Constructing Physics

Elementary Physics Course Notes

Adam Johnston
Department of Physics
Weber State University


Special relativity.
A warning: Things are about to get very strange.
Remember the two electrons that you bought a couple of weeks ago?  Imagine that you are bring them home with you on a train . . .
What are the forces that your two electrons exhibit from your point of view?
From the point of view of someone standing outside of the moving train?
You measure different amounts of time for the same event, since you observed different amounts of force on the two electrons.  This is a big problem for physics.  What's wrong?
Nothing.  We just have to change the way we think about time itself.
Furthermore, Michelson and Morely suggest the constancy of the speed of light without really knowing it.  Einstein uses constancy of the speed of light to develop special relativity and explain how those two electrons in the box can behave in two different ways.  This leads to strangeness such as:

Time dilation.

Length contraction.

Relativistic mass.

Energy and mass equivalence.

Evidence for this stuff?  Muons.  Cesium clocks.  Michelson & Morely (before Einstein).
Adam is noticeably frustrated with lectures as of late.  Classes are best when things explode, move, smash, ignite, spark, etc.  But, in light of the conditions necessary for special relativity to be a noticeable effect, objects have to be moving at close to the speed of light!  Adam and his superhuman strength can attempt to throw things this fast, but there is still a limitation that is imposed by nature herself.
Once we've come to understand time dilation, we can see that a twin sister who leaves on a very fast traveling spacecraft can come back and will have actually aged less than the twin left behind on Earth.  For example, the astronaut twin might have aged 1 year, but the Earth-bound twin might have aged 10 years.  (You can figure out exactly how fast one has to travel to achieve this effect by using the time dilation equation, but we haven't been going into the details of solving these kinds of problems in this course.)  In this case, it's easy to tell who is moving and who isn't.  Why?  Because the astronaut twin came back -- she had to accelerate to come back.  Any frame of reference which undergoes an acceleration is known as a non-inertial reference frame.  Once this reference frame is determined, then we can figure out which of the two twins can claim to be younger when they meet up again.
If two twins are out in the middle of space with a relative velocity between them, neither can tell who is doing the moving (if either).  From each perspective, the other is moving.  So, from each perspective, the other is the younger twin.  This is what is known as the "twin paradox."  How do we resolve such a paradox?  The good news is that the two twins don't get back together to compare ages until one of them accelerates.  It isn't until there is some change in the motion (an acceleration) that one of them is in the non-inertial reference frame.  At this point, this determines which of the two is going to have aged less (the twin who accelerated).
Confused?  Perplexed?  Good.
Consider what happens to time dilation, length contraction, and relativistic mass quantities as a relative speed hits the speed of light.  Specifically, why can't you travel at the speed of light?  If we go back to the equations for time dilation, length contraction, and relativistic mass, we can make v=c (your speed equals the speed of light) in any of these equations.  When you do this, we showed how the values for mass, length, and time become unreasonable and non-real.  Essentially, this is telling you that the situation being described can't happen, which means that traveling at the speed of light can't happen either.
Why/how can light travel at the speed of light, then?
We then introduced general relativity.  This description of the nature of the gravity is based on the principle of equivalence.  This principle demonstrates how gravity can be simulated by an accelerating frame of reference.  If you are in a spacecraft that is accelerating upwards at the rate of gravity, objects will seem to "fall" in the spacecraft, because the spacecraft is always moving faster and faster upwards.  This doesn't just include balls, rocks, cats, etc., but also includes light.  A beam of light shot from left to right will seem to fall "down" as the bottom of the rocket catches up to the beam.  Strangely, this situation is equivalent to being in gravity; but this situation shows light "falling!"  So, it seems that gravity should actually bend light.  This is something that Newton's description of gravity wouldn't have accounted for (since light doesn't have mass), but seems to be the case.  
Einstein's general relativity describes space as something which can actually have a shape.  The shape of space isn't something that we can see, but it can be detect by observing how things "fall."  To get a picture of this, we might picture a place called "Flatland."  If you lived on a piece of paper, you would be able to move left-to-right or back-and-forth, but you would have no idea what "up" or "down" meant, because your piece of paper doesn't have space in that direction.  However, if someone started to roll your piece of paper or curve it somehow, you might feel yourself getting pulled in one direction or another, even though you couldn't see how the paper was any different (because you still only travel along the surface of the page, not knowing what is "up" or "down".)  In our 3-dimensional space, we might imagine that some big bully could bend/warp everything in a 4th dimension that we can't see, but we would certainly feel this because we would start to get pulled magically in one direction or another.
We modeled how this might explain gravity.  A giant marble on a rubber sheet (a 2-dimensional universe) makes that universe warp, so that anything else rolling by doesn't travel in a "straight line" anymore; instead, it either gets pulled into the pit of the giant marble, or it orbits around.  Hey!  That's exactly how gravity works!  And this is what Einstein is trying to say with general relativity:  The mechanism for gravity could be a warping of space, rather than a magical invisible pull.  If this is the case, then how could we test it?  We should be able to see all objects follow weird, warped paths around very massive objects.  Gravitational lensing of light is an example of this.  Light gets bent as it travels very close to the Sun or other (even more massive) big objects.  We can see this more and more all the time as our telescopic views of distant objects get sharper.  Multiple images of the same object seem to bent around a very massive object in the foreground.
Phew.  Yes, this is more bizarre, but it's also more explanatory than what Newton gave us.