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What is the difference between an Einstein-Rosen Bridge (wormhole) and a tunnel through a mountain? Obviously, light that travelled around the mountain would take longer to reach other side so that light travelling through the tunnel. So, in principle, a relativistic particle travelling at 0.9c through the tunnel could beat the light that travelled around the mountain. This does not violate any known laws of physics (i.e. causality), does not permit time travel, and tunnels definitely exist.

However, many physicists express doubts about the existence of an Einstein-Rosen Bridge (from a theoretically standpoint), which, from what I understand, is simply just a tunnel through space. I had one physics professor who dismissed it as science fiction because he thought it was allowing faster-than light travel, violating special relativity (I was in his special relativity class at the time). A prior, I don't see why the Einstein-Rosen Bridge would violate either causality or special relativity and permit time travel. After all, it would just provide two paths for light to take, one of which would be shorter than the other. This may appear to permit faster than light travel because the beam of light travelling through the wormhole would beat the beam of light travelling around it. How is this different from light travelling through a mountain tunnel beating light travelling around the mountain?

If it is not possible (in principle) to construct such tunnel through space, why? Would some law of quantum mechanics or quantum field theory restrict the topology of empty space so that there can only be no tunnels or holes like there are through matter? In particular, do some physicists believe, under physical assumptions about the stress-energy tensor, that spatial slices of spacetime must be simply connected (fundamental group is trivial; no holes)?

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    $\begingroup$ If a wormhole was just a tunnel, why would it have such a fancy name? Remember that in special and general relativity we don't consider time and space separately. A wormhole connects two spacetime points. $\endgroup$
    – Marius Ladegård Meyer
    Commented May 7, 2023 at 11:28

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The tunnel along the axis of the sphere connecting north to south pole of a 2-sphere can be passed slowly, nearly at velocity zero. In space time the sphere is a set of points each with a time axis. Moving slowly means to go small distances in large time intervals. These paths in space time are called timelike. Timelike paths are limited to velocity dx/dt < c in a relativistic space-time.

On the other hand, there are spacelike curves. They connect points far away in no time or just a bit of time. They are impossible paths for matter and light, or to say it more drastically, they are pure fantasy, because they exist only for a moment of time like a chain of lights just flashing synchronously once along a line, whose points move then forward for all times unnoticed.

The world lines connecting the two sheets of the Schwarzschild geometry, completed by following all spacelike geoedetics, are all spacelike. No communication is possible. Nevertheless there is an effect, because the spectrum of waves depend on the boundary of the space slices along the time axis.

The identification of the two small circles around the poles of the 2-sphere, on the third hand, produces onother case of topology: On may get in no time onto the other hemisphere. Most important in geometry is another model, the projective sphere, where antipodes are identified. This is a model of a hemisphere with antipodes on the equator identified or, the model, where points are represented by straight lines through the origin.

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While a tunnel in mountain is a "hole" in matter, a wormhole is a "hole" in fabric of spacetime. Although spacetime and matter are closely related via equations of general relativity, they are different entities. Because of that different physical laws apply.

What is more, in your scenario with the tunnel through mountains, on both path speed of light is never exceeded. Whereas in case of wormholes, you would be able to traverse distances which need velocities higher than speed of light. Breaking this "light barrier" leads to issues with causality and cause time travelling. In light cone such paths are called space-like. In contrast, the mountain tunnel scenario involves only so-called time-like paths (i.e. speeds are lower than $c$) where causality is not broken.

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  • $\begingroup$ So the physical problem with a wormhole is that it would permit spacelike paths? I originally thought it was just two different null-geodesics? Wormholes aside, can there be more than one null-like geodesic through spacetime; isn't this how gravitational lensing works? And can one be shorter than the other? $\endgroup$
    – Robert
    Commented May 8, 2023 at 0:12
  • $\begingroup$ @Robert: Yes, space-like paths breach causality, therefore, they are forbidden. Null geodesics are of course allowed, those are followed by light (or any particle with zero mass at rest). But wormholes allow for speed higher than $c$, hence space-like geodesics appear. And this is the issue. Second question: yes, you can have more paths between two objects, this is a same as taking two roads with different length. However, on all of them, speed of light is still $c$, so causality is not breached. $\endgroup$
    – Martin Vesely
    Commented May 8, 2023 at 7:22
  • $\begingroup$ Thanks, Martin! Could there still be a "tunnel" through space the same way there is a tunnel through a mountain? This would simply amount to light taking two different paths between points, one of which is shorter than the other. $\endgroup$
    – Robert
    Commented May 9, 2023 at 2:13
  • $\begingroup$ @Robert: yes of course, you can send light in many different ways from a source to a target. Think for example about interference effects which are based on light taking paths of different lengths. But on any path, speed of light cannot be exceeded. However, I would not call different paths a tunnel. $\endgroup$
    – Martin Vesely
    Commented May 9, 2023 at 5:50

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