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So spacetime exists beyond the Cauchy horizon, but it's discontinuous? What on earth would discontinuous spacetime involve? It sounds like a sort of shattered

So spacetime exists beyond the Cauchy horizon, but it's discontinuous? What on earth would discontinuous spacetime involve? It sounds like a sort of shattered chaos of torn-up bits of space.

nerdponx 8y ago
It's not discontinuous, its derivative (a function describing the rate of change of space time) is apparently discontinuous, or infinite, or something else equally hard to picture. Plenty of strange things can happen with derivatives, e.g. https://en.m.wikipedia.org/wiki/Cantor_function https://en.m.wikipedia.org/wiki/Cantor_function
- jordigh 8y ago
  I don't find failing to be differentiable all that strange, even if the failure is on a perfect nowhere dense set. What I find weird is that the derivative, if it exists, cannot have a jump discontinuity. That means that the only other kind of discontinuity it can have is infinite oscillation like sin(1/x) near zero. This one is a bit of an obscure property of derivatives, corollary to theorem 5.12 in Baby Rudin.
  - jerf 8y ago
    "What I find weird is that the derivative, if it exists, cannot have a jump discontinuity." You mean the derivative of a specific function you have in mind, like perhaps the field equations? Or do you mean something other than what I understand by a jump discontinuity in a derivative, such as one gets for f(x) = {-x for x<0, x for x>=0}? Tone: Clarification request for my own understanding, not a "gotcha" post; I strongly believe you are saying something true but there's just too many details elided because they are trivial to you for me to quite follow, and I'm intrigued enough to want to be able to follow up, if you'd be so kind as to indulge me.
    - qubex 8y ago
      What s/he is saying, if I understand correctly, is that two and only two kinds of discontinuities in derivatives (and second- and higher-order derivatives) can exist: jump derivatives and those exemplified by the pathology inherent in sin(1/x) at the origin thereof; given that spacetime continues beyond the Cauchy horizon but the derivative does not, the first kind (the abrupt ”vertical step” kind) must be ruled out, leaving only the second variety as a candidate. Graph that function, and it’s first and second derivatives around the origin, and you will begin to realise how bizarre this behaviour is and what strange implications it may have. (Then again, this is deep within a black hole, in an area where time and the radial dimension of space are switched, so it’s just a new kind of devilry in an already fraught area.)
      - jkabrg 8y ago
        No. The relevant theorem is called Darboux's Theorem. It states that if f(x) has a derivative f'(x) everywhere, then f'(x) satisfies the intermediate value property, meaning: If f'(A) < 0 and f'(B) > 0 then there exists a C between A and B such that f'(C) = 0. Why this is interesting is because f'(x) can be a discontinuous function. f'(x) being discontinuous is not the same thing as f(x) not having a derivative at some point; for example, the absolute value function |x| is simply not differentiable at x=0. The following function IS differentiable at x=0 but its derivative at x=0 is discontinuous: f(x) = x^2 sin(1/x) if x != 0 else 0 You can verify that this function satisfies Darboux's theorem. Darboux's theorem implies in particular that if f'(x) exists at some point x and is discontinuous at that x then the discontinuity is not a jump discontinuity. Link to Darboux's theorem: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis) https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)
        jerf 8y ago
        Thank you.
        qubex 8y ago
        I stand corrected.
    - 8y ago
musgravepeter 8y ago
I need to read this more carefully, but a discontinuity in the derivatives of a system of equations at an interface is fairly common. If there are two regions obeying Maxwell's equations with a difference in the derivatives of the E/B fields at the boundary, this means the boundary has some delta function of charge or currents on the interface. Likewise in GR. There is a whole sub-field of joining solutions to Einstein equations together and determining the material at the interface. Wormhole solutions can be made with this cut-and-paste approach. (The best book on this is Eric Poisson's "A Relativist's Toolkit"). The paper referenced here is 200+ pages. The abstract states in part "We prove that for all such data, the maximal Cauchy evolution can be extended across a non-trivial piece of Cauchy horizon as a Lorentzian manifold with continuous metric.", which is what disproves strong cosmic censorship. I have no grasp of the details (my PhD in GR junction conditions is from the 90's and I left the field).
btilly 8y ago
Discontinuous derivative. If you want a visual analogy, think of the transition from a smooth fluid flow to turbulence where fluid goes from moving in a simple direction to going every which way at once. See https://www.ethz.ch/content/specialinterest/mavt/fluid-dynamics/fluid-dynamics/en/research/group-kleiser/transition-to-turbulence.html https://www.ethz.ch/content/specialinterest/mavt/fluid-dynam... for some pictures of that.

