Intro
There
are many fundamental relativity discussions that are buried deep inside
related threads and hence very difficult to access at a later stage. I
will try to elevate some of those issues to the level of a separate Blog
entry.
Ralfcis' problems
To start with, deep inside a thread on "Conventionality of Relativity", Ralfcis wrote:
"One can derive the length contraction formula from the time dilation
formula so why even bring in the redundant concept of length
contraction, they're the same thing. However, you said time dilation is
also not real (because both frames see time dilation in the other) but
using the criteria of the twin paradox, one can measure an age
difference consistent with time dilation that remains once the relative
velocity ends."
Jorrie replied:
"This is the crux of the matter. There is coordinate dependent time
dilation and then there is proper time dilation. There is coordinate
dependent Lorentz contraction, but there is no proper Lorentz
contraction. In this sense, the two are different, not 'the same thing'.
However, Lorentz contraction is real in the sense that when you make a
real measurement of the length of a passing spaceship, you get a
Lorentz contracted value. In this sense Lorentz contraction is 'real',
but then, 'real' means different things to different people  a debate
that I have no intention of entering.
As long as you have standard synchrony, i.e. Einstein synchronization of
clocks, you have Lorentz contraction and the limiting oneway speed is
c. The limiting 2way speed is c, irrespective of the clock sync
convention used."
To which Ralfcis replied:
"I'm lost here, totally blank. What you said may be beyond my ability to understand. I can't even formulate a question."
I
think the problem is that Ralf has developed his own (evidently flawed)
verbal framework for understanding relativity and he attempts to fit
every bit of related information acquired into this framework. Some bits
just do not fit into his framework and he gets stumped by it, or worse,
he contorts the information to fit in.
Since
I have spend thousands of words in an attempt to turn Ralf's framework
into the mainstream direction and evidently failed, I recommended that
he opens other threads to get more input and direction. This has apparently also failed to give any satisfaction.
Lately, Ralf has come around to a more conventional framework, but
since I have mostly completed this Blog entry, I will post it anyway. It
may be useful for other members of this forum  readers that have long
since unsubscribed from that lengthy prior discussion.
Let us
restrict the discussion to special relativity (SR) only, since without a
solid SR foundation, it is useless to discuss general relativity (GR).
Also, let us leave quantum physics and philosophical considerations out
of it. SR does not answer "why" or "how" questions, only "what"
(observable) questions.
The main unanswered questions seem to be about reciprocal time
dilation, relative elapsed time, Lorentz contraction and the isotropy of
the speed of light. In the latter, it is more specifically the oneway
speed of light being the same in every inertial frame that trips up many
a student of relativity.
a) The oneway speed of light
It seems appropriate to get this one out of the way first. There is
no magic about the oneway speed of light being the same in all inertial
frames. Einstein has simply declared it to be so as a convention^{[1]}
and then based the whole of his amazing theory of relativity on this
assumption. The fact that relativity works flawlessly within its
applicability, justifies Einstein's assumption without any shadow of a
doubt.
Importantly, this also seems to be what nature prefers. Physics would have been very 'ugly' if any other clock sync scheme was
used, e.g. the GPS system would have been all but useless, because the
speed of light would have been different in different directions.
This assumption determines the method for synchronizing clocks
throughout every inertial frame. In its simplest form, if we know the
distance d between two clocks that are permanently at rest
relative to each other and we send a time stamped signal, the receiving
clock simply adds a propagation delay Δt = d/c to the time stamp and sets its time accordingly.
Because this is such a simple and universal scheme, many present day
scientists simply accept that the oneway speed of light is c in
every inertial frame and never give it a second thought. This sometimes
leads to heated debate between scientists and "the rest", who are
attempting to understand the reasoning behind the principle.
b) Reciprocal time dilation
The fact that when A and B are in uniform relative (inertial) motion,
A observes B's clock to 'lose time' and B observes A's clock to 'lose
time' is directly related to the above convention about the oneway
speed of light. It comes about due to the way clocks are synchronized,
using the convention.
It does not determine who ages slower or faster, but just how the one
observer observes the others clock. Time dilation can be viewed as
simply a change in 'spacetime observation angle'  each
views the others time vector at an angle in spacetime, which depends on
their relative speed, which is reciprocal.
This does not make time dilation "an illusion" or "not real". When
proper scientific measurements of time are made between two inertial
observers in relative motion, the results are as real as any measurement
can be; but, it is reciprocal and hence coordinate dependent and not
absolute.
c) Relative elapsed time
This is where "aging slower or faster" comes in. Every inertial
object follows a trajectory through spacetime, called a 'worldline'.
