BEFORE ANSWERING PLEASE READ THE TEXT!
The problem posed here is to shorten the time measured by a traveller in his rest-frame, beyond the shortenning ensured by the Special Relativity. It seems that electromagnetic fields may be of help. The question is whether enough strong fields can be realized in practice.
The question whether a worm-hole can be created in our 4D space-time was asked frequently and it was proved that the answer is negative. It would necessitate a 5th dimension, and no such dimension is known to us. As an analogy, between two points on the surface of a simple sphere - a 2D surface - one could dig a worm-hole by passing through the body of the sphere, i.e. exploiting the fact that the sphere is imbedded in the 3D space.
The relativistic interval between two neighbor points P1 and P2 in the flat 4D space-time is given by
(1) F = (c dt)2 - Σi=1,2,3 (dxi)2
If F > 0 it can be denoted as (c dτ)2 where τ is the proper time, the time recorded by a traveller between P1 and P2, in his rest-frame of coodinates. Obviously, in this case (dτ)2 < (dt)2. However, for shortenning dτ significantly with respect to dt, the velocity V of the traveller has to be of the order of magnitude of the light velocity.
The problem posed here is whether there is a possibility to shorten dτ even more, by means of strong fields. Indeed, Einstein's Field Equations (EFE) contain in their RHS, fields. They change the metric from (1) to
(2) F' = Σi,j=0 3 gi,j dxi dxj
If F' is positive and smaller than F, one can denote F' = (c dτ')2, and F - F' = (dϕ)2, and get
(3) (c dτ')2 = (c dt)2 - Σi=1,2,3 (dxi)2 - (dϕ)2.
Thus, the effect of ϕ is analogous to a 5th dimension.
The equation (3) can be written in the more useful form
(4) (dτ')2 = { 1 - V2/c2 - [dϕ/(cdt)]2 } (dt)2.
Fields that render F' < F can be generated, but the question is whether enough big intensities can be obtained for [dϕ/(cdt)]2 to be significant.
For instance, the contribution of the gravitational field is is not significant. In the Schwarzchsild metric [dϕ/(cdt)]2 = 2GM/c2R. At the surface of the Earth this is of the order 10-11.
The electromagnetic (e.m.) field seems to offer a better value. For instance in the Reissner–Nordström metric [dϕ/(cdt)]2 is proportional with Q2/(2 c2 M) where Q is the electric charge confined to a mass M. It does not take into consideration magnetic fields.
Can somebody indicate a reference which calculates the metric in the complete e.m. field? Also, how much charge Q can be confined, say, in a box of dielectric of 1m3 volume?
Dear Sofia,
we must differentiate between general relativity as the (current, prevalent) theory of spacetime and gravitation, and alternative theories.
1.) General relativity knows no 5th dimension. It defines spacetime as a 4d differentiable manifold with a pseudo-Riemannian metrics that leads to a causal structure. The Einstein field equations are differential equations, i.e. as defining local entities like e.g. the metric, by themselves they do not exclude wormholes or other topologic peculiarities from the beginning. However, for all this to work, no embedding in a higher-dimensional space is needed. This is all intrinsic geometry as already regarded by Gauss 200 years ago.
2.) Alternative theories exist that do not exclude higher dimensions. I do not have an expert overview of these theories, but they are altogether of marginal importance, at least from a current point of view.
The important message here actually is: for wormholes or other non-trivial topologies to exist, no *fundamental* modification to general relativity or any higher dimensions are needed.
But here is a caveat: besides general relativity, theoretical physicists generally assume the validity of several additional principles which come "on top", and which we would actually not like to see violated. One of these is the dominant energy condition that says that in certain spacetimes like an asymptotically flats space-time, the local mass/energy density as measured by any observer is non-negative. In general relativity, all known wormhole scenarios somehow violate conditions like this or similar. If you want to google further: look for "Alcubierre drive". They are therefore regarded as "unphysical" solutions of the Einstein field equations.
