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Nariai universes

  1. Nov 2, 2007 #1
    I read in a NewScientist article recently that, according to a study by a mathematician at UCL, black holes may contain tiny universes inside them called Nariai universes, which expand in only one dimension as opposed to three. If this is the case, if matter were to enter the black hole, would it enter the Nariai universe intact? Or would it be reduced to subatomic particles and then enter the new universe?
     
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  3. Nov 2, 2007 #2

    Chris Hillman

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    What is the Nariai lambdavacuum?

    Well, I have no idea what New Scientist is on about this time :rolleyes: but I can tell you what the Nariai lambdavacuum is in gtr and be pretty confident that your UCL mathematician must be talking about the same thing. (Do you recall a name? And is that University College London?)

    But first: you should know that most real math/physics types have canceled their subscriptions to NS, which a dozen years ago was a fairly respectable publication, due to an unending spate :yuck: of sensationalized and highly misleading articles in recent years, such as the now notorious article on an alleged device which would violate conservation of momentum if it really operated as its inventor claims; see this discussion.

    OK, enough of that!

    Many Lorentzian manifolds are Cartesian products of a signature [itex](-1,1)[/itex] manifold (some would say signature zero) with a Riemanian two-manifold. Very very few of these are also solutions of the Einstein field equation. The Nariai lambdavacuum is one of the exceptions. Physically speaking it is a model spacetime in which the only source of the gravitational field is a Lambda term, i.e. that mysterious thing which goes by the name of "dark energy". As you probably know, Lorentzian manifolds can be described using many distinct coordinate charts, and when written in terms of different charts, the metric tensor may appear very different. Nonetheless such different ways of writing a solution really are describing the same geometry, and thus physically speaking (in terms of gtr), the same physical scenario or model universe.

    So here is one way of writing down the Nariai solution:
    [tex]
    ds^2 = -\sinh(kx)^2 \, dt^2 + dx^2 + \sinh(kz)^2 \, dy^2 + dz^2,
    -\infty < t, \, x, \, y, \, z < \infty
    [/tex]
    On the domain
    [tex]
    -\infty < t,y < \infty,
    \; 0 < x < 1/2 \, \operatorname{arccosh}(\sqrt{2}),
    \; 0 < z < \infty
    [/tex]
    we can adopt the frame field
    [tex]
    \vec{e}_1 =
    \frac{1}{\sinh(kx)^2} \, \partial_t
    + \frac{\sqrt{2-\cosh(kx)^2}}{\sinh(kx)} \, \partial_x
    [/tex]
    [tex]
    \vec{e}_2 =
    \frac{\sqrt{2-\cosh(kx)^2}} {\sinh(kx)^2} \, \partial_t
    \frac{1}{\sinh(kx)} \, \partial_x
    [/tex]
    [tex]
    \vec{e}_3 =
    \frac{1}{\sinh(kz)} \, \partial_y
    [/tex]
    [tex]
    \vec{e}_4 = \partial_z
    [/tex]
    Here, [itex]\vec{e}_1[/itex] is a timelike unit vector field, whose integral curves represent the world lines of a family of observers, while the other three are spacelike unit vector fields and represent the "spatial directions" defining a "local Lorentz frame" for each of these observers, at each event on his world line (in the domain where the frame field is valid). These observers happen to be inertial observers (their world lines have vanishing path curvature, i.e. vanishing acceleration vector, so they are freely falling in the gravitational field modeled by our spacetime geometry). Furthermore, their world lines form a vorticity-free timelike geodesic congruence, so they define a unique family of spatial hyperslices orthogonal to the world lines, and our frame is nonspinning, meaning that the spatial vectors are not rotating with respect to the spin axes of gyros carried by our observers. Such nonspinning inertial frames are the closest we can come in general relativity to the global Lorentz frames of special relativity (which are also nonspinning inertial).

