Prologue: The Laws of Physics Transformations

Prologue: The Laws of Physics Transformations Before we get lost in details, let’s remind ourselves about some of the Physics “Things” we believe. Then I’ll share an Answer.

physics things that we believe…  Newton’s Laws  conservation of momentum (linear and angular)  conservation of energy  laws of thermodynamics  Maxwell’s equations (E&M theory) even if you do not understand the physics of these “things” in detail, over your lifetime you have developed an intuition for how they work these are the “really big” ideas of classical physics

Transformations Now let me share an Answer. You have had the freedom to choose axes (reference frames) throughout Physics 23 and Physics 24 (or equivalents). If the laws of physics are valid, you must be able to transform from your coordinate system to someone else’s coordinate system, and get the same results. Relativity deals with transformations between reference systems which are moving with respect to each other. If two reference frames are moving with respect to each other, but not accelerating, Newton’s laws are valid in both.

Mathematically, how do you transform from one such coordinate system to another? Let’s take the simplest example: a reference frame x’y’z’ moving with a fixed velocity relative to reference frame xyz, such that the reference frames coincide at t=0, and the relative velocity is along the x-axis. This is called the Galilean Transformation. The coordinates x’y’z’t’ give the location of a stationary point in the fixed reference frame as measured from the moving reference frame. (Also, an observer in the fixed frame would say that the point is located at x’y’z’t’ relative to the moving frame.)

Newton’s laws are invariant under a Galilean transformation. The Galilean Transformation is “the common sense relationship which agrees with our everyday experience.” (hyperphysics*) It is not how the universe works! It is not the law! *I’ve paid for my “license” to use hyperphysics material in my teaching.

Maxwell’s equations are not invariant under the Galilean transformation. This was recognized by physicists of the 1800’s as a serious problem to be overcome. Lorentz discovered the transformation under which Maxwell’s equations are invariant. The Lorentz transformation (simplified version). This is the Law (and my Answer).

The Lorentz transformation is not intuitive. Why should the laws of nature be invariant under it? That is not a question for science to ask. Nature follows Lorentz transformations. Live with it. I am about to “derive” some equations of relativity using thought experiments. It is appropriate for you to be suspicious of thought experiments. The same equations can be derived using the Lorentz transformation. No dubious thought experiments. Just pure mathematics describing nature. Hyperphysics has a table showing the ideas and experiments that DEMAND relativity.

Relativity Special Relativity What does it mean to measure something? If we both measure the same object with the same tools, should we get the same result? What does it mean to know something? Should the laws of physics be the same for everybody? You’re in a spacecraft and a comet zips by. Are you moving or is the comet moving? What does it mean to be in motion?

We have this idea that “physical reality,” whatever that is, ought to be independent of who/when/where/how a measurement is made. Electromagnetic theory was perfected by Maxwell and others in the late 1800's. Water waves propagate through water, sound waves propagate through air. It is not critical to electromagnetic theory, but it was believed that electromagnetic waves propagated through the “ether,” relative to some universal reference frame.* The ether, being ethereal, proved very difficult to detect! *Imagine the ether attached to this universal reference frame. If you are moving relative to it, you experience an “ether drift.”

T1 Newtonian Relativity Theory You swim 200 meters downstream in a river, turn around, and swim 200 meters upstream. It takes a time T1. You swim 200 meters perpendicular to the river bank, turn around, and swim 200 meters back. It takes a time T2. T2 T1 and T2 are different. Newtonian “relativity theory” shows you how to calculate T1 and T2 (add the vector current velocity to your vector velocity). river current If you make the “same” measurement on light moving through the ether, you ought to get the “same” result; T1 and T2 are different.

Michelson and Morley built an interferometer capable of making such a measurement. mirror “partial” mirror A light source mirror    B Half the light follows path A. hypothetical ether drift Half the light follows path B. detector The dashed line portions of the paths are oriented differently relative to the ether drift.

If the times to travel paths A and B are the same, the two light beams arrive in phase and interfere constructively. If the times are different, the beams interfere destructively. Measurement of changes in interference fringe shifts allows you to deduce the time difference. A  B hypothetical ether drift

But wait! you object. It is impossible to make paths A and B exactly the same length. An observed fringe shift might be due to the path length difference, or it might be due to the different orientations of the path relative to the ether drift. So you take a measurement, rotate the apparatus 90 degrees in the horizontal plane, and take another measurement. The difference between the two measurements allows you to very precisely measure the time difference due only to the ether drift.

Michelson and Morley did the experiment in July, 1887. They found nothing. No ether drift. (Less than 5 km/s; current upper limit is 15 m/s.) No ether.* They tried it again later, in case during the July measurement the earth was coincidentally at rest with respect to the ether. They got the same results. *No universal frame of reference?

I’ve seen sources that say this result wasn’t terribly bothersome, because the “ether” was a conceptual convenience, and was not required to make E&M theory work. I’ve seen other sources that say this was “devastating” at the time. It certainly created a problem. In fact, if you believe Michelson and Morley and Maxwell, you are forced to conclude that the speed of E&M radiation is the same in all non-accelerated reference frames, regardless of the motion of the radiation source. A bit difficult—no, make that extremely difficult—to accept!