Best regards
Oliver
Dear Oliver,
First of all, thanks for the explanations. As I said, I am no expert in general relativity. I tried to get into the mathematics of the Einstein Field Equations (EFE) and I found very difficult and time-consuming to follow it. I am very interested, by my time is terribly limited.
Please don't laugh if I'd tell you that what lead me to ask this question was the issue of flying saucers. There are too many reports on them, for saying that it's just imagination. Further, since these objects seem to come from very far away, such journeys would lack any benefit for who planned them, if they would last millions of years. This is why I thought that they might use worm-holes.
Now, the fact that the EFE regards the space-time as a 4D manifold, may be just a choice, similarly to the model of Maxwell's equations. You see, on the RHS of the EFE appear fields (analogously to the sources in Maxwell's equations). They do modify the space-time geometry, so that they do supply, in a way, the η of which I asked in my equation (2). For instance, I know for sure that in a gravitational field, the proper time changes comparing with the one in the flat space. I suppose that in a very strong electro-magnetic (e.m.) field, it would also change.
So, you see, the EFE include a couple of entities: space, time, and fields. The space-time is embedded in fields - there might be, in principle, room for additional dimensions. So it seems to me. But, to may regret, I am not able to handle those EFE equations, solve them, and check whether there is a possibility to obtain from them a relation similar to my equation (2).
I find very interesting your words
"the dominant energy condition that says that in certain spacetimes like an asymptotically flats space-time, the local mass/energy density as measured by any observer is non-negative. In general relativity, all known wormhole scenarios somehow violate conditions like this or similar. If you want to google further: look for "Alcubierre drive". They are therefore regarded as "unphysical" solutions of the Einstein field equations."
I will look at that "Alcubierre drive" - thanks for indicating it - just I hope that I would be able to follow the mathematics.
With thanks and kind regards,
Sofia
Dear Oliver,
At your advice I read about the Alcubierre drive. I understood that there is a problem in creating an environment with negative energy.
And though, I'd like to tell you that even here, on the Earth, there seems to exist some evidence of an ancient relocation technique. You see, the Egyptian pyramides at Geeza were built from huge blocks of stone, though no quarry was near. It also seems that the pyramides are older than the Egyption dynasties. Archeologists have no plausible explanation of how the construction was done. By the way, such constructions are not only in Egypt.
Of course, the explanation should come from physicists, but they have none. It's an open question.
With kind regards,
Sofia
Dear Sofia,
we are deviating of course from the discussion on theoretical physics here in general, and wormholes in especial. Nevertheless, without being a historian or an expert in prehistoric engineering myself, I am pretty sure there is a completely classical explanation for how the Egypts built the pyramids. Just around that time, Stonehenge was built by Neolithic people, and nowadays the experts in the field do have quite a good view on how they did it. E.g. from quarries 200 miles away they transported the so-called "blue stones" from Wales to Salisbury plains, via the RIver Avon. I therefore surmise that the River Nile also has played an important role for the logistics issues...You should go and have a look at the Stonehenge musem, with the Stones quite nearby. It is very impressive and well-done...
Best regards
Oliver
Dear Oliver,
Yes, you are right, this is a deviation from the discussion and I wouldn't want to go too far with it. Just a small comment: you see, the pyramids construction preceeded the Egypt dynasties. No names of kings are inscribed in the interior chambers. Though, the construction details prove high knowledge in the geometric data of the Earth, and in astronomy. It's obvious that those who planned the pyramides were not primitive people, it was an advanced civilization (probably a disappeared one).
Now, returning to my question, please look at my equation (2). If two events occur at the same place, therefore L = 0, the equation becomes
(Δt)2 - η2 = τ12.
Could this equation have a meaning if Δt denotes the time that would be recorded between the two events in the flat space, and the value η being the effect of the fields? In other words, could it be that an observer traveling inside some field, record on his wristwatch an interval between two events shorter than it would take in the absence of the fields?