    There is a way of writing down a dual coframe
    [tex]\sigma^1, \; \sigma^2, \; \sigma^3, \; \sigma^4[/tex]
    which consists of mutually orthogonal unit covector fields and gives an equivalent description of the same family of observers, but I won't bother to write it out here. As an exercise in computing curvature following the method of Cartan (connection one-forms, curvature two-forms), the Nariai lambda vacuum is good practice. The answer is remarkably simple:
    [tex]
    {\Omega^{\hat{1}}}_{\hat{2}} = -k^2 \, \sigma^1 \wedge \sigma^2, \;
    {\Omega^{\hat{3}}}_{\hat{4}} = -k^2 \, \sigma^3 \wedge \sigma^4
    [/tex]
    which gives the Riemann tensor (dropping the hats sometimes used to stress that we are computing "physical components", i.e. components wrt a frame field, not a coordinate basis--- in this post, all components are taken wrt our frame field)
    [tex]
    R_{1212} = k^2, \; R_{3434} = -k^2
    [/tex]
    In other words, we have here the constant curvature Lorentzian manifold [itex]H^{1,1} \times H^2[/itex] of "radius" [itex]1/k[/itex]. (This is analogous to the Riemannian manifold [itex]H^2 \times H^2[/itex], the Cartesian product of two hyperbolic planes.) From the Riemann tensor we can compute the Ricci tensor and then the Einstein tensor, which turns out to be
    [tex]
    G^{ab} = k^2 \; \operatorname{diag} \left(-1,1,1,1 \right)
    [/tex]
    which is diagnostic of a "Lambda" term giving constant negative energy and equal but positive isotropic pressure (in any frame!). The Riemann tensor (considered as a linear operator on bivectors) has one double eigenvalue, [itex]-a^2[/itex], and one quadruple eigenvalue, zero, which is again a reflection of the highly unusual nature of the Nariai lambdavacuum (a product of two two-dimensional manifolds).

    Whenever one is confronted with a timelike congruence in gtr (such as the world lines of our nonspinning inertial observers), one should compute its kinematic decomposition (acceleration vector, expansion tensor, and vorticity vector). In this case we already noted that the acceleration and vorticity vanish and the expansion tensor is very simple:
    [tex]
    {\theta \left[ \vec{e}_1 \right]}_{ab} = \frac{-k \, \cosh(kx)}{\sqrt{2 - \cosh(kx)^2}} \;
    \operatorname{diag} \left( 1,0,0 \right)
    [/tex]
    The tidal tensor is
    [tex]
    {E \left[ \vec{e}_1 \right]}_{ab} = k^2 \, \operatorname{diag} \left( 1,0,0 \right)
    [/tex]
    (The kinematic decomposition and the Bel decomposition produce three-dimensional tensors; the brackets are meant to stress that these decompositions are taken with respect to a particular timelike congruence. Such a procedure is analogous to the way that the EM field tensor is decomposed into electric and magnetic vectors, which are three-dimensional vectors.) This means that our observers are drawing apart along the [itex]\vec{e}_2[/itex] direction, due to a constant tidal tension along that direction which each of our ideal observers can in principle measure with sensitive strain gauges attached to the skin of his spaceship. The magnetogravitic tensor vanishes (no spin-spin forces on gyroscopes carried by our observers!), and the Riemannian curvature tensor of our spatial hyperslices shows that these manifolds are all isometric to the Cartesian product [itex]R^1 \times H^2[/itex] where the spherical factor has "radius" 1/k.

    I seem to be spending more effort than usual warning various PF posters to be cautious about phrases like "space is expanding". As I keep repeating, almost always what this really means is that some family of observers is expanding in the technical sense of expansion tensor. Indeed, we see very clearly from the above analysis that our inertial observers are drawing apart along a preferred direction. (I left as an exercise the task of determing the Lie algebra of Killing vectors, but anyone who knows what that means can probably write down generators from the information that as a Lorentzian manifold we are dealing here with [itex]H^{1,1} \times H^2[/itex].)

    Without more information (such as a citation of the paper by your UCL mathematician) I cannot answer your question about how pieces of such a Nariai lambdavacuum are supposed to be lurking inside black holes, but I can say this: suggestions that the interior of black holes might contain regions which can considered "phenomonologically" (there's that word again!) as something like pieces of the Nariai lambdavacuum (or pieces of the de Sitter lambdavacuum) are nothing new: this rather vague idea has been kicked around for at least a decade. As I said, New Scientist is known for not being very scrupulous in its reporting :rolleyes:
     
    Last edited: Nov 3, 2007
  4. Nov 8, 2007 #3

    Chris Hillman

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    Hope I didn't frighten off the OP...

    It's bad form in general to follow up on one's own post, but I just noticed that of the two forms of the Nariai lambdavacuum (AdS and dS) I probably gave him the wrong one.

