On 1994, Miguel Alcubierre proposed a solution of the Einstein equations (see here) describing a space-time bubble moving at arbitrary speed. It is important to notice that no violation of the light speed limit happens because is the space-time moving and inside the bubble everything goes as expected. This kind of solutions of the Einstein equations have a fundamental drawback: they violate Weak Energy Condition (WEC) and, in order to exist, some exotic matter with negative energy density must exist. Useless to say, nobody has ever seen such kind of matter. There seems to exist some clue in the way Casimir effect works but this just relies on the way one interprets quantum fields rather than an evidence of existence. Besides, since the initial proposal, a great number of studies have been published showing how pathological the Alcubierre’s solution can be, also recurring to quantum field theory (e.g. Hawking radiation). So, we have to turn to dream of a possible interstellar travel hoping that some smart guy will one day come out with a better solution.

Of course, Alcubierre’s solution is rather interesting from a physical point of view as it belongs to a number of older solutions, like wormholes, time machines and like that, yielded by very famous authors as Kip Thorne, that arise when one impose a solution and then check the conditions of its existence. This turns out to be a determination of the energy-momentum tensor and, unavoidably, is negative. Then, they violate whatever energy condition of the Einstein equations granting pathological behaviour. On the other side, they appear the most palatable for science fiction of possible futures of space and time travels. In these times where this kind of technologies are largely employed by the film industry, moving the fantasy of millions, we would hope that such futures should also be possible.

It is interesting to note the procedure to obtain these particular solutions. One engineers it on a desk and then substitute them into the Einstein equations to see when are really a solution. One fixes in this way the energy requirements. On the other side, it is difficult to come out from the blue with a solution of the Einstein equations that provides such a particular behaviour, moving the other way around. It is also possible that such solutions are not possible and imply always a violation of the energy conditions. Some theorems have been proved in the course of time that seem to prohibit them (e.g. see here). Of course, I am convinced that the energy conditions must be respected if we want to have the physics that describes our universe. They cannot be evaded.

So, turning at the question of the title, could we think of a possible warp drive solution of the Einstein equations without exotic matter? The answer can be yes of course provided we are able to recover the York time, or warp factor, in the way Alcubierre obtained it with its pathological solution. At first, this seems an impossible mission. But the space-time bubble we are considering is a very small perturbation and perturbation theory can come to rescue. Particularly, when this perturbation can be locally very strong. On 2005, I proposed such a solution (see here) together with a technique to solve the Einstein equations when the metric is strongly perturbed. My intent at that time was to give a proof of the BKL conjecture. A smart referee suggested to me to give an example of application of the method. The metric I have obtained in this way, perturbing a Schwarzaschild metric, yields a solution that has an identical York time (warp factor) as for the Alcubierre’s metric. Of course, I am respecting energy conditions as I am directly solving the Einstein equations that do.

The identity between the York times can be obtained provided the form factor proposed by Alcubierre is taken to be 1 but this is just the simplest case. Here is an animation of my warp factor.

It seen the bubble moving as expected along the x direction.

My personal hope is that this will go beyond a mathematical curiosity. On the other side, it should be understood how to provide such kind of perturbations to a given metric. I can think to the Einstein-Maxwell equations solved using perturbation theory. There is a lot of literature about and a lot of great contributions on this argument.

Finally, this could give a meaning to the following video by NASA.

Hi Marco,

Came across your post while looking for existing literature.

I am trying to figure out if the new term in eq. 25 of Li-Xin’s “Theory of Electromagnetic and Gravitational Interactions” (arXiv:1511.01260v2) could lead to negative energy density under certain (EM) conditions.

This would be quite elegant, since it would not require to “twist around” with QFT and produce a result from purely classical QED / GRT considerations (hope).

Any thoughts?

– Rick

Dear Rick,

It seems a rather interesting paper. Anyway, it works in five dimensions and this would require an infinite tower of scalar excitations that have not been seen so far.

On the other side, it is possible that, as it also happens for the Higgs field, there are scalar fields permeating space-time (e.g. quintessence) that could play a role. This kind of fields has the property to have negative energy density but one could not escape QFT. E.g., see my paper https://arxiv.org/abs/1409.3775.

In classical GRT, warping factors can arise quite naturally. E.g., see https://www.mdpi.com/2624-8174/2/4/39. This could be easily put to a test.

Marco

Dear Marco

Thanks for your feedback, I will go through your papers and try to find a lead (I am a bit rosted, I have to say, so it will take some time, but motivated).

I was looking at this site https://www.uaptheory.com/ the other day. It looks at IR sensor data in Chile and at the US Navy “leak”. The Maldacena argument at the beginning seemed a bit like a “one way street”, but the picture analysis / observational part looks quite pertinent (i.e. lensing effect around the objects).

Since the US Navy confirmation, we have some reason to believe that those videos are not fake. Question is, if the observed effects are sensor or video compression artifacts, but having some experience working with mil. EO/IR sensors, I do not think so.

– Rick

Dear Rick,

Yes, I know that talking about this matter one enters into the lion turf. I am following all the fuss around this matter in US. My take is that, as customary, Pentagon has a new toy and is trying to mislead other contenders. When this will come out in full only they possibly know.

Marco