Please see, I saw in Internet reports of aircrafts caught in strange clouds, and reaching the destination before the usual time, but one cannot know whether it is true, or just stories.
With best regards,
Sofia
Dear Sofia,
if you refer to wormholes about the reason why flying object, which the A. Airforce pilots also recently detected, I don't see any solution with those.
It is sure that GR cannot have a clue about it..you certainly need a unified theory with quantum mechanics which can exploit the properties of the quantum vacuum in order to make things travel with such accelerations which would destroy everything inside a space ship. A free fall in a black hole would generate such accelerations as motion in time. Not the acceleration but the tidal effects in such case would destroy the space-ship.. According to the experiences of the pilots It should be possible to induce the quantum vacuum to behave like that without the tidal effects, and at the same time be in a sort of a "bubble" to avoid the friction with air, but this still remains the Physics of Star Trek...
A unified theory should have an additional degree of freedom which is needed in the attempts to arrive to a model of the Dark Matter.
Dear Stefano,
As I told to Oliver, I think that the UFOs cannot be denied - there are too many reports. Many people take them as stories, but I consider them seriously, all the more that most of the reports describe similar facts.
Our question is whether these objects are sent by an intelligent civilization, and where this civilization might be located. In our solar system none of the planets is habitable. So, I understand that these objects should come from far.
I can't imagine living beings which live millions of years. Please see, a missile flying with such a tremandous speed as 3km/s, needs 0.4 millions of years to reach Alpha Centauri. You can easily check that for a creature on the missile, the travel time (i.e. the proper time τ) would be practically the same. This is why I assume that there is a way to reduce the proper time. It should be something acting as a 5th dimension and providing the η in my equation (2) - see the question.
In your comment you speak of the Alcubierre drive. I read that it is not practical as far as our technology is considered. You also speak of an extraordinary acceleration. But I am asking whether there could be another solution, which does not need accelerating the missile so terribly, a solution based on fields which would change the space-time geometry and provide the η.
It is reported that when our aircrafts try to get close to a UFO, the functionning of the electromagnetic apparatuses of our aircrafts stops. If these reports are true, then the UFOs use very high electromagnetic fields.
You see, I am not able to perform calculi with the EFE, I am not competent in this domain. I would like to know how strong has to be an electromagnetic field, for changing the space-time geometry as much as to influence the value of τ.
With kind regards,
Sofia
The first model describing a wormhole was written by Einstein and Rosen. Thus, this model was also called Einstein-Rosen bridge. You can easyly convince yourself that this model is unphysical. Any observer reaching the bridge will gain the velocity of light and the force of gravity will be infinitely high at this location.
Dear Rainer,
Of course that the bridge is non-physical. For shortenning the path between two points on a spherical surface - such a surface is 2D - the surface has to be imbedded in a 3D space and the shortenned path should pass through the interior of the sphere. The interior does not belong to the surface of the sphere.
This is why I asked about a 5D space. Staying withing the 4D space it is obvious that one cannot shorten the path. Whatever is known to us today is that our spacetime is 4D, no 5th dimension is known to us.
But there seems to exist something that in certain cases may replace the 5th dimension: the geometry of the 4D space changes under the influence of the fields which appear in the RHS of the Einstein Field Equation. General relativity is not my domain of competence, and a trial to solve the EFE is too complicated mathematically for me. Though I pose the problem, absolutely in principle, whether this equation cannot be written as a 5D equation of a flat space, in which ome functional of the field would behave as a 5th dimension.
Wormholes are a feature of the Schwarzschild geometry. In the case of the Einstein-Rosen-bridge the geometry is mirorred, ie Flamm's paraboloid is doubled. Schwarzschild geometry can be embeded into a 5 dimensional flat space by using 6 variables, two variables lying in the same dimension. Thus, is the bridge a geometric object that can be drawn, if you supress the superfluous dimensions. You can find the figure in the internet or in textbooks on General Realtivity. The extra dimension is not a real physical entity, it is itroduced for better understanding the problem.