    Let me back up and try again. One can envision basically four possible products of two dimensional spaces of uniform curvature which form Lorentzian manifolds, which one might hope will turn out to be reasonable solutions of the EFE. The Riemannian factor can be the sphere or the hyperbolic plane, [itex]S^2, \, H^2[/itex] respectively, while the semi-Riemannian factor can be [itex]S^{1,1}, H^{1,1}[/itex] respectively. You can envision the last two as the hyperboloid of one sheet embedded in [itex]E^{1,2}[/itex] with constant time surfaces corresponding to longitudes and latitudes respectively. That is, in [itex]S^{1,1}[/itex], timelike geodesics are great circles, initially parallel timelike geodesics converge, and time is "cyclic". OTH, in [itex]H^{1,1}[/itex], initially timelike geodesics diverge, and the spacelike direction is "cyclic".

    Of the four products, only three work as solutions of the EFE:
    • [itex]S^{1,1} \times H^2[/itex] corresponds to the Nariai AdS-lambdavacuum; a natural family of inertial observers experience constant axial tidal compression and they converge axially; their hyperslices are isometric to [itex]R \times H^2[/itex],
    • [itex]H^{1,1} \times S^2[/itex] corresponds to the Nariai dS-lambdavacuum; a natural family of inertial observers experience constant axial tidal tension and they expand axially; their hyperslices are isometric to [itex]R \times S^2[/itex],
    • [itex]S^{1,1} \times S^2[/itex] corresponds to the Bertotti-Robinson non-null electrovacuum, the natural family of inertial observers measure an axial electrostatic field and constant axial tidal compression and they converge axially; their spatial hyperslices are isometric to [itex]R \times S^2[/itex]; this spacetime is conformally flat (identically vanishing Weyl tensor) and occurs for example in the interaction zone of a certain colliding plane wave (CPW) model,
    • [itex]H^{1,1} \times H^2[/itex] doesn't give a solution of the EFE (the Einstein tensor doesn't correspond to anything obtainable from stress-energy tensor of matter plus plausible non-gravitational fields).
    The point is that Frolov observers in the "future interior" of a Schwarzschild vacuum are also inertial, experience axial tension and expand axially*, and have orthogonal hyperslices isometric to "cylinders" [itex]R \times S^2[/itex]. This suggests trying to match across a "constant time" hyperslice to part of a Nariai dS lambdavacuum. But this introduces a "crease", unlike say matching the Schwarzschild static spherically symmetric perfect fluid across the sphere of zero pressure to the Schwarzschild vacuum solution to form an idealized modelf of an isolated star. All the attempts I have seen to introduce a bit of dS or Nariai-dS inside a black hole model suffer from essentially this same problem--- as the authors of the eprints in questions generally recognize. As a rule, such authors argue that more or less speculative considerations from QFTs suggest this is permissible in an "effective field theory" formulation of gtr as a classical approximation to some as yet unknown quantum theory of gravitation.

    (*But unlike our inertial observers in the Nariai dS-lambdavacuum, the Frolov observers also experience compression orthogonal to the axis and they converge orthogonally to the axis, which is one way of understanding why the "crease" is unavoidable.)

    It is possible to give a simpler frame field for the Nariai dS-lambdavacuum than for Nariai AdS-lambda vacuum, namely:
    [tex]
    \vec{e}_1 = \partial_t, \;
    \vec{e}_2 = \frac{1}{\sinh(k t)} \, \partial_z, \;
    \vec{e}_3 = \partial_r, \;
    \vec{e}_4 = \frac{1}{\sin(k r)} \, \partial_\phi
    [/tex]
    which gives the line element
    [tex]
    ds^2 = -dt^2 + \sinh(k t)^2 \, dz^2 + dr^2 + \sin(k r)^2 \, d\phi^2,
    [/tex]
    [tex]
    0 < t < \infty, \, 0 < r < \frac{\pi}{k}, \, -\pi < z, \, \phi < \pi
    [/tex]
    Then the acceleration and vorticity vectors of the timelike congruence defined by the unit timelike vector field [itex]\vec{e}_1[/itex] vanish, while the expansion tensor is
    [tex]
    {\theta \left[ \vec{e}_1 \right]}_{ab} =
    \coth(k t) \, \operatorname{diag} \, \left( 1,0,0 \right)
    [/tex]
    which shows constant expansion in the [itex]\vec{e}_2[/itex] direction. The tidal tensor ("electric part" of the Riemann tensor) is
    [tex]
    {E \left[ \vec{e}_1 \right]}_{ab} =
    -k^2 \, \operatorname{diag} \, \left( 1,0,0 \right)
    [/tex]
    which shows constant tidal tension in the the [itex]\vec{e}_2[/itex] direction.