How do you reconcile the lack of a universal reference frame with the idea that everybody's measurement of the same thing ought to produce the same result? Einstein, 1905, Special Theory of Relativity Special relativity treats problems involving inertial (non-accelerated) frames of reference. We believe the laws of physics work and are the same for everybody. Experiment demands that the speed of light be constant. Let’s make these two things our postulates and see where we are led. The validity of the postulates will be demonstrated if the predictions arising from them are verified by experiment.

Postulates of special theory of relativity:  the laws of physics are the same in all inertial reference frames  the speed of light in free space has the same value for all inertial observers* The first “makes sense.” The second is required by experiments but contradicts our intuition and common sense. *Independent of the motion of the source or relative speeds of observers!

Time Dilation – a consequence of the two assumptions Let’s begin with a definition, and then construct a clock. The time interval between two events which occur at the same place in an observer’s frame of reference is called the proper time of the interval between the events. We use t0 to denote proper time. You’ve been chosen to be a timer at a track meet, so you go stand by the finish line. You start your stopwatch when you see the puff of smoke from the starter’s gun at the starting line, and stop it when the first runner crosses the finish line. Did you measure the proper time for the sprint? Suppose you are timing an event by clicking a stopwatch on at the start and off at the end. In order for the stopwatch to measure the proper time, the “start” and “stop” events must occur at the same place in your frame of reference.

mirror L0  laser with built-in light detector Now let’s make a clock. tick tock “It's not that I'm so smart , it's just that I stay with problems longer.”—A. Einstein

How long does a “tick-tock” take? mirror time = distance / velocity t0 = 2L0 / c L0 “I measure proper time because the light pulse starts and stops at the same place.”  laser

Now put this clock in a transparent spacecraft and observe as it speeds past. “I don’t measure proper time because “tick” and “tock” occur in different places.” L0 tick tock   entire clock moves with speed v

How long does a “tick-tock” take? Let the total time be t. distance = velocity · time geometry says: ( L02 + (vt/2)2 )1/2 ( L02 + (vt/2)2 )1/2 L0   vt/2 vt/2 v

According to the second postulate of special relativity, light travels at a speed c, so D = ct. We also know the proper time from our “stationary clock” experiment: t0 = 2L0 / c

Solving (1) and (2) for t and replacing L0 using (3) gives: Note that (1-v2/c2)1/2 < 1 so t > t0. It takes longer for an event to happen when it takes place (is timed) in a reference frame moving relative to the observer than when in takes place at rest in the observer's reference frame. Time is dilated.* This applies to all clocks. *How to remember what “dilated” means. Pupils in your eye can dilate or contract. Dilate must be the opposite of contract, so “dilate” must mean take on a larger value.

A moving clock ticks slower. If I time an event which starts and stops in in my frame of reference, I measure t0. If I use my clock to time the same event as it takes place in a reference frame moving relative to me, I measure t>t0. In the latter case, I claim my clock, which measured t, is correct, so that an identical moving clock, which would measure t0 in the moving reference frame, is slow. An event must be specified by stating both its space and time coordinates.

Example: the Apollo 11 spacecraft that went to the moon traveled a maximum speed of 10840 m/s. An event observed by an astronaut in the spacecraft took an hour. How long does an earth observer say the event took? Problem Solving Step 0. Think first! Always ask: what is the reference frame of the event? Is the observer in this reference frame or moving relative to it? The event took place in the spacecraft. The proper time t0 is the time measured in the spacecraft. Thus, t0 = 3600 s. The “observer” in this problem is the person on earth, not the astronaut! The earth observer measures t.

Problem Solving Step 1. Draw (if appropriate) a fully-labeled diagram. (Include values of known quantities.) If it helps you to draw a sketch of the earth, a spacecraft, and a couple of stick figures, do so! Include values of known quantities. c = 3x108, v = 10840, t0 = 3600. If you use SI units throughout, your answer will be in SI units, and I only need to see units with your final answer. If you mix systems of units, show the units at each step.* You can always show units at each step if it helps you. Sooner or later, if you mix units, you will suffer pain.

Problem Solving Step 2. OSE. So far, we only have one relativity OSE: “Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. THAT'S relativity.” – A. Einstein.

Problem Solving Steps 3 and 4. Solve algebraically first, then substitute values. The algebra is already done in this case. t = 3600.00000235 s. Not a big difference, but it is measurable. The actual experiment has been done with jets flying around the earth, and the predicted time dilation has been observed.1 As expected, the earth observer measures a bigger number for the time. The moving clock on the spacecraft measured a smaller number. The moving clock ticks slower. 1J. C. Hafele and R. C. Keating, Science177, 186 (1972).

What if v>c? It can't happen. We’ll see later that it is not possible to accelerate an object up to the speed of light. What about time running backwards? Sorry, time always runs forwards. What about seeing an event before it happens? Can't, because c is finite. However, because of time dilation, events which appear to be simultaneous in one reference frame may not appear to be simultaneous in another reference frame. “The only reason for time is so that everything doesn't happen at once.” – A. Einstein

Prologue: The Laws of Physics Transformations