...two puzzles...
between circular and hyperbolic angles.
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On starting relativity with the metric equation rather than Lorentz transforms, Steve Gilham wrote:
I approve heartily of this proposal. Having had during my time as a
graduate student (working on fluid flow near black holes) unlearning
the quasi-Lorentzian approach normally taken to SRT, and only years
later finally having that "Aha!" moment when I realised what the
coordinate free geometric approach was all about, I think that
starting with the geometric approach would be a great improvement all
round.
Many [who find relativity counter-intuitive (my insert)]
seem to base their approach and objections on the classical two-frame
style of layman's relativity overviews, which cutting directly to the
metric would avoid.
Note: Although we cut to the metric here, we aren't shy about choosing a specific reference frame. Such shyness is something inherited from early relativity's rejection of a physically special frame, with respect to which the medium carrying lightwaves (or ether) was at rest. Rejecting this doesn't prevent us from making our own choice of a physically arbitrary reference frame, in terms of which to describe motion. J. S. Bell in Speakable & Unspeakable in QM (Cambridge, 1987) argues this, & drawings of the simplest embedding diagrams in general relativity (cf. Kip Thorne's Black Holes & Time Warps, W. W. Norton, 1994) require that this be done.
On our 2-clock relativity paper for teachers, and new approaches to relativity in general, relativity-author William Shurcliff wrote:
I wish you had pointed out this neat fact: because proper
speed can be infinite, improper speed is necessarily limited --
cannot exceed 3x10^8 m/sec. If someone asks: "Why is the speed
of light limited?", we answer: "It is not limited if defined in
a manner that requires no synchrony, i.e. if defined as a ratio
of distance (defined most simply) and time duration (defined most
simply).
It's a wonderful subject, fun to think about. You and I
believe it could be explained better, understood more simply.
But my firm and gloomy impression is that most of the professors
who teach the subject regard it as old, finished -- no need for
any further thought or new terms. "Its finished. Please don't
bother me with any new ideas."
About the Andromeda Problem here, e-mail correspondent Robert Williams wrote:
What's up man? I am a senior in High School taking my second year of Physics.
One of the privileges we have is to make a "scavanger hunt" for the regular physics students.
It is composed of impossible problems with crazy calculations and conversions.
I know this probably sounds unheard of to you, but, I think this is one of the coolest
problems I have ever seen. I must admit I don't have a clue as how to do this and
probably won't for a couple more years. But what I was wondering is if you could give me
just the answer. NO work just a number. I need that to put this problem on the scavenger
hunt. I am dying to see the student's minds meltdown. IF you won't do it I will
understand but I would truly appreciate it. I will even give you credit for the problem.
I need the answer by this coming Thursday so if you can do it please do it.
This would be truly awsome. Thanks alot man and you have my word that I would never let the
answer be known. Thanks. Hope to hear from you in a day or so.
On page 154 of Special Relativity by A. P. French (W. W. Norton, NY, 1968), one of the more respected older SR texts, Dr. French defines acceleration as the map-time derivative of coordinate-velocity, i.e. in our notation dv/dt, and then not surprisingly says:
"The main lession to be learned from the above calculations is that acceleration is a quantity of limited and questionable value in special relativity. Not only is it not an invariant, but the expressions for it are in general cumbersome, and moreover its different components transform in different ways. Certainly the proud position that it holds in Newtonian dynamics has no counterpart here."
One objective of these pages is to show that the proud position of acceleration in Newtonian dynamics does indeed have a counterpart in SR. That position is held not by coordinate-acceleration dv/dt, but by the frame-invariant proper-acceleration (ao=dw/dt=d(gamma v)/dt) as discussed in SpaceTime Physics by Taylor & Wheeler (W. H. Freeman, NY, 1963/92). Professor Taylor, when a way to solve relativistic constant acceleration problems with help from Galileo's* equations was first mentioned, wrote:
"It is always interesting how much each older theory telegraphs the structure of the new theory -- but only after one understands the new theory!"
Although we need a Galilean chase-plane to find out what Galileo's acceleration equations tell us about relativistic motion, Newton's equations provide considerable insight at high speeds if we only consider a proper-time clock along with the coordinates used to describe motion classically. We must also know how to define acceleration at high speeds. As shown elsewhere for the unidirectional case, at high speeds it is the second derivative of position with respect to time only in the Galilean kinematic, but it is the rate of energy gain, per unit distance-traveled per unit mass, in three co-existing kinematics! For unidirectional motion, proper-acceleration is also the coordinate-time derivative of proper-velocity, but in (3+1)D it is not the second derivative of position with respect to any time (cf. gr-qc/9607038). Although ``2nd time-derivative'' acceleration returns when we make the leap to 4-vectors, frame-invariance for acceleration (like that in Newtonian physics) stays with the proper-acceleration 3-vector alone.