    Because there are so many nice-looking charts on the factors, there are dozens of alternative charts one could use instead, some of which have advantages. For example
    [tex]
    ds^2 = 4 a^2 \, \left( -\sec(u+v)^2 \, du \, dv + \frac{dx^2 + dy^2}{\left( 1+x^2+y^2 \right)^2} \right),
    [/tex]
    [tex]
    -\pi/2 < u+ v < \pi/2, \; -\infty < x, \, y < \infty
    [/tex]
    is a double null stereographic chart. There is a lovely connection to elementary algebraic geometry here: the manifold [itex]H^{1,1}[/itex] as an emedded surface in [itex]E^{1,2}[/itex] (the hyperboloid of one sheet asymptotic to a light cone) is ruled by two families of null lines, which are the null geodesics in the surface! Thus, the coordinates u,v are easily visualized in this picture.
     
    Last edited: Nov 9, 2007
  5. Nov 14, 2007 #4

    Chris Hillman

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    Yet another chart for the Nariai dS-lambdavacuum

    Hi, hammertime, are you still out there somewhere? I'm still curious who the unamed UCL researcher is. Would this by any chance be Christian Boehmer, a researcher who is indeed at University College, London? I ask because I just noticed this new arXiv eprint which does employ the Nariai dS-lambdavacuum. However, if this is the eprint allegedly described in the New Scientist article you mentioned (which I haven't read), then I think either your description was inaccurate, or their article was inaccurate.

    The eprint in question studies the question: what happens to static spherically symmetric perfect fluids when we interpret the Einstein tensor computed from the metric as including a positive Cosmological constant term as well as the stress-energy tensor of a perfect fluid? The authors study two examples in some detail, the Whittaker ssspf (1968) which is the nonrotating case of the Wahlquist perfect fluid solution, and the Tolman IV ssspf (1939) which I mentioned above. In the first case, they find that for Lambda less than a critical value, they can match across the zero-pressure surface of a dS-lambdafluid ball to a dS-lambdavacuum exterior and then to a second dS-lambdafluid ball, to obtain a singularity free solution. For Lambda equal to the critical value, they match two dS-lambdafluid balls to the Nariai dS-lambdavacuum. In this case, they say, the spatial hyperslices have the form of a cylinder with two hemispherical caps. For larger values of Lambda they find unavoidable past and future strong spacelike curvature singularities (Big Bang and Big Crunch, if you will).

    This doesn't seem to have much to do with black hole interiors, but rather with highly artificial cosmological models with the intention of exploring how positive Lambda can change the global character of simple stellar models.

    The authors use yet another chart for the Nariai dS-lambdavacuum which is probably more perspicuous than any I used above (sorry!), so let me briefly discuss the Gaussian chart they use.

    Start with the frame field
    [tex]
    \vec{e}_1 = \frac{1}{\cos(z)^2} \, \partial_t + k \, \tan(z) \, \partial_z, \;
    \vec{e}_2 = \frac{\tan(z)}{\cos(z)} \, \partial_t + \frac{k}{\cos(z)} \, \partial_z,
    [/tex][tex]
    \vec{e}_3 = k \, \partial_\theta, \; \vec{e}_4 = \frac{k}{\sin(\theta)} \, \partial_\phi
    [/tex]
    This gives the line element
    [tex]
    ds^2 = -\cos(z) \, dt^2 + \frac{dz^2 +d\theta^2 + \sin(\theta)^2 \, d\phi^2}{k^2},
    [/tex][tex]
    -\infty < t, < \infty, \; -\pi/2 < z < \pi/2, \; 0 < \theta < \pi/2, \; -\pi < \phi < \pi
    [/tex]
    Note that the constant time slices are obviously globally isometric to [itex]R \times S^2[/itex] in this chart; since they are all identical to one another, we can visualize them all as a single ordinary finite length cylinder (mentally replace circles with two-spheres).

