Hello everybody. I would like to kindly ask your help with a hypothetical hairy question about which I think a lot recently.(adsbygoogle = window.adsbygoogle || []).push({});

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

ds^{2}= -c^{2}(1 - r_{S}/ r) dt^{2}+ (1 - r_{S}/ r)^{-1}dr^{2}+ 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.

ds^{2}= -c^{2}dt^{2}+ (1 - r_{S}/ r)^{-1}dr^{2}+ dΩ

Such metric is consistent with stress-energy tensor

T_{μν}= - c^{4}r_{S}/ 8 π G r^{2}(r - r_{S}) dr^{2}+ c^{4}r_{S}/ 16 G π r (dΘ^{2}+ sin^{2}Θ dφ^{2}).

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):

T_{μν}= x / r^{3}(dx^{2}- dy^{2}- dz^{2}) + y / r^{3}(dx dy + dy dx) + z / r^{3}(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 T^{μν}~ p u^{μ}u^{ν}. 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 T^{μν}~ p (u^{μ}u^{ν}+ g^{μν}) + ρ u^{μ}u^{ν}, 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 T^{μν}~ - ρ u^{μ}u^{ν}, 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 T^{μ}_{μ}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. :-)

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# Exotic stress-energy tensor and causality

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