Tag Archives: Relativity

Massive episode about relativity

21 Mar

And finally, the conclusion of our four part primer on relativity. Here’s a recap on the topics we’ve covered so far; the laws of physics depend on the frame of reference; the faster an object moves, the slower time moves in that frame; the faster an object moves, the shorter that object appears from a resting frame.  In this episode, we’re look at questions about one more property of an object. Specifically, how does motion affect mass?

Well, if there is one equation that people remember about special relativity, it’s

52c7687643df1c12231b39e324850586

And we’ve established that the speed of light, c, is a constant, so mass just depends on the amount of energy. Shortest fun fact ever…

Unfortunately not. E = mc² is a special solution of the more general equation

f653e9c4c421742eebeca629813279d0

where m should be the mο and p is the object’s momentum (inertial mass [m] x velocity [v]). When the velocity of the object is zero, the momentum expression disappears and we are left with the now famous equation; an equation which describes how the energy stored in an object is related to its resting mass. Hmm. What the heck is inertial mass, and how is it different from rest mass?

Inertia refers to the tendency for an object to continue traveling and resist a force that attempts to change it’s motion.  The rest mass is a measurement of the energy stored in the mass in a resting frame.  The relationship between the rest mass and the inertial mass (sometimes called the effective or relative mass) is

images

where mο is the rest mass and m is the effective mass.  As a result the effective/inertial mass of an object mass increases as the object travels at increasing speed.

Now, let’s put it all together.

  • An object can’t travel fast than the speed of light.
  • As an object’s velocity get’s closer to the speed of light, it’s effective mass increases.
  • In order to conserve resting mass in the Energy-Momentum relation,  energy and momentum vary proportionally.

With these conditions, adding energy to an object and attempting to increase it’s velocity results in an increase in it’s effective mass that grows exponentially as the velocity approaches c.

exponential-growth-graph

This is what exponential growth looks like. Small change initially; infinite change as v approaches c.

One cool implication of this is that if you keep putting energy into an object, it could achieve infinite apparent mass. Before this point it would form a black hole, but it’s still cool to think about.

I think that’s enough mind blowing stuff about relativity. Don’t worry if it seems a little confusing. To adapt a phrase from Niels Bohr, “Anyone who is not shocked by special relativity has not understood it.” If there’s nothing more that you take from these last few episodes, I hope you at least will accept that the most famous equation in physics really ought to be written as Eο = mοc2.

A short post on length contraction

20 Mar

Here’s a quick recap of the things we know about relativity; the perception of time is dependent upon velocity, and the speed of light is a constant regardless of the frame.(check out the full episode #1 here and #2 here). Alright, so we know something about the time and we know something about the speed. If you’ve ever dealt with rates before, you might be wondering “What’s happening to the distance?”

That’s a fair question, but it’s a little more complicated than it appears on the surface. Let’s think back to our train problem. If we want to measure the train at rest, we can hold a tape measure up to the ship and record the length. That’s works because the ship and the tape are both at rest with respect to one another, but what if we wanted to measure the space ship while it’s moving?

Now we have a problem. We could hold the tape measure still and wait for the train, but we’d have to be able to make a length measurement instantaneously as the train reached the tape. It’s not a good plan. We could move the tape measure at the same speed as the train, but then we’re taking another measurement with the tape at rest with respect to the train. Hmm. Let’s try something different. Bring in the timer gates.

image095

Here’s a quick refresher on timer gates. They are a gate with a laser between them (here represented by the dashed lines). When an object passes through the gate and breaks the laser, an electrical impulse is sent down the black wire to the computer which records the time. That’s all there is to it.

How can we use these? If we set up two along the tracks of the moving train, we can use one to record when the back of the train goes by and the other to record the arrival of the front of the train. You could do some math to determine the length of the moving train as a function of the time difference and the distance between the gates…but you probably don’t like math. The other option is moving the gates until the back of the train goes by one gate at the exact time that the front of the train arrives at the other.  At that point, the distance between the gates is the length of the moving train.

Alright, so I do that experiment and I find that the length of the moving train is related to the resting length of the train by this equation

image009This means that the moving object appears to be shorter than the object at rest by some factor related to the speed at which it’s moving. Actually, that factor seems kind of familiar. Didn’t we see that when we were dealing with time contraction?image008Yep. Same factor. That’s because it needs to be in order to balance out the equations that makes the speed of light a constant. Cool.

Two last notes about length contraction.