    The geodesic equations are very simple and immediately yield the first integrals
    [tex]
    \dot{t} = \frac{E}{\cos(z)^2}, \;
    \dot{z}^2 = A + \frac{k^2 \, E^2}{\cos(z)^2}, \;
    \dot{\phi} = \frac{L}{\sin(\theta)^2}, \;
    \dot{\theta}^2 = B - \frac{L^2}{\sin(\theta)^2}
    [/tex]

    The given frame describes a family of inertial observers, including a static sphere who remain at [itex]z=0[/itex]. The expansion tensor of the timelike geodesic congruence defined by [itex]\vec{e}_1[/itex] is
    [tex]
    {H \left[\vec{e}_1\right] }_{ab} = k \, \operatorname{diag} \left(1, 0, 0 \right)
    [/tex]
    which shows constant axial expansion, so we can visualize our observers as "latitude rings" on the cylinder which move away from the equatorial ring [itex]z=0[/itex] with coordinate speed [itex]dz/dt = k/2 \, \sin(2 \, z)[/itex], which for small [itex]|z|[/itex] is about k z. This is exactly what should happen for constant axial expansion! The tidal tensor is
    [tex]
    {E \left[\vec{e}_1\right] }_{ab} = -k^2 \, \operatorname{diag} \left(1, 0, 0 \right)
    [/tex]
    which shows constant axial tidal tension.

    The axial plane wave null geodesic congruence is
    [tex]
    \vec{k} = \frac{E}{\cos(z)^2} \, \partial_t + \frac{k \, E}{\cos(z)} \, \partial_z
    [/tex]
    which has vanishing optical scalars (as should happen for a plane wave). We can visualize its wavefronts as latitude rings (really two-spheres, of course) which travel up the cylinder with coordinate speed
    [tex]
    \frac{dz}{dt} = k \, \cos(z), \; -\pi/2 < z < \pi/2
    [/tex]
    As you can see, at the boundaries of our chart we have coordinate singularities, and the vanishing of the coordinate speed of our plane wave there shows these coordinate singularities comprise two "cosmological horizons". Needless to say, the physical velocity of our plane is everywhere unity (the speed of light in our geometrized units).
     
    Last edited: Nov 14, 2007
  6. Nov 15, 2007 #5

    Chris Hillman

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    Clarifying the global structure of the Nariai dS-lambdavacuum

    Come to think of it, the chart in my preceding post might also be a bit confusing in this context. So here's another one:
    [tex]
    ds^2 = \frac{-d\eta^2+dz^2}{\cos(\eta)^2/a^2} + \frac{-d\zeta^2+d\phi^2}{\cosh(\zeta)^2/a^2},
    [/tex][tex]
    -\pi/2 < \eta < \pi/2, \; -\infty < \zeta < \infty, \; -\pi < z, \, \phi < \pi
    [/tex]
    which the reader will recognize as the product of a Mercator chart on S^2 with an analogous chart on [itex]H^{1,1}[/itex] (speaking of MTW, see the "arc parameter" in their discussion of the FRW dust with S^3 hyperslices, where a very similar chart arises).

    Looking just at the [itex]H^{1,1}[/itex] factor
    [tex]
    ds^2 = \frac{-d\eta^2+dz^2}{\cos(\eta)^2/a^2}, \; \pi/2 < \eta < \pi/2, \; -\pi < z < \pi
    [/tex]
    you can immediately draw the Carter-Penrose diagram (each point representing a sphere of radius 1/a in the Nariai dS-lambdavacuum) with lower and upper boundaries [itex]\eta=\pm \pi/2[/itex] and with sides [itex]z = \pm \pi[/itex] which we expect to identify. To confirm that this represents [itex]H^{1,1}[/itex], consider the embedding in [itex]E^{-1,2}[/itex] (signature -++):
    [tex]
    X(\eta,z) = \left[ \begin{array}{c}
    a\, \tan(\eta) \\ a \, \sec(\eta) \, \cos(z) \\a \, \sec(\eta) \, \sin(z)
    \end{array} \right]
    [/tex]
    So the Penrose diagram makes this conformal to part of a cylinder with -+ signature. The upper boundary is future null infinity and the lower boundary is past null infinity. Notice this is a bit different from the diagram for de Sitter-lambdavacuum, which is discussed in Hawking and Ellis!

