems are valid GR systems.

For the region of spacetime that both coordinate systems cover, yes, this is true. However, if Adam's coordinate system covers a portion of spacetime that Eve's does not (in the scenario on Egan's web page, Adam's coordinates cover the entire spacetime, but Eve's only cover the wedge to the right of the horizon), then Eve's "point of view" will be limited in a way that Adam's is not.

in modern GR many people reject "induced gravitation" and agree that we can discern the difference between gravitation and acceleration.

References, please? In "modern GR", people recognize that the word "gravitation" can refer to multiple things. If it refers to "acceleration due to gravity", then "modern GR" agrees with "1916 GR" that "gravitation" can be turned into "acceleration" by changing coordinates, so both are "relative" in that sense. I don't know of anyone in "modern GR" who claims we can distinguish between "gravitation" in this particular sense and acceleration.

But if "gravitation" refers to "tidal gravity", then "gravitation" in that sense is *not* relative; it is spacetime curvature, which is a coordinate-independent thing. "Modern GR" *does* claim that "gravitation" in the sense of spacetime curvature can't be removed by choosing coordinates. However, it can be made negligible in a sufficiently small patch of spacetime by choosing coordinates appropriately. "1916 GR" said the same thing, so again "modern GR" is no different than "1916 GR" in this sense.

We thus distinguish in that example that Eve's acceleration is real, and that her gravitational field is only apparent because the effect is not caused by the nearby presence of matter. For that reason I think that we should prefer Adam's interpretation. Similarly, in case of a real gravitational field that we ascribe to the presence of matter, it is Eve's interpretation that we should prefer.

But Eve's interpretation doesn't cover all of the spacetime. That's obvious in the scenario given on Egan's web page, but you still don't appear to realize that exactly the *same* reasoning applies to the case of a black hole.

In the Adam-Eve scenario, Eve can easily compute that the proper time along Adam's worldline from when he steps off the ship to when he reaches the Rindler horizon is finite. She can also easily compute that there is nothing physically present at the Rindler horizon that would cause Adam's worldline to end there. Finally, she can compute that, once Adam reaches the Rindler horizon, he can't get back out into the region of spacetime "above" it, because to do so he would have to move faster than light. So Eve can conclude that there must be a region of spacetime beyond the Rindler horizon, where Adam's worldline goes, even though she can't see it (light rays from it can never reach her).

If Eve were hovering above a black hole, and Adam stepped off the ship and fell in, *exactly* the same reasoning would apply. You can even draw a spacetime diagram of that scenario that looks almost identical to Egan's diagram; I did it in a recent post in the other thread that we have running on this topic. So just as in the case of Egan's scenario, in the black hole scenario we can see that Adam's coordinates cover a region of spacetime that Eve's don't. *That* is the reason that Adam's interpretation is "preferred", to the extent that it is--in the region of spacetime that both Adam's and Eve's coordinates cover, neither one is "preferred"; they can both be used to describe events and calculate physical quantities, and both will give the same answers. But Eve's is limited in coverage in a way that Adam's is not.

Now, it is still not clear to me if O-S used what Einstein called a Gaussian coordinate system

Every coordinate system that I've ever seen in any relativity paper is a Gaussian coordinate system by Einstein's definition; it's a very general definition. All of the coordinates used in the O-S paper are certainly Gaussian.

So, I want to make sure that their model is self-consistent. I guess that it is; the only difference between their inner and outer region modelling is the presence of matter - correct?

Yes. The key constraint that needs to be enforced to make the model consistent is basically that the metric and its derivatives match at the boundary; the technical term is "junction conditions". (I'm not sure that specific term appears in the paper; I think it was coined later on. But I think they talk about matching at the boundary.)

 

[Mike Holland]

How can one determine the difference between gravitation and acceleration from inside a spaceship/lift? In Eve's case, she could experiment and find that the gravitation force is uniform, not focussed on a point below her, but that just means the mass is very great and very far away - within the limits of her measurements.

