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A Look At Space Using Special Relativity

November 17, 2011

The other day I mentioned how I struggle to understand what “objective” space is, if such an idea even makes sense.  I figured it’d be nice to bring you guys in on my struggle, so I’ll be posting some real world examples using modern physics, showing you guys just how weird and bizarre thinking about space-time really is.  Before I begin though, if anything in this post is incorrect, please let me know.  I’ve noticed that some of you who read this blog are physicists, and have contacted me by email in the past, so feel free to email me or comment on any errors in this.  I’ll fix them, and hopefully you’ll help me understand all of this better!

In my last post, I embedded a Youtube video from the BBC program called What Is Reality.  In it Dr. Tegmark of MIT wrote some equations on a glass window, pointing out that these mathematical equations are a window which we can use to look at and understand the world at its deepest levels.  One of them was the Lorentz transformation, which describes how to deal with fast moving things (near light speed).

It relates the coordinate system and flow of time between two “reference frames”.  For example, say I’m standing here on Earth, and I exist in 3D space with the coordinates x, y, and z, and time flows according to t.  t = 1 second, 2 seconds, 3 seconds, and so on, moment by moment, just like we’re used to.  Initially you’re beside me in a space ship on a long runway, but then you rapidly accelerate and by the time you lift-off, let’s assume you’re moving 80% the speed of light (0.8c).  (A crazy example, but just bear with me).

As you accelerated relative to me up to 80% the speed of light, time itself was slowing down for you relative to me.  The electricity flowing through your brain, the blood flowing through your veins, the very flow of your conscious thought, all slowed down relative to me.  At the same time, from your perspective in the space ship, the big super-long runway is scrunching up shorter and shorter in the direction you’re accelerating.  It’s very weird, and once I actually work an example for you, you’ll see just how weird all of this is.

Let’s write out the Lorentz transformation in equation form.

[latex]

\centering

\begin{equation}

x’ = \gamma_{v}*(x-vt)

\end{equation}

\begin{equation}

t’ = \gamma_{v}*(t – \frac{v}{c^2})

\end{equation}

where…

\begin{equation}

\gamma_{v} = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

\end{equation}

[/latex]

The variable v is the speed between the two reference frames and c is the speed of light.  x and t are position and time in the first frame, and x’ and t’ are position and time in the second frame.  Let’s now work a problem and examine it.  Here’s the problem:

According to Jason on Earth, a distant uninhabited Planet Y is 5 light years away.  Steve is in a spaceship moving away from Earth at 80% the speed of light, or 0.8c.  Steve’s on his way to check out the planet, but unfortunately for him, Greg didn’t care much for Planet Y, so he blew it up.  According to Jason’s astronomical logs on Earth, this happened 2 years after Steve passed Earth.  (We mustn’t forget that Jason had to wait a while for the light from the explosion to reach him.)  Call the the passing of Steve and Jason time zero for both.  (a) According to Steve, how far away is Planet Y when it explodes?  (b)  At what time did it explode?

Note that in this example, Steve doesn’t take off from Earth but is assumed to have been flying by at 0.8c from the get go.  To solve this problem we simply plug the numbers into the Lorentz transformation equation, but the answer is quite intriguing.

[latex]

\centering

\begin{equation}

x’ = \gamma_{v}*(x-vt) =

\end{equation}

\begin{equation}

= \frac{1}{\sqrt{1-(0.8)^{2}}}*((5 ly) – (0.8c)(2 yr)) = 5.67 ly

\end{equation}

\begin{equation}

t’ = \gamma_{v}*(t – \frac{v}{c^{2}}*x) =

\end{equation}

\begin{equation}

\frac{1}{\sqrt{1-(0.8)^{2}}}*(2 yr – \frac{0.8c}{c^{2}}*(5 ly)) = – 3.33 years

\end{equation}

[/latex]

A NEGATIVE time?  You may be thinking that’s a typo, but no, it’s not.  That’s really NEGATIVE 3.33 years.  What does all this mean?  It means that if after Steve sees the explosion, he were to calculate when the explosion happened from his frame of reference, it would have happened over three years before the event of him passing by me on Earth.   But I calculated it out and inferred that it happened two years after Steve passed by me Earth.  This may sound impossible but it’s actually correct. His reference frame is “aware” of the explosion over three years before my frame of reference is.

Let’s break all of this down, working it all out from both my perspective on Earth, and next for Steve on his spaceship.

Jason’s Perspective

From my perspective on Earth, everything is “normal”, so none of the math is terribly confusing.  Most of this problem is simple distance = rate * time stuff.  The planet explodes at t = 2 years after Steve’s passing by me.  At this time, Steve will have moved (0.8c)*(2 years) = 1.6 light years out toward the planet.  I see a distance between Steve and Planet Y of 5 ly – 1.6 ly = 3.4 light years and a relative velocity between Steve and the light of the explosion of 1.8c  (he’s moving at 0.8c toward the planet, and light is coming toward him at 1.0c from the planet).  So the light from the explosion will reach Steve in another t = 3.4 ly / 1.8c = 1.89 years — at the time, according to me, Jason, of 2 + 1.89 = 3.889 years.   The distance I now see between Steve and Planet Y (or the center of the debris) is 5 ly – (0.8c)*(3.889 years) = 1.889 ly.  In the meantime, I will know that less time has gone by on Steve’s clock.  In particular, [latex] \begin{equation} \sqrt{1 – (0.8)^{2}}*3.889yr = 2.33yr [/latex].

Steve’s Perspective

Now on to Steve’s perspective.  The distance the planet is from him when he realizes that it has been blown away is shorter for him.  How much shorter?  You apply the Lorentz contraction:  [latex]  \begin{equation} \sqrt{1-(0.8)^{2}} * 1.889 ly = 1.133 ly [/latex]  Now the big question we have to solve is this:  If Planet Y moves toward Steve at 0.8c, and light from the explosion moves at him at 1.0c, how long will it take light to get 1.133 ly ahead of the planet?  The relative velocity between Planet Y and the light from the explosion is 0.2c, and [latex]0.2c\Delta t’ = 1.133ly[/latex]  Solving for [latex]\Delta t'[/latex], we get 5.667 yrs.   If Steve’s clock reads 2.33 years when he first notices the explosion, and it took 5.667 years for that information to travel to him, the explosion must have happened at t’ = -3.33 years.

From this example you can see that objective events happen in different orders depending on your frame of reference (how you’re moving relative to one another).  In Steve’s frame there’s the explosion, and then he passes by me on Earth.  In my frame, Steve passes by, and then two years later Greg blew up the planet.  Both of us are correct.  When you accelerate, the flow of time changes, as well as properties of the space around you.

Currently I’m undergoing two different pursuits, but hope to tie them together.  The first pursuit is to figure out how the brain creates the subjective of sense of space from images falling on our eyes, and information being processed within our brains.  On the other hand, I work at understanding problems like the thought experiment above.  I hope to understand it well enough eventually to fuse the two together as best I can.  Most of the time, I just spin in circles, confused out of my mind.

I’ve mentioned many times on this blog that the way our brain represents space and the flow of time isn’t quite right.  This example illustrates that problem pretty well.  I struggle in vain to picture four dimensions in my mind, and see this fabric of space-time, and see the light-rays moving toward Steve, and also toward me, but I can’t even come close.  And when I try to picture things from each person’s perspective, I always want to assign a common time between the two of frames.  Some sort of “global” absolute time.  In the end, I just have to work these equations, and work more and more problems, in more and more varied situations, and get a better and better “feel” for what’s going on.

I’ve finally figured out how to embed math equations in my posts, so I want to start adding thought experiments like this in areas related to modern physics.  I’d like to maybe go into some of the basics of quantum mechanics, particle physics, nuclear physics, and all that good stuff.

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