When two inertial objects in free space are at rest relative to each
other, they follow equivalent (not necessarily identical) worldlines, so
they age identically.
If they are not at rest relative to each other, they are following
nonequivalent worldlines and they may age differently. They can
synchronize their clocks when they move past each other and after that
the one that experience the largest change of inertial frame will age
less. This is why the traditional "awaytwin" always ends up younger
than the "hometwin".^{[2]} If neither of them experiences any change of inertial frame, we cannot tell who ages more or less than the other.
It just so happens that in mostly quasiinertial cases (where the
acceleration phase is short relative to the inertial phases), the aging
difference is approximately the same as that given by the SR time
dilation formula. This fact has led to a lot of confusion in the popular
literature. There is no difference in elapsed times unless there has
been a difference in the change of inertial frames, which requires
acceleration of at least one of the two clocks.^{[3]}
d) Reciprocal Lorentz contraction
Like reciprocal time dilation, reciprocal Lorentz contraction is also
caused by the Einstein clock synchronization convention. If A and B are
in relative motion, each observes the others lengths to be contracted
in the direction of relative motion. When they are brought to relative
rest again, the reciprocal length contraction disappears  this is
unlike the case of relative aging, which is a lasting effect.
Like relative time dilation, Lorentz contraction can be viewed as
simply a change in 'spacetime observation angle' (each views the others
length at an angle in spacetime), which depends on relative speed. The
formula for Lorentz contraction is essentially the same as the time
dilation formula. Both these effects are contained in the Lorentz
transformations^{[4]} as special cases.
Twin "paradox" Variant
Using the above information, the classical 'twin paradox' can be twisted to a slightly more challenging one. Alice sets off from Earth on
her long fast journey, with Bob staying at home. Some years after Alice
have left Earth, she and Bob each opens a secret envelope, where they
for the first time get instructions on how to complete the mission.
The instructions could be either (i) for Alice to return to Earth and for Bob to stay put; or (ii) for Alice to coast on and for Bob to leave Earth fast enough so that he can catch up and join Alice in space.
Without doing any math, firstly, who would have aged less in each of
the two cases? Secondly, just before Alice and Bob opened their
respective envelopes, who would you say have aged less up to that point?
@ralfcis: Before attempting to answer these questions, first make
sure that you understand the discussion leading up to it. If not, keep
on asking questions, but please stick to the baseline given. I do not
want to waste time by analyzing some or other fancy relativistic
scenario that you can dream up  such time can be more effectively spent
by discussing the stated principles better.
Jorrie
[1] Einstein's 2005 paper "On the Electrodynamics of Moving Bodies", Section I, $1:
"We have so far defined only an "A time" and a
"B time." We have not defined a common "time" for A and B, for
the latter cannot be defined at all unless we establish by
definition that the "time" required by light to travel from
A to B equals the "time" it requires to travel from B to A."
The twoway speed of light obviously does not suffer from this clock
sync convention problem, because we need only one good clock and a
definition of distance to measure the round trip (average) speed of
light. The modern definition of distance is however dependent on the
speed of light in vacuum, so there is some degree of conventionality in
the modern value of the twoway speed as well. There is no doubt that is
the same in all directions, though.
[2] The rigorously correct statement would require them to meet again
in order to unequivocally establish who has aged less. It is however a
sufficient requirement that the two twins must just be at rest relative
to each other again in order to establish beyond reasonable doubt who
has aged less. For example, they can each observe an event that is
equidistant from the two of them and report their respective clock
readings at the time of observing the event.
[3] It must be clearly stated that acceleration per se does
not cause time dilation. Acceleration is required to change the inertial
frame and the time spent in the new inertial frame determines the
amount of elapsed time difference accrued  in other words, acceleration
is the cause of the different spacetime paths.
Instead of acceleration, their could be a "time handoff" by the
awaytwin to a third inertial observer, flying in the opposite
direction, who then completes the home leg. The calculated elapsed time
difference is still valid, because there is the same difference in
spacetime paths.
[4] The Lorentz transformations are more general than just time
dilation and length contraction. It gives the equations for Lorentz
covariance, which in simpler terms means converting time and space
intervals from one inertial frame to another in a consistent way. Space
intervals and time intervals between two specific events together form
the spacetime interval between the two events, which is the same for all
inertial frames. When you have more space interval between events, you
have less time interval and viceversa, but in a squared (not linear)
fashion.
PS. If there are any other relativity issues that you want raised to this 'Bloglevel', please p/mail me with the request.
