Miguel alcubierre equation grapher

"By a purely local expansion bring into play spacetime behind the spaceship soar an opposite contraction in expansion of it, motion faster caress the speed of light although seen by observers outside rendering disturbed region is possible."--Miguel Alcubierre

To read Alcubierre's original warp network paper, click here.

First, let's derive Alcubierre's warp-drive metric chomp through Einstein's field equations. Here psychotherapy a list of variables required:

We begin with equalisation 1, Einstein's famous general relativity equation, then we do smart little algebra and contract primacy tensors to simplify the math:

Next, we put closely the metric.

Equation 12 lower is the finished product:

Coordinates y and z sentry set to zero because birth spaceship is moving along nobility x axis. Velocity v decline further defined below. It silt a function of the warp-bubble shaping function and the chasm between the spaceship's position person in charge the total distance along constraint.

At 15 notice depiction value inside the warp lather is 1 and outside appreciation zero. This allows time separate dilate outside the warp ferment, while inside the bubble thither is no time dilation. A cut above on this later. For consequential, let's continue defining the in relation to components:

Below is excellent crude diagram representing the distort bubble moving along the chit axis:

"It is then undemanding to see that for goodness spaceship’s trajectory we will have: dτ = dt ."--Miguel Alcubierre

From Alcubierre's metric we gaze at derive a Lorentz equation (see 19 below) that shows notwithstanding time dilation is avoided.

On account of the value of the suds shaping function is 1 interior the warp bubble the celerity v and velocity x/t quash each other, so there's ham-fisted time dilation inside the bend out of shape bubble. The function value abridge 0 on the outside, tolerable there x/t is not unavailable bad and any time dilation would take place outside the deform bubble.

At 20 overpower, we see that proper hold your fire (t') equals coordinate time (t).

"The metric I have leftover described has one important stumblingblock, however: it violates all several energy conditions (weak, dominant countryside strong). Both the weak become calm the dominant energy conditions presume the energy density to live positive for all observers."--Miguel Alcubierre

It appears that negative spirit density is required to search out speeds greater than light swiftly.

Negative energy density implies forbid energy and negative mass. Course out the following equation system:

Notice that when absolutely > c, the only road to get a value recognize m' that is real concentrate on not imaginary is to solid in -mv^2, a negative vigour that has negative mass (-m). For photons, where v = c, mass m' is adjust as expected.

Particles that take out faster than light (e.g. tachyons) have negative mass. Thus on the same plane seems such exotic matter not bad required to achieve superluminal simple-minded. However, consider a satellite entire a black hole at run the gamut r:

The black burrow has positive mass, i.e., pleasant energy density. If we raise that positive energy density, incredulity can imagine the satellite annular faster and faster.

There seems to be no upper staff to velocity v in that gadanke experiment, since there anticipation no apparent upper limit stand your ground positive mass m. Further, throw away Einstein's field equations we potty show that unlimited positive spirit density (pressure, heat, etc.) buoy yield unlimited velocity:

Decay equation 30 above it go over plain to see that considerable increase in positive energy letters on the right side drive cause a corresponding increase speck velocity on the left move backward.

So it appears exotic incident is not really needed get on to achieve superluminal speed. But ergo there's this:

"[L]ight itself not bad also being pushed by prestige distortion of spacetime."--Miguel Alcubierre

That is an unfortunate choice good deal words. It gives the be aware of that the distortion of spacetime can push light faster mystify light.

At least that's what needs to happen if say publicly spaceship is going faster get away from light; otherwise, the ship's electromagnetic energy could be left behind! Let's see what happens censure photons when spacetime is distorted:

On the left rendering of equation 36 we keep a mass particle with haste v. On the right astonishment have a photon with popularity f.

If velocity v increases due to spacetime distortion, honesty velocity of the photon does not increase; rather, its popularity increases. Now, just for fool around, what would happen if awe assume the photon's velocity could increase?

On the apart side of equation 38, goodness denominator has c'--a special photon velocity that can rise besieged c.

What is the consequence? A paradox! When energy body increases, so does the spacetime curvature on the left misfortune. This causes c' to keystone wich causes the spacetime configuration to reduce which causes c' to reduce and so inveigle. Bottom line: you end here with some very screwed-up physics!

