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I am writing a SF story, in this story, a life-form(You can just call them alien) from another universe comes to earth. They used "imaginary velocity" to make their time elapse faster than all beings on the earth. When the alien is using "imaginary velocity", they eat humans.

From SR, the formula for time dilation is: $$t^{'} = \frac{t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\gamma t$$

if I plug in a imaginary number for $v$, for example $2ic$($i$ is the imaginary unit), then I get: $$t^{'} = \frac{t}{\sqrt{5}}$$ $\gamma$ is less than 1.

this means the time for the objects with imaginary velocity elapse faster than the objects in inertial frame.

In the story i will make it $10^{16}$ faster than inertial frame.(human's $10^{-16} s$ = alien's $1 s$, when the alien spent one day in their reference frame, only $86400*10^{-16} = 8.64*10^{-12} s$ passed from human's reference frame) So no people can see what's happening, also no machine can detect the actions of aliens.(in the latter part of the story, the situation will change)

Is their any contradiction with the theory of relativity or other physics?

ps. There is a known problem that all observable must be real numbers. However, I decided to ignore it. Since tachyons have imaginary mass and some SF story also used this in their story.

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$v=2ic$ means the object move $2ic$ m in 1 s, the distance is also imaginary. So, it is not moving in the real spacetime. You will see that object as "not moving" even if it has $2ic$ speed.

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    $\begingroup$ I don't understand the physics either, but if the upshot of it is that time passes 10^16 times faster for the aliens, then i don't see how that prevents them from being detected by any machine or human. For example if they open a door to go through it, are you saying that they will open and close the door so quickly that (for example) an alarm wouldn't be triggered? Have you thought about the stress that would be placed on a door to be opened and closed so quickly, at billions of times the speed of sound? The door might just explode with the force acting upon it. $\endgroup$ – Max Williams Jun 26 '15 at 12:13
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    $\begingroup$ In fact, just the action of moving within our atmosphere would generate massive shock waves. I think that they basically can't interact with earth at all without just smashing it into bits by accident. $\endgroup$ – Max Williams Jun 26 '15 at 12:16
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    $\begingroup$ First point: Without further qualification imaginary velocities are fairly meaningless. Second point: At the magnitudes you are considering the aliens will not need any time dilation effects. If they can move an operate at 0.5c humans will not have time to register them - Other than in the way they interact with their environement ie. what-if.xkcd.com/1. $\endgroup$ – Taemyr Jun 26 '15 at 12:56
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    $\begingroup$ @fairytale plus, all of Physics is a mathematical "trick". GR is a huge mathematical trick that makes no sense in the real world at all....unless we count the transit of venus, time dilation, gravitational lensing etc... $\endgroup$ – Aron Jun 26 '15 at 18:32
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    $\begingroup$ @Taemyr I would say that "classically" TDSE on a "moving" particle, velocity of the particle is the group velocity of the particle....If we apply the same logic to a 4+0 particle and we rotate the particle on the time axis....OMG.... $\endgroup$ – Aron Jun 26 '15 at 18:37
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It is difficult to answer this question without you telling us what a velocity of $2ic$ means.

However my 5c is that this will not work.

Examine an alien moving from Human A to Human B, with a distance of $d$, at a constant speed of $2ic$. Lets ignore for now the fact that this statement is meaningless.

From the perspective of the humans the alien will spend time $t=\frac{d}{2ic}$ to cross this distance. Also, from the perspective of the human the alien will age $t'=\frac{d}{2\sqrt{5}ic}$.

From the perspective of the alien the humans are moving at a velocity of $2ic$. He will need to cross a distance $d'=d\sqrt{5}$ and will do so in time $\frac{d\sqrt{5}}{2ic}$. Meaning the formulae does not provide consistent results in the presence of imaginary velocities.

TLDR: If the alien is moving at $2ic$ with respect to the humans the humans are moving at $2ic$ with respect to the alien

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    $\begingroup$ Right. Ask yourself at what time Oxford stops at the train. $\endgroup$ – a CVn Jun 26 '15 at 12:58
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    $\begingroup$ @fairytale i is a constant, you can't choose its value to suit your needs (make γ less than 1). Also, i times anything becomes a complex number, and the <, > operations are not defined for complex numbers (although you can compare their magnitudes, or define a lexicographic ordering) $\endgroup$ – zovits supports GoFundMonica Jun 26 '15 at 13:36
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    $\begingroup$ @fairytale "less than" and "greater than". $\endgroup$ – Taemyr Jun 26 '15 at 13:45
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    $\begingroup$ @fairytale By "<, > operations" I meant the comparison operators (functions with two real numbers as input and a boolean output). They are canonically not defined for complex numbers, which aren't scalar (one dimensional) values like real numbers, but rather 2D vectors, consisting of a real and an imaginary component. $\endgroup$ – zovits supports GoFundMonica Jun 26 '15 at 13:47
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    $\begingroup$ @zovits complex numbers are scalar. $\endgroup$ – Taemyr Jun 26 '15 at 13:50
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  1. Instead of making time a complex dimension, you could have another (real) time dimension. That allows the state to evolve without moving forward in our time.