If we throw out the principle of equivalence, doesn't most of GR goes with it?.

Mike

 

[PeterDonis]

How can one determine the difference between gravitation and acceleration from inside a spaceship/lift?

One can't.

In Eve's case, she could experiment and find that the gravitation force is uniform, not focussed on a point below her, but that just means the mass is very great and very far away - within the limits of her measurements.

Actually, even the word "uniform" has to be carefully defined in this case. If we consider a family of observers who are at rest at different spatial locations in Eve's coordinates, they will not all feel the same acceleration; observers further away from the Rindler horizon than Eve is will feel less acceleration than Eve, while observers closer to the Rindler horizon will feel more.

The correct way to distinguish the case of Eve from the case of an Eve-like observer who is accelerating above a gravitating body is by looking at spacetime curvature. Eve can compute the components of the curvature tensor in her coordinates just as Adam can in his; both of them will get zero, indicating that the spacetime they are in is flat, so no gravitating mass is present. An Eve-like observer accelerating above a gravitating body will compute a non-zero spacetime curvature; so will an Adam-like observer who is falling towards the body. This indicates that gravitating mass is present. But these computations can't be made "locally"; that is, they can't be made just using data acquired at one event (or in a small local patch around one event). They have to be made based on measurements made at different spatial locations, and/or at different times, so that the data covers a large enough portion of the spacetime for curvature to show up (where "large enough" depends on the accuracy of the measurements).

 

[Mike Holland]

Thanks. That's what I thought. I would be really upset if anyone disproved the principle of equivalence - I think it is the most brilliant insight ever! But I am quite happy with it only working for observations made in small lifts and spaceships (including observations through the windows).

Mike

 

[harrylin]

For the region of spacetime that both coordinate systems cover, yes, this is true. However [..]

I introduced here a primer of what I want to discuss in the parallel thread to make myself understood; I will continue that part of our discussion there (and there is almost too much to catch up with there!). What I wanted to get clarified here, as it is precisely the topic:

[..] Yes [= their model is self-consistent [..]; the only difference between their inner and outer region modelling is the presence of matter]. The key constraint that needs to be enforced to make the model consistent is basically that the metric and its derivatives match at the boundary; the technical term is "junction conditions". (I'm not sure that specific term appears in the paper; I think it was coined later on. But I think they talk about matching at the boundary.) [..]

OK, thanks for that clarification - it tells me that my first impression of their paper was correct.

How can one determine the difference between gravitation and acceleration from inside a spaceship/lift? In Eve's case, she could experiment and find that the gravitation force is uniform, not focussed on a point below her, but that just means the mass is very great and very far away - within the limits of her measurements. If we throw out the principle of equivalence, doesn't most of GR goes with it?
[...] I am quite happy with it only working for observations made in small lifts and spaceships

In fact, nowadays "blind" Earth sensors can be made (detecting the field non-uniformity) that fit inside a "picosatellite" of 10x10x10cm.

GR does not depend on technical limitations of measurement nor does it forbid people to measure on more than a single point - that would make it an invalid theory from the outset.
Schwartzschild and Oppenheimer used non-local coordinates because GR does not require a "local" reference system. That does in no way affect the Einstein equivalence principle. As I cited earlier:

"K' [..] has a uniformly accelerated motion relative to K [..] [This] can be explained in as good a manner in the following way. The reference-system K' has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to K'."
- https://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity

Also:
"This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the Earth (in its entirety) vanishes.
[..]
Even though by no means all gravitational fields can be produced in this way [= from acceleration], yet we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind. "
- starting from section 20 of: https://en.wikisource.org/wiki/Rela...ument_for_the_General_Postulate_of_Relativity

 

[Mike Holland]

I had no idea we had such sensitive instruments! OK, so all we need is an infinite mass an infinite distance away. No problem!

Thanks for the references.

