- #1

harrylin

- 3,875

- 93

**Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's GR?**

This topic is a spin-off from a number of recent discussions:

"Are "flowing space" models compatible with GR?"

"Schwartzchild and Synge once again"

"Oppenheimer-Snyder model of star collapse"

"Notions of simultaneity in strongly curved spacetime"

"limit of Rindler coordinates"

As well as my post with more links in How do black holes grow?:

https://www.physicsforums.com/showthread.php?p=4165405

According to Einstein, the "Schwarzschild singularities" do not exist in physical reality. And it seems that most

^{1}black hole specialists think that Einstein was simply wrong. But if the models of Finkelstein (Physical Review 1958) and Kruskal resemble in any way those of Hamilton, then I wonder how they can be compatible with Einstein's GR - with which I mean Einstein's theory of gravitation as expounded in the period 1916-1920.

Due to different meanings of words by different schools of thought it is difficult to estimate if their theory merely

**sounds**different or really

**is**different from his. I would like to get to the bottom of this, hopefully with the help of GR experts on this forum.

Before elaborating that question I will start with a short historical introduction including some disambiguation and definitions:

-----------------------------------------------------------------

"General relativity" started out as a theory about a "general PoR", which gave it its name; that was not directly a theory about gravitation but about relative motion. However, as Einstein put it:

*the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can "create" a gravitational field by a simple variation of the co-ordinate system.*-E. 1916

Most people (and as I now found, even Einstein) - abandoned this 1916 flavour of GR. I think that it just cannot work, and one of the reasons will come up here below. Compare the physics FAQ:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html

Einstein's concept of transforming acceleration into a "real" gravitational field had to be dropped; and typical for him, he did not shout it from the rooftops . What remained was his theory about the effects of real gravitational fields, based on a subtly rephrased EEP. In this discussion I will label that version of his theory as "Einstein's GR", and accordingly with "EEP" I will mean his final version.

Terminology that is incompatible with that of Einstein obfuscates a correct understanding of his theory. In this discussion I will therefore stick to his definitions, and kindly request anyone participating to do the same. I found that the following terms require precision:

**relative motion**: the difference of the motion of two entities, as measured with a reference system

- §3 of http://www.fourmilab.ch/etexts/einstein/specrel/www/

**inertial**: in uniform motion (same as in classical mechanics:

**not**geodesic)

- https://en.wikisource.org/wiki/A_Brief_Outline_of_the_Development_of_the_Theory_of_Relativity

**Special Relativity**(term created by Einstein): Theory based on the PoR and the light principle, relating to coordinate systems relative to which isolated, material points move uniformly in straight lines. Does not include the EEP.

- https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

**Einstein Equivalence Principle**(

**EEP**, 1935 definition):

*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

It has a footnote that is relevant for this discussion, see further.

**gravitational field**(simplified definition from already

^{2}ca.1919): effects due to the neighbourhood of matter (the equivalent effect due to acceleration is an apparent field).

- "The geometrical states of bodies and the rates of clocks depend in the first place on their gravitational fields, which again are produced by the material systems concerned - https://en.wikisource.org/wiki/Time,_Space,_and_Gravitation

- "we shall have to take account of the fact that the ponderable masses will be the determining factor in producing the field, or, according to the fundamental result of the special theory of relativity, the energy density" - https://en.wikisource.org/wiki/A_Brief_Outline_of_the_Development_of_the_Theory_of_Relativity

**proper acceleration**: not found. And https://en.wikipedia.org/wiki/Proper_acceleration is incompatible with the EEP.

-------------------------------------------------------------------

And now (finally!) a retake of earlier discussions, some of which are still going on.

pervect, DrGreg and Atyy suggested that we take a look at Rindler coordinates and there has been quite some discussion on this.

In referral to http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html, it is obvious that Adam and Eve have different views of physical reality: for example when Adam "falls away" according to Eve, she ascribes the frequency difference from her two clocks fully to the effect of a gravitational field which makes her clocks go at different rates; while in contrast, Adam ascribes the frequency difference that Eve observes to the Doppler effect from her acceleration. But when Eve discovers that in reality she is not in a gravitational field, then she will agree with Adam; her changed conception of reality also changes her conception of what really happens with Adam.

The difference started with the realisation (perhaps already in 1918, as fall-out of Einstein's Twin paradox paper) that "induced gravitational fields" are an illusion that do not really work. Thus the equivalence principle cannot swap physical reality as is the case with SR's "relativity of simultaneity". The moment that Eve realizes that her "gravitational field" is an illusion, she will agree with Adam that she is looking at the Doppler effect instead of clocks running at different rates in a gravitational field.PeterDonis said:Then what's the difference in Schwarzschild spacetime? That's the whole point of the Rindler horizon analogy: that if Eve does not think Adam never crosses the horizon, Eve' who is hovering above a black hole horizon should not think that Adam', who drops off her spaceship and falls into the hole, never crosses the horizon either.

