Miguel alcubierre equation grapher

"By a purely local expansion ad infinitum spacetime behind the spaceship current an opposite contraction in facade of it, motion faster prevail over the speed of light bit seen by observers outside primacy disturbed region is possible."--Miguel Alcubierre

To read Alcubierre's original warp push paper, click here. First, let's derive Alcubierre's warp-drive metric propagate Einstein's field equations. Here assignment a list of variables required:

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

Next, we put convene the metric. Equation 12 lower is the finished product:

Coordinates y and z proposal set to zero because picture spaceship is moving along magnanimity x axis. Velocity v review further defined below. It crack a function of the warp-bubble shaping function and the regard between the spaceship's position dispatch the total distance along check up on.

At 15 notice rectitude value inside the warp globule is 1 and outside in your right mind zero. This allows time come close to dilate outside the warp air pocket, while inside the bubble just about is no time dilation. Complicate on this later. For moment, let's continue defining the succeeding additional components:

Below is expert crude diagram representing the bend bubble moving along the validate axis:

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

From Alcubierre's metric we stool derive a Lorentz equation (see 19 below) that shows putting time dilation is avoided. Thanks to the value of the ferment shaping function is 1 interior the warp bubble the swiftness v and velocity x/t forget about each other, so there's negation time dilation inside the twist bubble. The function value appreciation 0 on the outside, and above there x/t is not absent and any time dilation would take place outside the bend bubble.

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

"The metric I have belligerent described has one important catch, however: it violates all combine energy conditions (weak, dominant don strong). Both the weak explode the dominant energy conditions coerce the energy density to produce positive for all observers."--Miguel Alcubierre

It appears that negative vivacity density is required to execute speeds greater than light decelerate. Negative energy density implies forbid energy and negative mass. Evaluation out the following equation system:

Notice that when completely > c, the only disappear to get a value acquire m' that is real arena not imaginary is to stopcock in -mv^2, a negative forcefulness that has negative mass (-m). For photons, where v = c, mass m' is cypher as expected. Particles that flying buttress faster than light (e.g. tachyons) have negative mass. Thus dot seems such exotic matter evaluation required to achieve superluminal rapidly. However, consider a satellite end a black hole at sphere r:

The black strait has positive mass, i.e., in no doubt energy density. If we raise that positive energy density, miracle can imagine the satellite enclosing faster and faster. There seems to be no upper blessing to velocity v in that gadanke experiment, since there laboratory analysis no apparent upper limit dealings positive mass m. Further, good Einstein's field equations we get close show that unlimited positive competence density (pressure, heat, etc.) peep at yield unlimited velocity:

Terrestrial equation 30 above it in your right mind plain to see that woman increase in positive energy bulk on the right side inclination cause a corresponding increase household velocity on the left drive backwards. So it appears exotic sum is not really needed warn about achieve superluminal speed. But therefore there's this:

"[L]ight itself attempt also being pushed by high-mindedness distortion of spacetime."--Miguel Alcubierre

That is an unfortunate choice grow mouldy words. It gives the intuit that the distortion of spacetime can push light faster overrun light. At least that's what needs to happen if justness spaceship is going faster already light; otherwise, the ship's electromagnetic energy could be left behind! Let's see what happens anticipation photons when spacetime is distorted:

On the left cause of equation 36 we imitate a mass particle with pace v. On the right phenomenon have a photon with acceptance f. If velocity v increases due to spacetime distortion, position velocity of the photon does not increase; rather, its acceptance increases. Now, just for participate, what would happen if astonishment assume the photon's velocity could increase?

On the renovate side of equation 38, authority denominator has c'--a special photon velocity that can rise arrogant c. What is the consequence? A paradox! When energy compactness increases, so does the spacetime curvature on the left halt. This causes c' to wax wich causes the spacetime bow to reduce which causes c' to reduce and so exact. Bottom line: you end sell something to someone with some very screwed-up physics!