Mike

 

[pervect]

I ran across this on the springerlink site I mentioned in another post. The link will turn into a pumpkin - I mean get hidden behind a paywall - after Nov 30, however.

I thought it gave a good overview. My short summary & interpretation of the main point. "We know better now".

http://dx.doi.org/10.1023/A:1022919909683

One of the great conundrums in the history of general relativity is certainly constituted
by the “Schwarzschild solution.” Also to a person with a marginal interest
in the history of this discipline, the noun immediately recalls to the mind this
puzzling circumstance: during more than four decades since the discovery of the
“Schwarzschild solution,” the overwhelming majority of the relativists harbored
the conviction that the region within the “Schwarzschild radius” was physically
meaningless, and strove to show that it could not be accessed from the outer
space. During the subsequent four decades, after a seminal and nearly forgotten
paper [1] [Synge, J. L. (1950). Proc. R. Irish Acad. 53A, 83.] that Synge wrote in 1950, an equally overwhelming majority of them
came to the conviction that the same region was physically meaningful and accessible
“without a bump” along geodesics. This major theme, for the time span
1915–1955, has undergone a very accurate historical scrutiny [2, 3, 4].
{
[2] Eisenstaedt, J. (1982). Arch. Hist. Exact Sci. 27, 157.
[3] Eisenstaedt, J. (1986). Arch. Hist. Exact Sci. 35, 115.
[4] Eisenstaedt, J. (1987). Arch. Hist. Exact Sci. 37, 275.
}
The subsequent years, in particular the crucial sixties, still await for a like historical work.

I should add that after this introduction, the authors go on to look at a different issue, the differences between Scwarzschild's original paper and the usually quoted "SC" coordinates. The two are not the same.

 

[PeterDonis]

I should add that after this introduction, the authors go on to look at a different issue, the differences between Scwarzschild's original paper and the usually quoted "SC" coordinates. The two are not the same.

I had seen mention of this before, but this paper does a good job of explaining what was going on.

Another interesting thing I saw in this paper is the claim that restricting the range of the standard Schwarzschild r coordinate to 0 < r < infinity is an "arbitrary restriction". They reference a 1989 paper by Abrams. The argument goes like this: we start with the general line element (in slightly more compact notation than the paper uses)

ds^2 = H(r) dt^2 - F(r) dr^2 - G(r) d\Omega^2

with 0 < r < infinity because the "r" here is supposed to be the "standard" r of spherical polar coordinates with its standard range. Then we rescale the r coordinate to eliminate the function G(r), by defining r^* = \sqrt{G(r)}, so that we can rewrite the line element as

ds^2 = H(r^*) dt^2 - F(r^*) {dr^*}^2 - {r^*}^2 d\Omega^2

But the paper claims that, since G(r) was an arbitrary function, we can no longer be sure that the range of r* is 0 < r* < infinity, since we can't assume that G(0) = 0.

The reason this jumped out at me is that it is not the derivation I'm used to seeing of the standard Schwarzschild line element. The standard derivation (as given, for example, in MTW) starts by *defining* the r coordinate such that the area of a 2-sphere at r is given by 4 \pi r^2. That definition ensures that the angular part of the line element is r^2 d\Omega^2, with no other factors present.

The only reason I can see to work with a more general form of the line element with an extra function G(r) in the angular part would be if one wanted to use a *different* radial coordinate, such as the isotropic radial coordinate, for which the area of a 2-sphere at "r" is *not* 4 \pi r^2. But if you are just trying to derive the standard Schwarzschild line element, I don't see the point of doing that; it's easy to show (as MTW do) that there is no loss of generality in defining the radial coordinate as I described above as long as the spacetime is spherically symmetric, and that definition obviously requires 0 < r < infinity. I haven't seen anything in any other literature I've read about that being an "arbitrary restriction"; has anyone else?

 

[harrylin]

I had no idea we had such sensitive instruments! OK, so all we need is an infinite mass an infinite distance away. No problem!

Thanks for the references.

