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Was the Schwarzschild geometry acceptable

  1. Oct 29, 2006 #1
    Today, as I guess, there are good indications that black-holes are a reality.
    But let us go back in time and pretend we are physicists in 1916 or a few years later.
    Schwarzschild lectures us about its static and spherical solution to the Einstein's equation.
    The consequence is striking: any communication is cut off between the universe and the interior of the Schwarzschild sphere.

    How could you digest such a striking consequence?
    Would you not reject the Einstein's equations?
    Would there be good or compelling reasons to trust the Einstein's equations?
    Would there be no alternaive but to accept the Einstein's equations and its consequence?

    I hope the reasons of the physicist in that time have not been lost!
    It could help me to digest GR!


  2. jcsd
  3. Oct 29, 2006 #2


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    Historically, the first thing you do is mis-understand the coordinate singularity at the event horizon as a physical singularity, rather than a coordinate singularity. (An example of a coordinate singularity is the North pole of the Earth. The geometry of the Earth is perfectly normal there, but every direction is "south".)

    I'm not sure how much "help" this will be, except as something to avoid! But perhaps that's useful.
  4. Oct 29, 2006 #3
    At the time I think Schwarzschild expected his solution to be valid only outside of a planet or star (just like the 1/r^2 solution of Newtonian gravity is invalid beneath the earth's surface) and, making some presumptions about the density of physical matter, would not have interpreted black holes to be a consequence of Einstein's equations. Unfortunately, I doubt he had time to give lectures on the topic.
  5. Oct 30, 2006 #4

    I have read about the "Kruskal-Szekeres" coordinates in Landau-Lifchitsz and on Wiki. It is clear that the Schwarzschild metric is not singular.

    Nevertheless, when a mass approaches the Schwarzschild radius, something very special happens, specially for those who observe it from outside (the black hole). The event horizon of a black hole really separates two regions of space.

    Maybe for some time this aspect of the Schwarzschild metric was considered as an extreme -maybe unphysical- application of this metric, I don't know. For sure, its use was mostly for less extreme conditions (perihelion, light deviation) and for discovering the solutions of the Einstein's equations.

    However, other metrics, not satifying the Einstein's equations, could have shown very nearly the same behaviour for r>>rg without showing any event horizon. (simply taking gtt = 1+2U/mc², and modifying U for small r to avoid the appearence of any event horizon)

    I know from reading Wheeler or Landau how consistent the foundations of GR are. Also how beautiful they are. However, the step taken by GR, as illustrated by the Schwarzschild metric, is a huge extrapolation from the knowledge available at that time. Therefore, I am quite convinced that there were solid reasons, even at that early time, to keep confidence in GR despite the surprises. And a few decades later, these surprises could even be turned in extraordinary predictions by GR, increasing the confidence in GR.

    I am missing some understanding about why it was preferred to keep the Schwarzschild metric instead of modifying (slightly) the Einstein's equation. Or may be that was studied and I would like to know about the debate and the arguments.

    It would help me a lot if the possibility of an event horizon (black hole) could be made clear from more simple arguments than solving the Einstein's equation. It would help me also if I could understand why a metric like (gtt = 1+2U/mc², ...) could be valid even when |U/mc²| is not small anymore, which is quite an extreme situation.

    Thanks for your help,

    Last edited: Oct 30, 2006
  6. Oct 30, 2006 #5


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    I'd like to point out that the Rindler metric, the metric of an accelerating observer in a perfectly flat space-time, ALSO has an event horizon. So I'm not sure I see the point.
  7. Oct 30, 2006 #6

    Thanks that you remind me about the Rindler metric.
    My point is about the extrapolation associated with GR.
    But indeed, the Rindler metric can hardly be doubted since it is a consequence of SR (I am right?).
    Am I right to say that our main experimental knowledge are in situations where U/mc² << 1 , or that at least that was the case before WWII ? Therefore the extrapolation up to black holes seems very bold to me. For example a theory allowing for a small modification to the Newtonian potential could remove black holes completely (a transformation like 1/r ==> 1/(r+r_Schwarzschild) is negligible indeed when U/mc²<<1). So my question: is the structure of GR so elegant and convincing (before WWII) that the extrapolation was easily accepted. What other evidences came later?

  8. Oct 30, 2006 #7


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    Basically, a lot of the evidence that supports GR has come in only recently, in large part due to the devolpment of atomic clocks.

    In the early days, there were only a few "classical tests" of GR - the precession of the perihelion of mercury, a somewhat indirect argument as other pertubations due to Newtonian gravity also caused precession, and a single measurements of gravitational light deflection. Now we have much better measurements of light bending, precision measurmenets of the Shapiro effect, and with atomic clocks a direct terrestrial confirmation of the effect of gravitational potential on time rate.

    While most of our tests are for what one might call the "weak field", through PPN (post post Newtonian) formalism and measurement of the PPN parameters, we have been able to eliminate or greatly constrain many other theories of gravitiy as contenders. For instance, Branse-Dicke theory has an adjustable parameter w - the theory is still viable, but the value of 'w' must be very large, a region in which the theory gives results that are essentialy equivalent to GR.

    see also

    With GP-B, we will be able to either eliminate more theories, or falsify GR.

    We don't currently have many strong field results (one pair of inspariling binary pulsars, which also seems to confirm GR).

    Where does this leave black holes and event horizons? We don't have a lot of evidence at the current time, but what evidence there is seems to be mostly consistent with GR. Open issues in gravity and cosmology do exist but don't have much to do with black holes. Event horizons are intrinsically difficult to confirm, but we do have some evidence that suggests that black holes tend to be "black", a key prediction of the theory. But it's hardly conclusive evidence at this stage.

    See for instance http://arxiv.org/abs/gr-qc/9803057 for some discussion of "black holes being black".

    [add]For one criticism of the above POV see http://arxiv.org/PS_cache/astro-ph/pdf/0308/0308171.pdf [Broken] - it agrees with the point that black holes appear to be black, but disagrees that the current models of how this happens are accurate.
    Last edited by a moderator: May 2, 2017
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