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Space as a 4-D MRI Scan?

by MattRob
Tags: scan, space
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MattRob
#1
Dec13-10, 09:19 PM
P: 100
I've heard again and again that time is more or less the same as any other spatial dimension. I'll refer to them as X, Y, Z, and W, where you can pick any three to be space and a fourth to be time, since they're the same. W will be time for ease of reference.

Except, if that's the case, and space is really X, Y, Z and W, then why do particles emit in the X, Y and Z dimensions but not the Z?

Also, If time really is a fourth dimension as surely as the spatial dimensions, then how come nothing can rotate on an axis that would change their relationship with this dimension, time?
If it's a Minkowski x1 x2 x3 x4 manifold (which I honestly have only a very basic understanding of, if any at all) as earlier described in another thread, then how come rotations and translations only apply to x1 x2 and x3 but not x4, in that sense, x4 truly must be unique. Because a force on one end of a 3-d object along the X axis can create a movement along the Y axis through angular velocity, i.e. rotation.

In the same since, since an object exists for any length of time, then it has a length in the W dimension. And if it has a length in the W dimension, why is it a force along any of the X, Y, or Z axes could not cause angular velocity so as to change it's speed along the W dimension?

Either that, or for some reason the universe is completely "flat" in the fourth dimension, so there is no + or - interactions along the W axis, while there are interactions along the X, Y, and Z axes. To imagine this, In this case, if you were to replace X for W, then the universe would look like a giant 2-d plane revealing a 3-d shape one 2-d cross-section at a time, like MRI imagery. Now add 1 to each of those numbers, and I think that's how modern science has it, if anyone can verify.
A giant 3-d plane revealing a 4-d shape one 3-d cross-section at a time.

Here's a question that's stinging me, then, is that same question about dimensional rotations. Does that mean it would be possible to "rotate" through time? Also, if this is the case, then particles don't emit "randomly", but their emissions are dependent on factors so chaotic that randomly generating numbers is just as accurate as anything else.

(i.e., try to imagine how 50 million ping pong balls will hit 50 million basket balls at different angles when they're all floating in space... Yeah. You're better off picking random numbers.)

Now what if all of them were perfectly aligned on a 2-d plane, so that there is NO downwards or upwards motion in-between them, so they all only bounce along the X and Y planes while their movement through the Z dimension is uniform and undisturbed at all by their X and Y motion. Now apply this to four-dimensional space, where they all interact along the X, Y and Z coordinates but not W.
But, if particles truly did emit randomly, then why would they only emit in the X, Y, and Z, dimensions and not the W dimension? As any emission or otherwise any motion at ALL in respect to the W dimension (time) would cause a cascade of reactions so objects would all fall out of our 3-d "plane", which is this very moment, so they'd be pushed into the past and future, since the plane is no longer perfectly aligned.

However, that picture only works if the center of mass of every object is perfectly aligned. However, since particles live a finite amount of time, and time is an object's length in the W dimension, then their center of masses can NOT be aligned, and the applied forces offset from the center of mass wouldcreate angular momentum along the W axis.

No matter how I look at it I can't get myself to see time as another spatial dimension. The only way I can picture that without a chaotic collapse of uniform time flow is if every particle in the universe came in and out of existence every quanta of time. In which case, everything would be fine and handy-dandy. What's wrong with that?

Well... that was sort of complicated and I'm sort of diving into the deep end here, so could someone

please tell how I'm wrong? Is my very assumption that time is a spatial dimension wrong?
I'm not even going to ask If I'm right...
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K^2
#2
Dec13-10, 09:36 PM
Sci Advisor
P: 2,470
You can have rotations around any valid axis in 4-space. The reason you might not realize that is because the metric is screwed up for the time-dimension, so the rotations that involve time-dimension come out as hyperbolic boosts instead. Lorentz boost is a rotation in the 4-space.

That pretty much covers all of your questions.
MattRob
#3
Dec13-10, 09:48 PM
P: 100
I need to dust off my math! I can say that reply was probably one of the most thrilling things I've ever read. That was all purely speculation. I expected an answer much different than that. So from the tiny bit of reading I've done in the past few minutes, am I right to assume that the Lorentz Boost is the reason it's impossible to reach the speed of light?

My Gosh, there really is something screwed up with the fourth dimension!

Is there anywhere were I can do some more reading into this?

K^2
#4
Dec13-10, 10:34 PM
Sci Advisor
P: 2,470
Space as a 4-D MRI Scan?

It's a little more complicated. The real reason is that everything always moves AT the speed of light. The angles change. But because of the way the time dimension works, the projection of the proper 4-velocity onto 3-space looks like normal 3-velocity with the norm capped at c.

Ok, maybe I should start with this one from the beginning. First of all, let me work in natural units, where c=1. Also, lets drop 2 of the spacial coordinates, so that we have just two coordinates, one of which is time. The two coordinates are x(τ) and t(τ). τ is proper time, which you can think of as just a parameter in a parametric equation. More interestingly, if you have a clock that follows x(τ), t(τ) path, τ is the time that the clock will read. Hence the name.

Proper velocity will also have two components. vx=dx/dτ and vt=dt/dτ. The later is of particular interest at this stage, because it tells us how far an object will move in time, relative to our coordinate system, vs how much time passes according to the clock moving with the object. In other words, this is time dilation. We define dt/dτ=γ.

Proper velocity is not what we observe, however. What we are really watching is displacement with respect to our time. So v=dx/dt = vx/γ is what we are really looking at.

It so happens that proper velocity of every physical particle has magnitude of the speed of light. Keeping Minkowski metric in mind, we have

[tex]v_t^2 - v_x^2 = c^2 = 1[/tex]

Since we have vt=γ, and vx cannot be negative, the above sets limits on γ, which is greater or equal to 1. It also gives us vx in terms of γ.

[tex]v_x = \pm \sqrt{\gamma^2-1}[/tex]

Since we already have v=vx/γ, we can easily do the substitutions.

[tex]v = \pm \sqrt{1-\frac{1}{\gamma^2}}[/tex]

Two things you should be able to see here. First of all, you can solve for gamma.

[tex]\gamma = \sqrt{\frac{1}{1-v^2}}[/tex]

This looks exactly like your standard Lorentz boost formula with c=1. Second part is that we can establish limits on v. When γ=0, v = 0. When γ goes to infinity, v goes to 1. There is no real value for γ that can make v>1. In the mean time, the proper velocity, vx goes to ∞ as γ goes to infinity. That tells you that it's possible to travel any distance in finite amount of proper time. Also, notice that vt can take positive or negative value for any given γ. Standard model suggests that vt for particles and anti-particles has opposite sign. The choice of which is which is arbitrary.


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