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Re: Special Relativity



Regarding John Clement's comment:

Sometimes it is stated that General Relativity is inevitably needed
in the analysis of the twin paradox. Hestenes (1990) discusses how
this misconception was initiated by Einstein himself! Hestenes goes
on and states that "he may have harbored some misconceptions of his
own relativity". Interesting indeed.

Interesting indeed.

...
As long as the analysis is entirely done from the point of view of
the stay at home twin there is no paradox.

There is no paradox no matter who's reference frame is used, as long
as it's done correctly.


Special relativity can not be used to
analyze the problem from the point of view of the moving twin

Sure it can. Saying you can't use SR to analyze the observations of
the moving twin is as silly as saying that you can't use Newtonian
mechanics to understand how a Coriolis force appears in a rotating
reference frame (assuming, of course, that all speeds considered here
as so slow compared to c that the Newtonian approximation is
adequate in the first place).

Just because a particle undergoes an acceleration is no reason to
declare SR invalid, and just because an observer also undergoes an
acceleration is nor reason to delcare SR invalid either. But it is
important to keep in mind when an accelerated frame of reference is
used in SR that the transformation equations to such a frame from
an inertial frame is *not* a Lorentz transformation. Lorentz
transformations only describe transformations between pairs of
inertial frames. Transforming to an accelerated frame merely
requires using the proper *non*Lorentz transformation. This is
*quite* similar to the situation in Newtonian nonrelativistic physics
where accelerated reference frames are perfectly valid and often are
quite useful. It's just that when using such a reference frame the
the transformation to it from an inertial reference frame is not a
simple Galilean transformation (as is the case between the members
of the equivalence class of inertial frames) and Newton's laws get
modified by the appearance of various kinds of inertial (or so-called
fictitious) forces. In SR when a non-inertial frame is used then the
relativistic equations of dynamics also are modified accordingly with
the appearance of relativistic analogs of the nonrelativistic
inertial forces.

because part
of the time he is not at a constant speed.

This is not a problem for SR. It is only a problem if you insist on
using a *single* LT to completely describe the relationship between
the moving twin's frame and the Earth-twin's frame at all times. If
you use a multiplicity of LT's for the different constant velocity
parts of the moving twin's journey, or if you use the proper
non-Lorentz transformation to describe the situation in the moving
twin's frame if his/her acceleration has a more continuous profile,
then all is well with an SR analysis of the situation from the moving
twin's frame.

General relativity supplies the
pieces needed to complete the analysis from the point of view of the
moving twin's acceleration.

General relativity has *nothing* to do with the problem except in the
trivial special case that SR is fully contained in GR as the proper
description of nature when there are no significant gravitational
effects. GR is only necessary when describing a relativistic
situation involving true gravitational (i.e. not simply inertial)
effects, or equivalently, when describing a situation involving an
intrinsically curved spacetime.

Unfortunately getting students to understand this is often fairly
difficult.

It's not always just students that have such problems.

Students will routinely apply equations such as
x=X0 + V0 t + 1/2 a t^2 in situations where the acceleration
changes. In other words they do not recognize that equations are
only applicable under certain conditions.

And sometimes they might even be tempted to try to understand the
twin paradox problem from the point of view of the moving twin by
blindly applying a single LT, or by blindly applying the effects
(time dilation, length contraction, etc.) derived from a single
LT, rather than using the proper equations that actually describe
the more complicated situation.

The idea that an equation is only valid in certain ranges or when
the constants are indeed constant is a difficult concept. Similarly
the equations of special relativity are not applicable during the
moving twin's acceleration.

Sure they are. But you just have to actually use the *correct*
equations of SR when analyzing the situation from the moving
twin's frame or reference. And the correct equations are not just a
single LT and the effects coming from it alone.

David Bowman

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.