  2. Are the aliens made of matter? Unless it's made of stuff other than our familiar protons and electrons, it could not operate differently than the normal rules of the universe.

  3. If they are made of stuff from another brane or somesuch, how could they eat meat made from normal matter?

Perhaps the time freeze is a technological ability, using an enveloping field to allow time to (in effect) pass faster inside. That is a staple of SF, but generally unexplained: #1 above could be used .

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  • $\begingroup$ Imaginary time? Science Fiction with Imaginary TIME?! Quick someone email Prof Hawkings, he's been waiting for this day! $\endgroup$ – Aron Jun 26 '15 at 17:52
  • $\begingroup$ The first point is a good suggestion. About point 2, I haven't decided yet. For 3, there are some settings that I didn't stated in this post. "Eat" is just an analogue. To be more detail, they absorb some kind of energy from human, then that human will vanish. $\endgroup$ – fairytale Jun 27 '15 at 10:27
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How can we get imaginary velocity? This is pretty obscure Physics I'm afraid.

Now lets first go to basics.

What is velocity?

$v = x / t$

Now as you already noted. If v is imaginary, then either x is imaginary or t is imaginary.

Let's assume we are in a 3,1 spacetime shall we? I know its pretty hard to imagine, but lets pretend ;).

If my understanding of the Wick rotation of 3, 1 spacetime over the time axis is correct, it would imply our friends with imaginary velocity inhabit a 4, 0 spacetime or 4 euclidean space.

Things get weird really quickly.

I can't seem to figure out how to type out my workings wrt to TDSE, but the jist of it is that, when you plug in imaginary time into TDSE, you get a particle that is confined to a single point in real time (or at least drops of exponentially from a single point in time), but is periodic on real space, covering the whole of space...

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  • $\begingroup$ what is the meaning of "covering the whole of space", do you mean the probability to find the particle at every point in the universe is equal? $\endgroup$ – fairytale Jun 26 '15 at 18:57
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    $\begingroup$ @fairytale I think it is periodic. There are those who believe that Santa is in fact made of particles that obeys such physics. It is why the effects of Santa are felt everywhere all at the same time, but when you try to observe him (stay up late), you collapse the wave form and he doesn't appear. It also explains how the Santa tunnels into people's homes. $\endgroup$ – Aron Jun 26 '15 at 19:00
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This is pretty way over my head as far as the maths goes, but I do just want to point out the difficulties of moving extremely fast. Namely, wind resistance, relative force and all sorts of other nasty physics effects explored in various forms of fiction to date.

An excellent example of this is in The Thief of Time by Terry Pratchett. In the book there are history monks, who can bend time as they wish. Various examples of the difficulties and advantages of this can be found in the book, including the simple difficulty of walking, talking and breathing in an athmosphere which does not want to behave at your speed.

Another good example is in the recent film Xmen Days of Future Past. Quicksilver has speed as his "mutation" and as you may remember, can cause people to summersault over by simply touching their face.

You'd therefore have to make your aliens superhumanly strong, able to operate in near vacuum, and incredibly resilient so they can simply survive in the high speed environments of imaginary velocity.

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Complex arithmetic is not a geometric construct, as much as it is a 'cheater' logic used to provide snobby academic types with means to impress and daze laypersons with negative answers to squared functions. All instances of complex arithmetic are trivial (in the philosophical sense, not the mathematical), and as such I find your attempt to actualize it in a work of SF refreshingly humorous!

I'm going to go ahead and assume we're using complex arithmetic to describe pan-dimensional motion. Let's insist the 'observable universe' exists in three dimensions and we give locations as vectors (v = [x,y,z]) and velocity as velocity over time (v/t). I want to make this an exercise in 3-manifolds rather than familiar euclidean space, but that makes me dizzy. It shouldn't matter over a small enough portion of space. Anyway, a human could be the classic example of Sagonian flatlander when viewed by our aliens. Or the aliens could be just as flat a we, yet able to project itself where our dimensions intersect. Many principles of flatland relativity might not 'scale' along the other dimensions, or 'proportions' of objects and energies change with respect to motion perpendicular to familiar references. Objects of this 'imaginary' universe might therefore react differently to gravitation, dilation, etc.

many questions arise, such as: if we accept motion with an 'imaginary' element to also satisfy complex arithmetic, doesn't that mean a force could decelerate what it might otherwise accelerate?

also: do we allow imaginary components to other aspects of reality, such as 'imaginary' mass, 'imaginary' charge, 'imaginary' gravity? Or are such imaginary components already provided by imaginary space?

Are our aliens pandi or transi? that is, are they Pan-dimensional bodies/intelligences, built of materials from both the real and 'imaginary' dimensions; or are they Trans-dimensional, able to, through some exploit, transition between our dimension and their home dimension. The former allow aliens to "eat" puny humans from every direction simultaneously, or even from within! Yikes!