    Now we can immediately read off an obvious frame field,
    [tex]
    \vec{e}_1 = \frac{\cos(\eta)}{a} \, \partial_\eta, \;
    \vec{e}_2 = \frac{\cos(\eta)}{a} \, \partial_z
    [/tex]
    [tex]
    \vec{e}_3 = \frac{\cosh(\zeta)}{a} \, \partial_\zeta, \;
    \vec{e}_4 = \frac{\cosh(\zeta)}{a} \, \partial_\phi
    [/tex]
    which is defined everywhere and which turns out to represent the physical experience of inertial observers; their world lines are the longitude lines in the embedding we gave above. Thus, they contract and then expand; specifically, the expansion tensor reads (components taken wrt our frame):
    [tex]
    {H \left[ \vec{e}_1 \right]}_{ab} =
    \frac{\sin(\eta)}{a} \, \operatorname{diag} \left( 1,0,0 \right)
    [/tex]
    However, from previous work we know there exists another inertial congruence which has constant axial expansion [itex]1/a[/itex]. To find it we boost the first two vectors like
    [tex]
    \vec{f}_1 = \cosh(f) \, \vec{e}_1 + \sinh(f) \, \vec{e}_2, \;
    \vec{f}_2 = \sinh(f) \, \vec{e}_1 + \cosh(f) \, \vec{e}_2
    [/tex]
    where [itex]f(\eta,z)[/itex] is an undetermined function, and demand both that the acceleration vector vanish and that the expansion tensor satisfy
    [tex]
    {H \left[ \vec{f}_1 \right]}_{ab} =
    \frac{1}{a} \, \operatorname{diag} \left( 1,0,0 \right)
    [/tex]
    This determines our function in the form
    [tex]
    \exp(f(\eta,z)) = \frac{\cos(z/2+\eta)+\sin(z/2)}{\cos(z/2-\eta)-\sin(z/2)}
    [/tex]
    The new frame is only valid on the inverted triangle in the Carter-Penrose diagram to which I drew attention in my last post. The two bounding null geodesics correspond to parallel null rays in the embedding above; as I said, these can be considered cosmological horizons analogous to those which arise in the de Sitter lambdavacuum. A better way of saying all this might be to say that the new congruence and the new frame are defined in the absolute future of the "point" [itex]\eta=-\pi/2, \, z=0[/itex] in our Carter-Penrose diagram. (Compare the similar discussion of dS in HE.)

    The new congruence defines a new family of orthogonal hyperslices, which turn out to also be isometric to [itex]R \times S^2[/itex].

    (BTW, in the preceding post, when I wrote
    I was guilty of writing lazily. Of course, I started with the obvious frame field read off the latter metric and followed the same procedure of boosting by an undetermined parameter which is a function of z, and then determining the unknown function by demanding that the acceleration vector vanish.)

    There is also a dual congruence which is "eternally contracting" with constant axial expansion tensor. Thus I should stress that our three congruences arise by imposing different boundary conditions to define different families of observers. These three families of observers are each inertial but have quite different histories in terms of "Hubble expansion"! Recall that in the de Sitter and Nariai lambdavacuums, unlike cosmological dust models, there are no preferred timelike geodesic congruences.

    BTW, in my Post #3, I should have mentioned that using the null coordinates u,v as parameters, our embedding becomes
    [tex]
    X(u,v) = \left[ \begin{array}{c}
    a\, \tan(u+v) \\ a \, \sec(u+v) \, \cos(u-v) \\a \, \sec(u+v) \, \sin(u-v)
    \end{array} \right]
    [/tex]
    This gives a notable double ruling by null geodesics on [itex]H^{1,1}[/itex].
     
    Last edited: Nov 16, 2007
  7. Nov 17, 2007 #6

    Chris Hillman

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    Illustrating the Nariai dS-lambdavacuum I

    In my last post, I mentioned a parameterization of [itex]H^{1,1}[/itex] as an embedded surface in [itex]E^{1,2}[/itex] given by
    [tex]
    X(u,v) = \left[ \begin{array}{c}
    a\, \tan(u+v) \\
    a \, \sec(u+v) \, \cos(u-v) \\
    a \, \sec(u+v) \, \sin(u-v)
    \end{array} \right]
    [/tex]
    You can check that this does indeed parameterize the hyperboloid of one sheet
    [tex]-x_1^2 + x_2^2 + x_3^2 = a^2[/tex]
    (Note that the case a=1 can be characterized (with tongue in cheek) as tachyonic velocity space, just as [itex]H^2[/itex] is the velocity space for ordinary test particles in [itex]E^{1,2}[/itex].)

    As I said, this gives a double ruling by null geodesics:
    [​IMG]

    [EDIT: I just learned from Evo that the img tags are not enabled in this forum at PF. But look at the thumbnails below; on some browsers, mousing over the leftmost thumbnail will pop up its name, "H11_doubleruling.jpg". In future I should probably use The Gimp to add captions to such figures to avoid possible confusion about which figure is which.]