* The kinematic equations describing constant acceleration, sometimes referred to as "Newtonian" since they follow from Newtonian dynamics, were developed geometrically for describing the motion of falling objects by Galileo (cf. Dialogue Concerning the Two Chief World Systems, University of California Press, 1962).
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:)Unfortunately, I think, confusion about this may have resulted in a failure to fully explore the benefits of addressing relativity from the point of view of someone with a single (user-chosen) inertial reference frame. Single reference-frame thinking, of course, is the kind of thinking done by most earth-dwelling students in introductory physics courses. Because of this confusion, students are not told the relativistic things one can do with that kind of thinking. We have taken up that challenge here.
For example, the concept of a "traveler-kinematic" relies on a traveler whose distance measurements are referred to a reference inertial frame that he or she may be moving with respect to. In other words, all equations use "map distances", dx, measured with respect to some user-defined reference frame, such as the earth's surface or the rest frame of the galaxy center. If we thus shamelessly [:)] pick a user-defined inertial frame for measuring distances, many advantages result.
For example:
Introductory University, College, and Conceptual Physics Courses basically cover (and provide) some or all of the foregoing in their sections on dynamics (i.e. not including sections on waves, condensed matter, and thermodynamics). We show here that with a minimum of new concepts and abstraction, one can take the next step in these same courses and either graphically or with equations...
In the long run, the foregoing material would then be needed only in review chapters of Modern Physics and dedicated Relativity Courses. These courses might then...
In Special Relativity and the early part of General Relativity Courses, one might then...
The next step is a big one. You will probably want General Relativity if you want to make a living in the relativity sciences, or if you love abstract applied math. Many ideas associated with Pythagoras theorem (so beautifully extended by Special Relativity) are lost altogether, in every place where the special relativist might simply estimate using a gravitational field. Quantities of interest not only vary from point to point (you do get to keep a differential form of Pythagoras Theorem), but most practical parameters have between 16 and 256 components! This makes the quick working out of numerical problems an intuitive challenge at best. Of course, exciting things happen as well when you...
However, to predict what happens to a twin who leaves on a voyage and then returns, only path-dependent elapsed-time need be understood. The rule is this: traveler time in context of a single inertial frame goes at a rate that is a factor of 1/gamma slower than that of a non-traveler, where gamma is defined as usual (see below). Even acceleration of the traveling twin can be ignored in solving the problem!
Let's do it quantitatively: Twins 1 and 2 are together at event A (say their 10th high school reunion) and back together again at event B (say their 20th reunion), but twin 1 spends most of the intevening time going at coordinate-speed v traveling who knows where (it doesn't matter as long as he makes it back for the 20th reunion), while twin 2 stays in town. At event B, twin 2 will be 10 years older, while twin 1 will be 10/gamma years older, where gamma = 1/Sqr[1-(v/c)^2] as usual.
We've ignored the accelerations and decelerations, and only require that our two twins be present at events A and B having taken different paths to get there. We've also solved the problem in that particular inertial frame for which events A and B occur at the same place. If you like, accelerations and decelerations can also be considered in such problems graphically and quantitatively without transforming to other frames, but that is not required to figure differential aging for the simplest twin-paradox adventures.
Consistency with other inertial frames (i.e. frames for which A and B are separated in space) is further questioned in the original twin paradox problem. As you probably know, with help from the Minkowski metric or Lorentz transforms you can solve the problem in any other inertial frame, and you get the same result for aging of the twins. For example, if twin 1 spends most of the first half of their time going at speed v to the left, with the remainder being spent going at speed v back to the right, you might pick a frame moving at speed v to the left. The x-ct diagram for this solution will look quite different, but the answers will be the same.
Anyone first seeing it might be confused by the counter-intuitive nature, when expressed in everyday parlance, of the strange linkage between space and time required to solve the problem in this second case. But this is not needed to get kids started, at least as far as relativistic-twin adventures are concerned, since you can get the answer simply by considering the path-dependence of time between events in the context of a single inertial frame.
If anyone has opinions in this regard about the various books out there now, or can suggest some other questions to ask as well or instead, let me know! If you know someone working on an SR or introductory physics text, share your impressions with them as well.
Four-Vector Form:
Note in the first set of equations that, when velocities are dressed up in units of c, times in units of c/ao, and distances in units of c^2/ao, the six equations involving x plot onto only 4 curves, since v/c = aot/c, and u/c = aob/c. For relationships between kinematics, note that the right-hand side of all 6 x-equations is gamma-1, so that you have gamma expressed separately in terms of all six kinematic variables: t, v=dx/dt; tau, u=dx/d(tau); and b, w=dx/db.
The simplest complete set of interkinematic conversion relations may be...
{Thanks. :) /pf}