If you think there is a difference, what's the difference? Why can't Eve' reason the same way that Eve does, to conclude that Adam' does cross the horizon?

Inversely, if Adam realizes that his "inertial motion" is an illusion and that in fact he is falling towards a black hole while Eve is staying put, then he should agree with Eve, which also means that his wristwatch is quickly slowing down compared to her clocks.

pervect came with a similar example, even with numbers:

It looks to me that Rindler's metric is standard EEP metric; and as some people think that Einstein was ignorant about those and would have changed his position if he had heard of this argument, it will be useful to elaborate on this. From Einstein's 1935 paper:pervect said:[..] The experiment involves launching a spaceship that accelerates at 1g for a year shiptime - or .1g for 10 years shiptime - or .001 g for 1000 years shiptime.

The spaceship observes the Earth through a telescope. The prediction of SR in this case [...] [note: this is SR as long as the EEP is not used] that the Earth appears to fall behind an event horizon. There will be some last event that the spaceship sees - say year 2100 exactly on the new years day celebration in Grenwich.

The metric from the accelerating spaceship looks like this, assuming the spaceship accelerates in the z direction. (There are some variant forms of the metric, this version is normalized so that g_uv = diag(-1,1,1,1) at the origin.

ds^2 = -(1+ gz)^2 dt^2 + dx^2 + dy^2 + dz^2

*It is worth pointing out that this metric field does not represent the whole Minkowski space but only part of it.*

Thus, the transformation that converts ds

into (1) is

ε

It follows that only those points for which ε

Thus, the transformation that converts ds

^{2}=-dε_{1}^{2}-dε_{2}^{2}-dε_{3}^{2}+dε_{4}^{2}into (1) is

ε

_{1}=x_{1}cosh αx_{4}, ε_{2}=x_{2}, ε_{3}=x_{3}, ε_{4}=x_{1}sinh αx_{4}.It follows that only those points for which ε

_{1}^{2}≥ε_{4}^{2}correspond to points for which (1) is the metric.Thus the wikipedia article http://en.wikipedia.org/w/index.php?title=Rindler_coordinates&oldid=522511984 describes EEP metric.

Thanks - and for completeness I will give a counter argument which I guess to be similar to Einstein's interpretation. Of course the Earth won't cease to exist at New years day 2100; your prediction is a misinterpretation of the idea that the Earth (and the whole universe with it) is falling into a gigantic black hole. If we model that our whole universe is falling into a giant black hole then according to Einstein's GR the Earth will not reach a local time of New Years day 2100; nothing exists forever and it will literally take forever to reach that day. In other words, that event will not happen.As the observer on the spaceship watches the Earth approach New Years 2100,, the spaceship sees the image grow dimmer and dimmer, and the Earth's clocks appear to slow down. Just as it would if the Earth were falling through the event horizon of a very large black hole, as g_00 falls towards zero at the critical value z = -1/g. (In non-geometric units, that's z = c^2/g). This is the critical value because g_00 goes to zero. I believe you call it something like "time stopping?" I forget how you referred to this condition.

Now, if we apply your argument, the Earth ceases to exist in the year 2100 at new Years in some philosophically meaningful sense. At the very least, something dramatic happens on that date, as "time stops".

My position is that it's pretty obvious the Earth won't cease to exist at New Years day on the year 2100 in any sort of meaningful sense. And that the people on Earth won't even notice this, or notice anything about "time stopping" or anythign like that. In fact, they'll find New Years day 2100 quite unremarkable.

^{3}

However, most of us believe that the idea that our universe is falling into a gigantic black hole does not conform to reality; we think that that black hole does not exist. And just as in the earlier example of Adam and Eve, our physical reinterpretation implies changing the redshift interpretation, with as consequence that now the Earth's predicted local time is simply that of SR.

Thus you used the same "map" as Einstein, but you interpreted it differently. Note that in both examples the correct prediction according to both entrenched positions is assumed to be based on physical reality, and not on mere appearances based on fictive fields that lead to contradicting predictions.

Anyway, this was just the introduction, the purpose of this topic is to get an understanding of Finkelstein's or Kruskal's reference systems, hopefully with the help of such examples.

Can someone explain this in a simple, qualitative way, if possible by means of an extension of pervect's numerical example? Of course that solution will only be 1 spatial dimension, but it should be feasible to next imagine it in 2D/3D. Then everyone can judge for themselves how compatible such solutions are with Einstein's GR.

1 Reading between the lines I think that Kraus and colleagues share Einstein's opinion

2 As early as 1919 - that was new to me! That may explain why no English version of Einstein's 1918 paper appeared at the time.

3. Even if infinite time could happen, people on Earth could not notice it; to think that they could is a mix-up of frames.

Last edited: