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It's All Relative

Posted by Akano , in Math/Physics Mar 22 2012 · 552 views

math physics Einstein relativity
Being a physics grad student has seen me be in quite the scientific mood lately, hasn't it? Well, unfortunately, I still don't have a new comic made (I'm sorry, everyone! ><), but I do have another idea for a blog entry. Last week, Pi day (March 14) marked Einstein's 133rd birthday, and since my Classical Mechanics course is covering the Special Theory of Relativity, I thought I'd try to cover the basic ideas in blog form.

According to the laws of physics laid down by Sir Isaac Newton, all non-accelerating observers witness the same laws of physics. This included an idea of spontaneity, the idea that someone traveling on the highway at 60 mph would witness an event occur at the exact same time as someone who was just sitting on the side of the highway at rest. The transformation from a reference frame in motion to one at rest for Newtonian physics is known as a Galilean transformation, where x is shifted by -vt, or minus the velocity times time. Under such transformations, laws of physics (like Newton's second law, F = ma, remain invariant (don't change).

However, during the 19th century, a man by the name James Clerk Maxwell formulated a handful of equations, known now as Maxwell's equations, that outline a theory known as electromagnetic theory. Of the many new insights this theory gleaned (among these the ability to generate electricity for power which every BZP member uses) one was that light is composed of oscillating electric and magnetic fields; light is an electromagnetic wave. By using his newly invented equations, Maxwell discovered what the speed of light was by formulating a wave equation. When his equations are used to describe electromagnetism, the speed of light is shown to be the same regardless of reference frame; in other words, someone traveling near the speed of light (as long as they weren't accelerating) would see light travel at the same speed as someone who was at rest. According to Newton's laws, this didn't make sense! If you're in your car on the highway and traveling at 60 mph while another car in the lane next to you is traveling at 65 mph, you don't see the other car moving at 65 mph; relative to you, the other car moves at 5 mph. The reason that light is different is because a different theory governs its physics.

This brought about a dilemma: is Maxwell's new electromagnetic theory wrong? Or does Newtonian mechanics need some slight revision? This is where Einstein comes in. He noticed the work of another physicist, Lorentz, who had worked on some new transformations that not only caused space to shift based on reference frames moving relative to each other, but also shifted time. Einstein realized that if light had the same speed in all non-accelerating reference frames, then objects moving faster experienced time differently than those that moved slower. This would come to be known as the Special Theory of Relativity.

How does this make sense? Well, if you have some speed that must remain constant no matter how fast one is traveling, you need time to shift in addition to shifting space to convert between both reference frames, since speed is the change in distance over the amount of time that displacement took place. If you have two reference frames with some relative speed between them, the only way to shift your coordinates from one to another and preserve the speed of light is if both frames experience their positions and times differently. This means that, if something moves fast enough, a journey will take less time in one frame than the other. Special relativity says that moving clocks progress more slowly than clocks at rest, so someone traveling in a rocket at a speed comparable to the speed of light will find that the journey took less time than someone who had been anticipating his arrival at rest. This also means that if someone left Earth in a rocket traveling near the speed of light and came back ten years later would not have aged ten years, but would be younger than someone who was his/her age before his journey took place. Weird, huh?

If you think this is crazy or impossible, there have been experiments done (and are still going) to try to confirm/reject the ideas of special relativity, and they all seem to support it. There's another relativity at play as well known as general relativity, which states that gravitational fields affect spacetime (the combination of space and time into one geometry). General relativity says that the higher up you are in a gravitational field, the faster clocks run (time speeds up). A proof of this theory is GPS; the satellites that help find your position by GPS are all higher up in Earth's gravitational field than we are, and thus their clocks run faster than those on Earth's surface. If general relativity weren't considered in the calculations to figure out where you are on Earth, your GPS would be off by miles.

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I'm actually just about to hit Relativity in my class as well. :P
I always found the subject so fascinating, it's part of why Optics is my field of choice. (that and high-energy laz0rs, but I digress).
When I first learned about relativity, it was through the "light clock" explanation. Where you have a cylinder 3.0 X 108 meters tall, and a beam of light bounces back and forth, marking a second at each contact. Then as the clock moves the light traces a sawtooth path, and using Pythagorean theorem, you could tell the new distance, along with the new time for the clock.

I always loved that explanation. It just seems so intuitive and counter-intuitive. :P
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I think the problem here is a confusing of time itself with abnormal changing. If a person prematurely ages to the point that they look like 80 when they're 40, they're still 40, no matter what their accidents (material properties) look like. The same principle also applies to clocks, and to persons. Just because this high speed makes the accidents of a thing speed up or slow down does not mean that time itself slows down or speeds up for a person.

Yes, if you could not tell, I firmly hold that time is universal, and cannot be altered for a particular person. :P
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Eeko: Optics and Acoustics (and wave physics in general, like quantum mechanics) are my favorite areas of physics. In fact, I'm planning to do research in AMO (Atomic, Molecular, and Optical) physics for my Ph.D. Also, that's a really cool explanation; I assume that the cylinder is moving perpendicular to the propagation of the light, correct?

Also, your avatar/sig combo is made of win.

Zarayna: The reason for your belief that time is independent of reference frame is because no human being has ever experienced travel at near-light speed (we're talking a noticeable fraction of the speed of light, such as 1/5 * c or something like that). Even when traveling in your car on the highway, if you were to speed at 100 mph, you would only be traveling one ten millionth of the speed of light, which is nowhere near fast enough to notice time dilation. For reference, the ratio of the speed of light to the speed of sound is approximately 881,000, and the fastest airplane (after a quick search) can only reach Mach 9.6, or 9.6 times the speed of sound, which is still about 1/88,000 times the speed of light, so time dilation is still unnoticeable at that speed. However, particles being smashed together in the Large Hadron Collider at CERN are traveling at speeds comparable to the speed of light, so relativity has to be taken into account.

And I'm not saying that people appear to age more slowly when they move fast; I'm saying that they physically don't age as fast compared to someone at rest because space and time are connected by the invariance of the speed of light. Also, the person traveling faster would say that his trip took less time than it did to someone at rest because their clocks would have measured different elapsed times, even if those clocks were governed by something incredibly regular like radioactive decay.

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Pinkie Pie laughs in the face of Physics. :P

The other cool thing about the light clock, is that you can geometrically derive Lorentz's equation for time dilation. It's a little tricky, but it can be done.

Something I was just wondering though:
Say you are traveling at .6*c, relative to say... the sun. Now say that someone else is moving .6*c in the opposite direction, also relative to the sun.
Wouldn't you see the other moving at 1.2*c, which should be impossible?
Or, similarly, say you were traveling 90% the speed of light, and fired a laser in front of you. So you see the laser moving away from you at 3.0 X 108. What keeps you from speeding up to 90% the speed of the light wave you just fired?

Just two scenarios I find confusing. Wondering if you knew the answer.
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Awesome questions! In this case you have three reference frames: one perfectly at rest, one moving right at 0.6c, and one other moving at -0.6c. The person at rest would see the two objects moving with equal and opposite velocities. However, the people in either moving reference frame would not see the other person moving toward them at 0.6c. So, I think to answer your question, you would first have to do a transformation from one of the moving frames to a rest frame, and then a subsequent transformation to the other moving frame. The addition of velocities in relativity is

β'' = (β + β')/(1 +ββ')

where β'' is the speed (divided by c) of one person as seen by the other, and β and β' are the speeds as seen by the rest frame. So, for two rockets traveling towards each other at 0.6c in the rest frame, the relative speeds of the rockets to each other is about 0.882c, or 15c/17.

For the second scenario, let's consider two reference frames, one at rest and one traveling at 0.9c. When the frame going 0.9c emits laser light, both that frame and the rest frame measure it going at exactly c. Well, Lorentz transformations say that time passes slower for the person traveling at 0.9c to the person at rest by a factor of about 2.29. However, a person traveling at relativistic speeds also experiences length contraction, which means that distances in the direction of motion appear smaller in the faster frame.

This is related to the barn/pole paradox: say you have a barn whose length is 10 m and your friend is holding a pole with length 12 m. At rest, the pole can't fit in the barn. If you are an observer at rest, there is a speed at which your friend can travel through the barn such that, in your reference frame, it will fit inside the barn. However, in your friend's reference frame, the barn's length shrinks, so the barn is significantly shorter than the pole in his reference frame (since the pole is at rest relative to him). So, moving at 0.9c and being at rest yield the same speed of light, c, in each reference frame. It is because of this idea that the speed of light is a sort of cosmic speed limit because the speed of light is the same in all reference frames, thus it would take you an infinite amount of energy to achieve light speed if you move at less than the speed of light to begin with.

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I didn't think about adding a third frame at rest. Makes sense though. :P
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Name: Akano
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