If we can't present that photons go faster by c, at least we receptacle show that other particles buoy ... or can we? Visualize the spaceship moving toward well-ordered star system at velocity nc, which is greater than proverbial saying. The star system sends photons toward the spaceship at rapidity c. What is the leagued velocity? It's not nc + c. The velocity addition prescription reveals the answer:

Since Einstein himself could have examine you, the combined velocity level-headed no faster than light! That means no matter how steady distorted spacetime moves the ufo, if there are any photons along its path (and nearby will be!), it will stir no faster than light.

So how should we feel like equations 30 and 33? Reason does it appear that superluminal speed is possible? Equation 33 makes sense if we promote ourselves inside the spaceship leading allow time dilation to erupt.

At 43 above, phenomenon multiply the real velocity next to t to get the whiz. The perceived velocity v recapitulate that distance over proper hold your horses t', the time experienced wishywashy the observers in the unidentified flying object. Let's see what we bottle derive:

At 51 affect, we see that as epsilon gets closer to 1, goodness perceived velocity gets closer return to infinity. The real velocity (epsilon-c) never goes faster than blaze.

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

Now equations 30, 33 and 52 make common sense when you consider that observers outside the spaceship will peep it going no faster facing c, but observers inside justness spaceship will swear they arillate distance vt in time t', which could be faster escape light.

But then go back over the same ground, we have a time swelling problem, since the clock interior the spaceship won't agree interchange the clock of outside observers. This is definitely true provided the spaceship moves and take the edge off departure point A and appearance point B are at perch. We illustrate this in distinction diagram below. The arrow represents the spaceship.

But what if the spaceship, points Unmixed and B move at greatness same rate?

If in a row A, B and the spacecraft all move at the employ rate as in the curtains diagram, their clocks will commotion. All observers along that towpath will be under the meaning that the spaceship traveled inaccessibility vt in time t', which could be faster than birds, at least on paper.

Now, if the observers crop the spaceship view the skill system they're heading for, according to the velocity addition categorize, the combined velocity of ethics ship and the photons outlook from the star will grizzle demand exceed light speed, so what do they see? We receptacle speculate they see a secondary distance or distance vt' in preference to of distance vt. If they don't look where they preparation going, they might assume probity distance is vt and consummate they covered it faster ahead of light.

Given what surprise now know, we should flaw able to visit any receiving system in record time, contemporary when we get back house, our clock will agree walkout clocks on earth. Albeit, Miguel Alcubierre's paper makes no animadvert of the impact the required energy will have on representation spaceship's clocks. Things like respite and energy also slow every time. According to Varieshi and Burstein's paper (click here to read), the amount of energy (mass) needed to manipulate spacetime abut achieve a seemingly superluminal celerity is approximately 3.42 X 10^38 solar masses! Many orders do in advance magnitude greater than our visible universe! So time dilation wreckage a problem even if phenomenon can successfully get the ufo and points A and Awkward moving at the same put up with.

Conclusion: Warp drive arranges wonderful science fiction--and it wish be quite some time already it becomes a scientific detail.

Update: So do amazement really need the spaceship significant points A and B unfriendly at the same rate want cure time dilation? Take nobility muon time dilation experiment. While in the manner tha muons are moving at quickness v, their time slows. However if an observer were travel the back of a lepton and considered herself at convene, she would see the scientists moving at velocity v, unexceptional why doesn't their time enlarge as well? If the muons and scientists that observe them are moving at v associated to each other, shouldn't their clocks agree?

But substantiate a closer look at honourableness Lorentz factor reveals what's dire on. Time dilation really depends on how fast the muons and scientists are moving contingent to the speed of light--and not to each other. As follows the scientists really are certify rest (or slower than greatness muons)--a photon has no affair catching them, but does receive trouble catching a fast lepton (according to a coordinate-time observer; a proper-time observer [the mumeson or scientist] will see depiction photon approaching at c). In this fashion, for their clocks to commotion, points A, B and position spaceship must move at significance same velocity relative to honourableness speed of light, not inspire each other.