So spacetime exists beyond the Cauchy horizon, but it's discontinuous? What on earth would discontinuous spacetime involve? It sounds like a sort of shattered chaos of torn-up bits of space.

nerdponx 8y ago

It's not discontinuous, its derivative (a function describing the rate of change of space time) is apparently discontinuous, or infinite, or something else equally hard to picture. Plenty of strange things can happen with derivatives, e.g. https://en.m.wikipedia.org/wiki/Cantor_function https://en.m.wikipedia.org/wiki/Cantor_function

jordigh 8y ago
I don't find failing to be differentiable all that strange, even if the failure is on a perfect nowhere dense set. What I find weird is that the derivative, if it exists, cannot have a jump discontinuity. That means that the only other kind of discontinuity it can have is infinite oscillation like sin(1/x) near zero. This one is a bit of an obscure property of derivatives, corollary to theorem 5.12 in Baby Rudin.
- jerf 8y ago
  "What I find weird is that the derivative, if it exists, cannot have a jump discontinuity." You mean the derivative of a specific function you have in mind, like perhaps the field equations? Or do you mean something other than what I understand by a jump discontinuity in a derivative, such as one gets for f(x) = {-x for x<0, x for x>=0}? Tone: Clarification request for my own understanding, not a "gotcha" post; I strongly believe you are saying something true but there's just too many details elided because they are trivial to you for me to quite follow, and I'm intrigued enough to want to be able to follow up, if you'd be so kind as to indulge me.
  - qubex 8y ago
    What s/he is saying, if I understand correctly, is that two and only two kinds of discontinuities in derivatives (and second- and higher-order derivatives) can exist: jump derivatives and those exemplified by the pathology inherent in sin(1/x) at the origin thereof; given that spacetime continues beyond the Cauchy horizon but the derivative does not, the first kind (the abrupt ”vertical step” kind) must be ruled out, leaving only the second variety as a candidate. Graph that function, and it’s first and second derivatives around the origin, and you will begin to realise how bizarre this behaviour is and what strange implications it may have. (Then again, this is deep within a black hole, in an area where time and the radial dimension of space are switched, so it’s just a new kind of devilry in an already fraught area.)
    - jkabrg 8y ago
      No. The relevant theorem is called Darboux's Theorem. It states that if f(x) has a derivative f'(x) everywhere, then f'(x) satisfies the intermediate value property, meaning: If f'(A) < 0 and f'(B) > 0 then there exists a C between A and B such that f'(C) = 0. Why this is interesting is because f'(x) can be a discontinuous function. f'(x) being discontinuous is not the same thing as f(x) not having a derivative at some point; for example, the absolute value function |x| is simply not differentiable at x=0. The following function IS differentiable at x=0 but its derivative at x=0 is discontinuous: f(x) = x^2 sin(1/x) if x != 0 else 0 You can verify that this function satisfies Darboux's theorem. Darboux's theorem implies in particular that if f'(x) exists at some point x and is discontinuous at that x then the discontinuity is not a jump discontinuity. Link to Darboux's theorem: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis) https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)
      - jerf 8y ago
        Thank you.
      - qubex 8y ago
        I stand corrected.
  - 8y ago

musgravepeter 8y ago

I need to read this more carefully, but a discontinuity in the derivatives of a system of equations at an interface is fairly common. If there are two regions obeying Maxwell's equations with a difference in the derivatives of the E/B fields at the boundary, this means the boundary has some delta function of charge or currents on the interface. Likewise in GR. There is a whole sub-field of joining solutions to Einstein equations together and determining the material at the interface. Wormhole solutions can be made with this cut-and-paste approach. (The best book on this is Eric Poisson's "A Relativist's Toolkit"). The paper referenced here is 200+ pages. The abstract states in part "We prove that for all such data, the maximal Cauchy evolution can be extended across a non-trivial piece of Cauchy horizon as a Lorentzian manifold with continuous metric.", which is what disproves strong cosmic censorship. I have no grasp of the details (my PhD in GR junction conditions is from the 90's and I left the field).

btilly 8y ago

Discontinuous derivative. If you want a visual analogy, think of the transition from a smooth fluid flow to turbulence where fluid goes from moving in a simple direction to going every which way at once. See https://www.ethz.ch/content/specialinterest/mavt/fluid-dynamics/fluid-dynamics/en/research/group-kleiser/transition-to-turbulence.html https://www.ethz.ch/content/specialinterest/mavt/fluid-dynam... for some pictures of that.