Sofia, you seem to think that the unphysicality of wormholes has to do with the fact that you need somehow 5 dimensions for the wormhole to tunnel through. This is not the case!
I can only repeat myself here: as I said already: you do not need an additional dimension in order to embed 4d spacetime, wormholes or not! You should give up the idea that what you need to draw on paper is what you need outside spacetime. In GR there is no 5th dimension. Actually it is quite common for people not used to Riemannian geometry that they always try to think in "embeddings", whih is fine at the beginning. But eventually, in order to fully grasp Riemannian geometry or GR, you need to give up this notion.
The unphysicality of wormholes, as I said, stems from the fact that you need to violate the dominant energy condition. Therefore, wormholes are a completely valid solution in GR, but they are unphysical from reasoning outside relativity. The same applies to Warp drives, as I have pointed you to, too.
Dear Rainer,
Please read what I am saying here: the 5th dimension can be mimicked by fields.
In the flat space-time, the proper time τ is given by
(1) (c dτ)2 = (c dt)2 - Σi=1i=3 (d xi)2.
In the presence of fields, the proper time is given by
(2) (c dτ)2 = Σi,j=03 gi,j dxi dxj.
Let's write the difference between the two equations as
(3) Δ(ds}2 = [(c dt)2 - Σi=1i=3 (d xi)2] - Σi,j=0. . . 3 gi,j dxi dxj.
With this notation we can rewrite the equation (2) for the proper time in presence of fields,
(4) (c dτ)2 = (c dt)2 - Σi=1i=3 (d xi)2 - Δ(ds}2.
It's obvious that Δ(ds}2 modifies the proper time. Doesn't it look similar to a contribution of a 5th dimension? The question is whether fields can be obtained such that Δ(ds}2 be positive and significant, not negligeable. In that case one got an effecr equivalent with a worm-hole. Of course, it is also a question, which consequence such fields can have on human beings, and what is the financial price of generating such fields.
I am not good at general relativity, but, for instance I saw in Wikipedia the Schwarzchsild metric, i.e. the metric in a gravitational field. The modification of the proper time with respect of the flat space depends on the difference
(5) 1 - 2GM/Rc2.
If this difference tends to zero, the proper time tends to zero. Unfortunately, the gravitational field offers no hope. The corrective term 2GM/Rc2 doesn't exceed the order of 10-10 even in the neighborhood of supernovae or black holes. Thus, the proper time is not affected by gravity.
One should try with other fields, maybe electromagnetic. I am not trained with the Einstein Field Equations and have no chance to solve them with electromagnetic fields. This is why I placed my question on RG with the hope that somebody could do it.
With best regards,
Sofia
Sofia, maybe you are hoaxing me.
You say you do not know much about GR, and that's totally OK! But I do, so please read my reply and try to understand it.
And I do have answered your question. If you say you keep hoping that someone competent could answer I take it you do not consider my replies competent enough. That's fine, I can live with that.
I am bailing out now, good bye.
Oliver,
I am not here for hoaxing people, but for trying to answer a question I consider interesting. About me thinking you not competent - please leave such things, we are grown up people. I DO think you competent. But I repeat, since no 5th dimension is known to us, I propose to consider FIELDS. Either you did not pay attention, or the times at which we place our posts are in anti-phase.
About bailing out - nobody is forced to talk with me, collaboration here is based only on mutual interest in the topic.
Thanks for contributing so far.
Dear Sofia D. Wechsler , I think you can find interesting the second tweet of this Twitter account: https://twitter.com/jjrodrip It's a prototype of a stellar ship (or ship) that will allow humans to overcome the distances between stars. Some of the other tweets are prophecies or predictions of the near future some of which have already been fulfilled but they are in spanish. Regards!
Dear Jacinto Jesus Rodriguez Peres,
I looked at the 2nd item as you said but I saw no prototype of stellar ship fit for overcomming distances. I saw just a globe of sparkles of light in the sky. All the best!
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