    In this figure, the straight lines in the surface are null geodesics of [itex]E^{1,2}[/itex] as well as of [itex]H^{1,1}[/itex]. As you can see, there are two rulings; one family consists of the curves [itex]u=u_0[/itex] and the other of the curves [itex]v = v_0[/itex]. Recall that this parameterization gives the line element
    [tex]
    ds^2 = \frac{-4 a^2 \, du \, dv}{\cos(u+v)^2}, \; -\pi/2 < u + v < \pi/2
    [/tex]

    Earlier I discussed another parameterization:
    [tex]
    X(\tau,\phi) = \left[ \begin{array}{c}
    a\, \sinh(\tau/a) \\
    a \, \cosh(\tau/a) \, \cos(\phi) \\
    a \, \cosh(\tau/a) \, \sin(\phi)
    \end{array} \right]
    [/tex]
    which gives the line element
    [tex]
    ds^2 = -d\tau^2 + a^2 \, \cosh(\tau/a)^2 \d\phi^2, \;
    -\infty < \tau < \infty, \; -\pi < \phi < \pi
    [/tex]
    From this we can read off an obvious frame field
    [tex]
    \vec{e}_1 = \partial_t, \; \vec{e}_2 = \frac{a}{\cosh(\tau/a)} \, \partial_\phi
    [/tex]
    Here, the timelike unit vector field gives a timelike geodesic congruence, the world lines of a family of inertial observers; our equatorial trig chart is comoving with these observers. As I said, these world lines appear as "longitude hyperbolas", and their orthogonal hyperslices appear as "latitude circles":
    [​IMG]
    I pointed out that when we construct the Nariai ds-lambdavacuum as the product [itex]H^{1,1} \times S^2[/itex], these properties are inherited, and the expansion tensor of our timelike geodesic congruence then shows axial contraction on [itex]\eta< 0[/itex] followed by axial expansion on [itex]\eta>0[/itex].

    A third parameterization is
    [tex]
    X(t,z) = \left[ \begin{array}{c}
    a\, (t^2-z^2-1)/(2 \, t) \\
    a \, (t^2-z^2+1)/(2 \, t) \\
    a \, z/t
    \end{array} \right]
    [/tex]
    which gives the line element of the lower half plane chart:
    [tex]
    ds^2 = \frac{-dt^2 + dz^2}{(t/a)^2}, \; -\infty < t < 0, \; -\infty < z < \infty
    [/tex]
    In this chart, the obvious frame
    [tex]
    \vec{f}_1 = \frac{-t}{a} \, \partial_t, \; \vec{f}_2 = \frac{-t}{a} \, \partial_z
    [/tex]
    (regarding the minus signs, recall that here t < 0!) is distinct from the first frame, as we'll see in a moment, but [itex]\vec{f}_1[/itex] gives a timelike geodesic congruence. Indeed, the expansion tensor of this congruence shows constant axial expansion, unlike the time varying expansion of the first congruence. In the LHP chart, the world lines of these observers appear as vertical coordinate lines, but our time coordinate is not the proper time of these observers--- so strictly speaking this is not a comoving chart! If we plot these world lines in the embedding, we obtain this picture:
    [​IMG]
    In this picture, notice that I have also drawn two null geodesics, one from each family in our double ruling, which happen to be parallel in [itex]E^{1,2}[/itex]. These two null geodesics intersect at past null infinity, as I said before. They split [itex]H^{1,1}[/itex] into top and bottom halves; our second family of observers (and the LHP chart) is only defined in the upper half. (Naturally there is also an UHP chart and it describes a similar family of eternally contracting observers; all these families experience constant axial tidal tension, as I said; part of the point here is to illustrate how to disentangle phenomena which derive from properties some congruence from phenomena which derive from properties of the spacetime itself.).

    In euclidean geometry, the simplest way of obtaining two-dimensional real projective space is to identify antipodal points on the sphere [itex]S^2[/itex]. In the same way, we can identify antipodal points on [itex]H^{1,1}[/itex] (which is the same as [itex]S^{1,1}[/itex] at the minimal required level of structure) in order to obtain a projective version.

    (I should say that these figures were drawn using Maple.)
     