Mike

You're welcome - but I wonder if you understood the references. Why do you think that you would need an infinite mass an infinite distance away? It's a bit similar to an inertial reference system (with which I mean a system in uniform rectilinear motion): we do not need any literal reference body like that. Theory relates to idealizations.

 

[harrylin]

[...] I thought it gave a good overview. [..]
http://dx.doi.org/10.1023/A:1022919909683

Thanks that looks interesting!

The author seems to answer an unanswered question that I posed here:
I guess that "extension "a la Synge" means that it was Synge who proposed the inside model.

 

[pervect]

Thanks that looks interesting!

The author seems to answer an unanswered question that I posed here:
I guess that "extension "a la Synge" means that it was Synge who proposed the inside model.

There's some related stuff at:
"Schwazchild and Synge once again"
"More on the early interpretation of the Schwarzschild Solution" (Arxiv, appaently published as well)
"On the Singularities of a Riemannian Manifold", Szerkes

Synge's paper is behind a paywall still (JSTOR). Some of the above will be behind a paywall soonish.

 

[Mike Holland]

You're welcome - but I wonder if you understood the references. Why do you think that you would need an infinite mass an infinite distance away? .

The idea is to get a uniform, linear gravitationan field, so that these very sensitive instruments cannot tell the difference between it and an accelerating frame. But I was joking - that's why I said "no problem" tongue-in-cheek.

I admit the maths in the first reference is way beyond me. The second one is largely what I have read in many popular books on GR.

 

[harrylin]

The idea is to get a uniform, linear gravitationan field, so that these very sensitive instruments cannot tell the difference between it and an accelerating frame. But I was joking - that's why I said "no problem" tongue-in-cheek.

I admit the maths in the first reference is way beyond me. The second one is largely what I have read in many popular books on GR.

The first reference does not only contain math, but also a clarification of the intended physical meaning of the math. And I still wonder why you continue to get the wrong idea... One last try (as it is off-topic):

"Principle of Equivalence: If in a space free from gravitation a reference system is uniformly accelerated, the reference system can be treated as being "at rest," provided one interprets the condition of the space with respect to it as a homogeneous gravitational field. - Einstein et al, Physical Review 1935

Now tell me, how does the equivalence principle pretend that gravitational fields must be uniform or linear?

Note: Interestingly he adds the footnote that It is worth pointing out that [the resulting] metric field does not represent the whole Minkowski space but only part of it. That could be fitting for discussion in a parallel thread.

 

[pervect]

Synge's paper is behind a paywall still (JSTOR).

I should add that if one's interest is in the actual physics, rather than just history, there are plenty of modern textbooks that explain the same thing Synge's paper did - and they're probably be easaier to read and written at greater length, as well.

Some can even be found online, e.g. Caroll's lecture notes.

 

[Mike Holland]

Now tell me, how does the equivalence principle pretend that gravitational fields must be uniform or linear?

My point was simply that with these very refined instruments, an observer in a lift could detect the differences in a gravitational field from the top of the lift to the bottom, and as a result he would know he as in a gravitational field and not accelerating. To make these differences too small for him to measure, we would need a very uniform field, which is why I suggested a very large mass a very large distance away.

As long as you are observing/measuring at one point, the principle works, but when you can take measurements at two or more separate points with sufficiently sensitive instruments you can tell the difference.

Mike

 

[PeterDonis]

To make these differences too small for him to measure, we would need a very uniform field, which is why I suggested a very large mass a very large distance away.

Or we could restrict measurements to a much smaller length scale. 10cm sounds pretty small by everyday standards, but it's still 14 orders of magnitude larger than an atomic nucleus and 34 orders of magnitude larger than the Planck length.

As long as you are observing/measuring at one point, the principle works, but when you can take measurements at two or more separate points with sufficiently sensitive instruments you can tell the difference.

Yes, that's true. But conversely, given a fixed sensitivity of instruments there will be some length scale small enough that we can make measurements at two points separated by that length scale and not detect the difference.