#1. The length contraction would be perceived by both the people in the moving object and the people not in the moving object. Each would consider themselves at rest and the other objects moving.

Length Contraction

On the top: Trains DEF seem short from the frame of ABC
On the bottom: Trains ABC seem short from the frame of DEF

#2. The length contraction only occurs in the direction of motion. So, in our last example, the height of the train would remain the same even while the length was apparently decreasing.

That’s a pretty good primer on length contraction. I’ll do one more post on relative mass and then we’ll take a little break from physics to look at some other cool stuff.

Keep moving, you’ll stay younger

16 Mar

So in the last episode, I explained very basically what relativity means. We saw that the same event could appear differently to two observers depending each man’s frame of reference. That was bizarre, and from there we could begin to ask other questions. For instance, what other measurements depend on our frame of reference?

Let’s start with one of the more basic measurements-time. We’re going to use another example to explain what’s happening, but this time we’re also going to use some algebra and a triangle. Bear with me, the math isn’t too bad.

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

Alright, so imagine that we’ve got a space ship that’s traveling at speed v, and there’s a flash of light that is bouncing up and down inside of the space ship. The speed of  the flash of light (and all light for that matter) is c, and that doesn’t change because the speed of light is a constant. So far so good? Great.

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

For an observer on the ship, we’re at rest with respect to the ship. Another way of saying that is that the ship is his rest frame. If we call the height of the ship L, then the total time it takes the light to travel from the floor to the ceiling is

image004

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

But looking up at the ship from Earth is a different story because we’re dealing with a different frame of reference. While the light is bouncing up and down, it also has to travel horizontally with the rest of the ship. The total distance it travels is the diagonal, D. As a result, the time it takes in this alternate frame is.

image005

Note that the speed of light, c, is the same for both of these equations. D, however, is not the same as L; D is longer than L.

Let’s stop right here to drive home that concept. The light travels less distance in the rest frame than it does in the moving frame. As a result, less time passes in the rest frame than in the moving frame. A clock on the space ship would go slower than a clock on Earth. That’s time dilation, and it’s not just a theory; It’s a fact!

… Now, if you wanted to find out how much slower, you’d have to do some math. We can find D using the Pythagorean theorem

image006

At this point the math get’s really elegant (don’t worry, I won’t show it). Suffice it to say, you plug the D into the Δt’ equation, square the whole expression, collect the Δt’ terms, isolate the Δt’, square root the whole expression, and you end up with this equation

image007

Which is cool, because then we can substitute in Δt to get

image008…the equation for how time in a moving frame relates to time in a rest frame. Boom.

So that’s time travel. Questions are totally appropriate at this time. Stay tuned for the next episode where I’ll show you how to shrink objects.

Framing the debate: relativity

15 Mar

Note: Albert Einstein’s birthday was yesterday, March 14th. I didn’t realize that until after I had decided to write a few facts about relativity. Sometimes you just get lucky.

I’ve seen a couple of links to a Popular Science article, “Warp Factor,” about a NASA engineer’s effort to develop an engine that could travel faster than the speed of light. I read it last night and I was rather skeptical, purely from a theoretical standpoint. Still, it did get me thinking about relativity and some of the funky things that happen at high speeds. Yet in order to talk about the weird stuff, you have to know what relativity means. Relative to what?

The most common way of explaining this, used by Daniel Frost Comstock in 1910 and Albert Einstein in 1917, is more or less this; imagine there is a man on a train moving past a man on the side of the tracks. Then, at the exact moment that the passenger and the bystander pass one another, a flash of light happens at the center of the train car. What does each man observe?

735px-Traincar_Relativity1.svgThe observer on the train experiences precisely what we’d expect him to see. He is traveling the same speed as the train, so relative to him the train is at rest. The flash of light happened at the center of the train, so the light reaches both ends of the train at the same time.

800px-Traincar_Relativity2.svg

The observer on the ground has a very different experience. The flash of light happened at the center of the train but relative to him the train was moving to the right. The left side of the train moved towards the light while the right side of the train moved away from the light. As a result, the light reaches the left side before it reaches the right side.

This raises a problem. One event happened, but there were two entirely different experiences. Which one is valid? The surprising answer is both! The key words in each of the explanations was the term relative to him and his frame of reference (hence the field of relativity).  Either of these can happen, depending on the observer’s frame of reference.

And that’s just the beginning of the weird stuff that happens in relativity (Still if you have any questions, now is a good time to ask). Stay tuned for the next episode where I’ll tell you all about time dilation.