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    Last edited: Nov 18, 2007
  8. Nov 17, 2007 #7

    Chris Hillman

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    Illustrating the Nariai dS-lambdavacuum I

    [EDIT: A bit of experimentation suggests that PF currently allows at most 3 attachments per post, so I have put the other two figures in this post :rolleyes:]

    The orthogonal hyperslices for the eternally expanding observers are seen here:
    [​IMG]

    Last but not least, the Carter-Penrose diagram for the Nariai dS-lambda vacuum looks like this:
    [​IMG]
    In this diagram, each point represents a two-sphere of radius a, and we should think of identifying [itex]\phi = \pm \pi[/itex], so that we have conformally mapped [itex]H^{1,1}[/itex] to a portion of a cylinder (with Minkowski signature). In the diagram, past null infinity corresponds to [itex]\eta=-\pi/2[/itex] while future null infinity corresponds to [itex]\eta=\pi/2[/itex].

    This diagram is essentially the chart on [itex]H^{1,1}[/itex] with line element
    [tex]
    ds^2 = \frac{a^2 \, \left(-d\eta^2 + d\phi^2 \right)}{\cos(\eta)^2},
    -\pi/2 < \eta < \pi/2, \; -\pi < \phi < \pi
    [/tex]
    which can be obtained from the parameterization
    [tex]
    X(\eta,\phi) = \left[ \begin{array}{c}
    a \, \tan(\eta) \\
    a \, \sec(\eta) \, \cos(\phi) \\
    a \, \sec(\eta) \, \sin(\phi)
    \end{array} \right]
    [/tex]
    In this diagram, note the two null geodesics which meet at [itex]\eta=-\pi/2, \, \phi=-\pi[/itex]. These are the two drawn above which as I remarked are parallel in [itex]E^{1,2}[/itex]. I stress again that they belong to different rulings in the double ruling! The figure showing theworld lines and orthogonal hyperslices of the eternally expanding observers has been oriented so as to suggest (I hope) its relation with the Carter-Penrose diagram.

    Since embeddings are not physically significant, according to gtr, why do I give all these parameterizations? The reasons are mainly mathematical:
    • the corresponding embeddings provide a quick and easy check that we have not misinterpreted the global structure of our spacetime,
    • by identifying the components in two parameterizations (of the hyperboloid of one sheet, [itex]-x_1^2 + x_2^2 + x_3^2 = a^2[/itex], in [itex]E^{1,2}[/itex]), we easily obtain transformations between the corresponding two coordinate charts.
     

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    Last edited: Nov 18, 2007
  9. Nov 18, 2007 #8

    cristo

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    Thanks for your informative posts, Chris. I can confirm that Chris Boehmer was indeed quoted on this NS page. I'm not sure if it's exactly the same as the on that the OP read, but it seems to be. Here's a few excerpts incase you can't read it:
     
  10. Nov 18, 2007 #9

    Chris Hillman

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    Probably not the same eprint after all

    Hi, cristo,

    I'm glad to see confirmation that someone is reading this!

    From the quotations you gave (indeed, I have not read the NS article), it seems to me unlikely that the NS article is discussing the arXiv eprint I saw. It looks like in the work discussed in NS, Böhmer and Vandersloot numerically simulated LQG inside a black hole model and believe they spotted evidence of constant axial expansion of some infalling particles inside the hole, and suggest that in some effective classical field theory interpretation with gtr, this should be modeled classically as some kind of matching of a Nariai-dS interior region to a Schwarzschild vacuum (to take the simplest case). But if all they spotted was axial expansion of freely falling observers.... well, see my discussion of the Frolov observers in the future interior region of the humble Schwarzschild vacuum! Hmm... "got rid of the singularity" suggests they saw numerical evidence of constant axial tidal tension, which would be more convincing.

    Even though I still have essentially no idea what NS was gushing about, I think it can be helpful to see illustrations of some basic computational skills, and I also feel that it is very beneficial to anyone working in gtr to have at hand a cornucopia of charts on [itex]S^2, \, H^2, \, H^{1,1}[/itex].

    I just noticed that way up above in my Post #2, I wrote
    Oops! I thought I had edited that out when I wrote my Post #3. The Riemann tensor was given correctly but of course this is [itex]S^{1,1} \times H^2[/itex] (aka Nariai-AdS lambda vacuum), not [itex]H^{1,1} \times H^2[/itex]. In my Post #3 I noticed that of the two forms of the Nariai lambdavacuum (corresponding to the two possible signs of [itex]\Lambda[/itex], I had probably given him the wrong one. Thereafter in this thread I have been discussing [itex]H^{1,1} \times S^2[/itex], aka Nariai-dS lambda vacuum. Sorry for any confusion!
     
    Last edited: Nov 18, 2007
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