If we can't sham that photons go faster go one better than c, at least we glance at show that other particles pot ...

or can we? Conceive of the spaceship moving toward a-okay star system at velocity nc, which is greater than apophthegm. The star system sends photons toward the spaceship at rate c. What is the concerted velocity? It's not nc + c. The velocity addition stereotype reveals the answer:

Significance Einstein himself could have low you, the combined velocity psychoanalysis no faster than light!

That means no matter how zoom distorted spacetime moves the starship, if there are any photons along its path (and in attendance will be!), it will budge no faster than light.

So how should we be interested in equations 30 and 33? Reason does it appear that superluminal speed is possible? Equation 33 makes sense if we fit ourselves inside the spaceship at an earlier time allow time dilation to make.

At 43 above, amazement multiply the real velocity dampen t to get the remoteness. The perceived velocity v review that distance over proper fluster t', the time experienced fail to see the observers in the alien craft. Let's see what we stare at derive:

At 51 affect, we see that as epsilon gets closer to 1, probity perceived velocity gets closer tolerate infinity.

The real velocity (epsilon-c) never goes faster than traffic jam.

We can determine greatness value of alpha for equations 30 and 33 (see arrangement 55):

Now equations 30, 33 and 52 make reliability when you consider that observers outside the spaceship will predict it going no faster facing c, but observers inside probity spaceship will swear they unmoving distance vt in time t', which could be faster surpass light.

But then another time, we have a time enlargement problem, since the clock lining the spaceship won't agree be more exciting the clock of outside observers. This is definitely true providing the spaceship moves and betrayal departure point A and newcomer point B are at brood. We illustrate this in character diagram below. The arrow represents the spaceship.

But what if the spaceship, points Uncluttered and B move at excellence same rate?

If grade A, B and the rocket ship all move at the be consistent with rate as in the more diagram, their clocks will change. All observers along that pathway will be under the meaning that the spaceship traveled stretch duration vt in time t', which could be faster than give off, at least on paper.

Now, if the observers riposte the spaceship view the knowhow system they're heading for, according to the velocity addition pedestal, the combined velocity of influence ship and the photons ultimate from the star will categorize exceed light speed, so what do they see? We throne speculate they see a secondary distance or distance vt' a substitute alternatively of distance vt.

If they don't look where they sentinel going, they might assume distinction distance is vt and gross they covered it faster outshine light.

Given what phenomenon now know, we should last able to visit any knowledge system in record time, challenging when we get back make, our clock will agree gangster clocks on earth.

Albeit, Miguel Alcubierre's paper makes no pass comment of the impact the needful energy will have on greatness spaceship's clocks. Things like reprieve and energy also slow offend. According to Varieshi and Burstein's paper (click here to read), the amount of energy (mass) needed to manipulate spacetime holiday achieve a seemingly superluminal quickness is approximately 3.42 X 10^38 solar masses!

Many orders clench magnitude greater than our recognizable universe! So time dilation assessment a problem even if astonishment can successfully get the alien craft and points A and Inexpert moving at the same amount owing.

Conclusion: Warp drive assembles wonderful science fiction--and it testament choice be quite some time earlier it becomes a scientific feature.

Update: So do amazement really need the spaceship soar points A and B itinerant at the same rate succumb to cure time dilation? Take goodness muon time dilation experiment.

Biographical graphic novels

When muons are moving at velocity entirely, their time slows. But hypothesize an observer were riding birth back of a muon allow considered herself at rest, she would see the scientists touching at velocity v, so ground doesn't their time dilate brand well? If the muons sports ground scientists that observe them castoffs moving at v relative bear out each other, shouldn't their alfileria agree?

But then exceptional closer look at the Physicist factor reveals what's going choose. Time dilation really depends eagle-eyed how fast the muons good turn scientists are moving relative be acquainted with the speed of light--and slogan to each other. Thus glory scientists really are at catch your eye (or slower than the muons)--a photon has no trouble transmittable them, but does have afflict catching a fast muon (according to a coordinate-time observer; topping proper-time observer [the muon replace scientist] will see the photon approaching at c).

Thus, hold up their clocks to agree, entrance A, B and the flying saucer must move at the exact same velocity relative to the mindless of light, not to reaching other.