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Exotic stress-energy tensor and causality

  1. Jul 15, 2014 #1
    Hello everybody. I would like to kindly ask your help with a hypothetical hairy question about which I think a lot recently.

    It is known fact, that it is not possible to construct a wormhole without exotic mass that violates the weak energy condition. It is also known that many quantum fields violate probably all known energy conditions to some degree, but this violation is usually so small that construction of a wormhole using this violation remains very implausible. I did some simple calculations to get some insight into this problem. While the Swarzschild metric

    ds2 = -c2 (1 - rS / r) dt2 + (1 - rS / r)-1 dr2 + dΩ

    gives zero stress-energy tensor (it is a vacuum solution), we may try to remove the curvature in the time-part and obtain "tunnel" in space without event horizon.

    ds2 = -c2 dt2 + (1 - rS / r)-1 dr2 + dΩ

    Such metric is consistent with stress-energy tensor

    Tμν = - c4 rS / 8 π G r2 (r - rS) dr2 + c4 rS / 16 G π r (dΘ2 + sin2 Θ dφ2).

    Such stress-energy tensor is traceless and consists of pure pressure, without rest mass.

    So my question is: Let us assume there is a form of matter that produces stress-energy tensor with only space-like components.
    a) Does this immediately lead to a causality violation?
    b) Does this inevitably lead to infinite "creation of matter from nothing"?

    Example of such stress-energy tensor of point source satisfying conservation laws in Cartesian coordinates (I think):

    Tμν = x / r3 (dx2 - dy2 - dz2) + y / r3 (dx dy + dy dx) + z / r3 (dx dz + dz dx)

    About point a)
    Here, I would like to avoid tachyons with space-like four-velocities creating the stress-energy tensor like Tμν ~ p uμ uν. Rather, I am asking if it is plausible that there would be some particle with negligible rest mass that produces strong field with stress-energy tensor like Tμν ~ p (uμ uν + gμν) + ρ uμ uν, where ρ ~ 0. Am I missing some implication of the stress-energy tensor on the interaction of the exotic matter with the regular matter?

    About point b)
    I know that there is this issue of infinite low potential energy. If one would allow negative mass, for example matter for which Tμν ~ - ρ uμuν, an empty space could spawn regular matter + this exotic matter and never stop. (Vacuum would be unstable, if you will.) But it seems to me that the pure space-like stress energy tensor does not cause this form of catastrophe. At least if the trace Tμμ is negative and therefore the same as for regular matter. Am I correct in this?

    Thank you for all your answers and for your patience with my speculations. :-)
  2. jcsd
  3. Jul 15, 2014 #2


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    I haven't looked at your equations, but from your verbal description it sounds like your example violates the trace energy condition (TEC) and the dominant energy condition (DEC). The DEC is the condition you need in order to guarantee that there is no flux of energy propagating at speeds greater than c. Therefore the form of matter assumed in your example probably could be exploited to violate causality.

    To answer this, just calculate the divergence of your stress-energy tensor. It should be zero.
  4. Jul 15, 2014 #3
    Thank you very much for your reply!

    I see, I wasn't aware of this. I can intuitively understand why DEC implies causality conservation. But does DEC violation automatically mean causality violation? Is there some theoretical example of classical matter that preserves causality and breaks the DEC? (Or some weaker condition guaranteeing the causality).

    This is not what I meant. Let us assume there are two matter fields T1μν and T2μν, which both satisfy

    T1μν = 0,
    T2μν = 0,

    but for which

    T1μν + T2μν = 0.

    So the divergence is zero for each of them, but the matter still emerges from nothing. This cannot happen for normal matter, but for some sort of exotic matter that has trace of stress-energy tensor opposite to normal matter, it can. I wonder if some pathological thing like that can happen to my matter. (It cannot be exactly the same thing, since Tμν of matter satisfying energy conditions (normal matter) and matter violating them (my matter) cannot sum exactly to zero, but I wonder if there is some similar pathology present..

    Thank you!
  5. Jul 15, 2014 #4


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    In your case the energy momentum tensor, since it has to have the pressure and energy density in its diagonal form, would imply that you have both opposite energy density and pressure.
    For the pressure it sounds normal to me (that's a feature of the vacuum)... But I don't really find a meaning in negative energy density.
  6. Jul 15, 2014 #5


    Staff: Mentor

    Well, if ##T^{\mu \nu}{}_{; \nu} \neq 0## for your matter, then your matter is physically impossible, so it's a moot point, isn't it? (In other words, you need to check that *first*, before even bothering to wonder what any other implications are.)
  7. Jul 16, 2014 #6
    This is a misunderstanding. My matter satisfies Tμν = 0. I was arguing that this condition is not enough to prevent creation of matter from nothing if (some particular form of) exotic matter is present, I never said I would like to violate energy/momentum conservation. I am interested in matter with following properties:

    1) Tμν = 0
    2) Tμνkμ kν ≥ 0 for every timelike vector field kμ (opposite of the weak energy condition).
    3) Tμμ ≤ 0 (same as for regular matter)

    (I am using convention -+++).

    By other words, the matter conserves energy and the stress-energy tensor is space-like and corresponds to negative pressure. Now, I am interested what pathologies emerge for such matter (what strange things will happen that probably shouldn't?) and if there can be some additional condition that would provide causality. (For example: We forbid waves in such matter, because they would be tachyonic. Or: The matter acts on normal matter only by some special forces.. I wonder if there is a condition that would make it causal.)
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