Before we attempt to validate any sort of equation we must consider: TIME is a perception of and an attempt to measure and count the CHANGE in state of a three dimensional, mechanical device (clock, brain, planetary motion, etc). As a strictly human invention it might not have meaning to the aliens at all. SPEED is a distance traversed across three dimensional space by such a given time. It's hard to consider an alien as an being with capable intelligence while operating without time. For this reason, assume we cannot allow an 'imaginary' time component. There is another reason I think we should disallow (t+it) from our equations. If we are measuring time by 'seconds' than are assumed to have a clock. If we move trans dimensionally, are we carrying this clock with us? or are we leaving this clock to return to it later? This also affects the way in which we're comparing the motion and the metabolic rates of the extra-planers. This is similar in principle to the reason time appears to dilate near the speed of light. Remember, the speed of light is invariant at any reference. The motion of mechanics within the clock are not changing, rather the energy of a system increases as it accelerates, thus the underlying reference for motion is changing.

We are implying a sort of 'imaginary' time my simply allowing an imaginary component for each each spacial dimension.

We describe velocity thusly:

v = [(x+ix), (y+iy), (z+iz)] / t

this is 'time invariant'; the alien motion is trans-dimensional, yet occurs within the same reference of time as any human observer. For example, a human could move a mile in the 'real' universe in a minute. Or an alien could move at the same speed, over the same amount of time, along the 'imaginary' universe and appear to NOT MOVE AT ALL. Also, it is possible for aliens to exist as 'purely imaginary' constructs, meaning they can phase in and out of human perceptive existence. They do this as a function over topography, rather than an increase/decrease of relative time scales.

[x,y,z]/(t+it) is a 'positionally invariant' case, meaning aliens occupy the same space as us, and are at all times fully accessible, yet can modulate the rate and scale of physical forces acting upon them.

The later case is more restrictive, in that a change upon t affects x,y,z uniformly. The former, I like because thinking about moving along the -ix and the +iy at the same time makes me want to stand on my head! This concept is illustratable by considering the 3-sphere model of the universe:

In the case where our 'observable' universe is a 3-fold we allow 4 dimensions and say the universe is the group where any point is given by |∀p| = SQRT(w^2 + x^2 + y^2 + z^2) = 1. 'imaginary' motion would not simply be a scalar, otherwise it might be possible our aliens are able to attain the 'Singularity'; where |∀p| = 0 although I don't know why they'd bother to go there or how they'd return. Or are they already and always there?

In either case, adding complex geometry is not simply scaling the existing dimensions (thus increasing and decreasing speed). it is the proposition of doubling the number of dimensions. If you observe, moving from one, two, to three dimensions, the origin, or singularity, doesn't change. It is always 0; 0,0; 0,0,0; respectively. It follows more dimensions would share our singularity at the same point. For every dimension element of our frame, we add an imaginary element. Alien 'imaginary' dimensions confer the same constraint in that |∀p+ip| = 1. Velocity in any direction is given as a Magnitude. |v|. Because we're in a complex plane, we are essentially allowing velocity to be negative. If so, motion along a supposed 'imaginary' dimension is different from motion along the traditional plane, in that the classical Newtonian 'equal and opposite force' becomes 'equal and potentially perpendicular force'

This could mean the aliens, when they exert themselves upon the 'real' universe, would appear in a vastly decelerated frame as a static, immutable thing, such as a tree or rock, while absorb incoming energy (by projecting it along the perpendicular reference of it's 'imaginary' aspect), or it might be super fast like a plasma and amplify every inbound energy (causing air to heat, voltage to multiply, etc).

'imaginary' motion could be interpreted as a change in SCALE. Increased/decreased gravitation: Our aliens would become LIGHTER in the presence of a GREATER gravity, and crushed by the empty expanse between distant galaxies. They might even phase from our detection altogether.

Your provided equation is okay, but I don't like the (c^2) bit because the speed of light is SUPPOSE to be a constant and invariant. I think what you're attempting to do is give velocity as a fraction of the speed of light. Don't do that! let c = 1 and you simplify that part of your equation to 1-v^2. Even a human where moving at 0.999c faster than an alien, it would still be 1c less than the speed of light!

I'd argue that the point at which our aliens intersect with with an earthling, IE, when they eat one, speed is not an issue, rather, the alien's path of motion is equivocally perpendicular to the human. His intersection upon our plane would take the form of an expanding/contracting sphere which would, as it became more distant along the imaginary plane, simply become insubstantial to the point of phasing out completely. It would be like a blackhole emerging from nowhere, to suck you in. Don't even bother to ask Alice what is happening to that entangled-particle friendship bracelet you're share/wearing, but I assume the effect would be similar to ripping her arm out of her socket in a direction perpendicular to everywhere.

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Even if there was nothing wrong with the imaginary numbers, there is another issue: moving $10^{16}$ times faster would cause very nasty side-effects as the air molecules cannot get out of the way.

There is an interesting explanation of this problem on xkcd:

http://what-if.xkcd.com/1/

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