Re: Ton of Bricks Paradox/Contradiction?

Alen says...

On Sep 20, 1:09 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:

No, it doesn't. The observers don't measure their
time in the moving frame, they only measure their own
proper time. How does the observers records show that
one observer accelerated at the same time as the other
observer? It *doesn't*. It shows that:

The left observer started accelerating when the left
clock showed time tau=0. (Call that event e_L)

The right observer started accelerating when the right
clock showed time tau=0. (Call that event e_R)

What record shows that they accelerated at the same time?
What record shows that e_L took place at the same time as
e_R?

The observers start off with synchronised clocks
in the stationary frame, so their own proper times
are identical with one another and with the stationary
frame time.

So e_L and e_R are synchronised events
in the stationary frame and in the observers frames,
which are all the same frame initially.

Yes, so as measured by the stationary frame, the
two clocks always remain synchronized.

But as measured by the moving frame, the
two events do *not* take place at the same time,
and so as measured by the moving frame, the
clocks are not synchronized ever.

Let's be more precise here. If, (1) as measured in inertial
frame F, two clocks are synchronized at time t1 (where t1 is the
coordinate time of frame F), and (2) at every time t > t1,
the velocity of one clock at time t is the same as the velocity
of the other clock at time t (where "velocity" and t are measured
in frame F), then for all times t > t1, the clocks are synchronized
according to frame F.

Yes. And, from the perspective of the frame
of the clocks,

There is no such thing as the "frame" of the clocks.
The clocks are *accelerating*. They are *changing*
(inertial) frames. They are moving from being at
rest in a frame in which their clocks are synchronized
to a frame in which their clocks are *not* synchronized.

Clocks don't "carry their synchronization with them"
if they are accelerating.

Yes, they do

No, they don't.

if they represent merely the same
experiment at different locations.

Different locations *and* different times.
In the moving frame, the two experiments
take place at different *times* as well
as different locations.

If I accelerate today, and you accelerate tomorrow,
then between those two times, you and I will be travelling
at different speeds.

An accelerated observer can be considered to
be changing from one inertial frame to another
at slightly different velocities. You can analyse
acceleration that way.

Not without actually doing calculus. Claims that are
true about an inertial coordinate system are *not*
true about slowly changing coordinate systems.

They don't *have* a frame. They are accelerating,
which means that they are *changing* frames.

They can be considered to occupy a frame
for an infinitesimal time dt, before transrerring
to another frame at slightly higher velocity.

Yes, they have one frame before accelerating, and
in that frame, the clocks are synchronized. They
have a *different* frame after accelerating, and
in that frame, the clocks are *not* synchronized.

So how can you say that the clocks are synchronized
"in their own frame"?

However, let's pick one particular coordinate
system that is useful for uniform acceleration.

Start with an inertial coordinate system with
coordinates x and t. Convert to new, accelerated
coordinates X and T via the transformation:

X = square-root(x^2 - c^2 t^2)
T = c/g arctanh(ct/x)

Then in terms of X and T, an observer at "rest"
at a constant value of X will "feel" an acceleration
of c^2/X. A clock at "rest" at a constant value of
X will tick at the rate

dTau/dT = gX/c^2

where g is a constant with dimensions of acceleration.

Note two facts about this accelerated coordinate system:
1. Clocks that are "higher" (larger value of X) tick
*faster*.

2. The acceleration "felt" by an object at rest (constant X)
*decreases* as X gets larger.

So from the point of view of someone at rest in this
coordinate system, it appears that there is a gravitational
force at work, with the gravity decreasing at higher altitudes.
Clocks that are at higher altitudes run faster than clocks at
lower altitudes.

Let's use this accelerated coordinate system to
describe our two observers. Let's assume that
the left observer feels acceleration g, and
let's assume that he remains at rest in the X,T
coordinate system. That means that in terms of
the X,T coordinate system, the left clock is
initially at X = c^2/g, and the right clock is
initially at X = c^2/g + D.

Here's the strange fact: If the right clock
is *also* accelerating at rate g, then the
right clock is *not* at rest in the X,T
coordinate system. Why not? As I said, the
acceleration "felt" at location X is given
by
a = c^2/X

So, if the right observer is to remain at rest
in the X,D coordinate system, then he must accelerate
at exactly

a = c^2/(c^2/g + D)
= g/(1 + gD/c^2)
which is less than g

So for the right observer to remain at rest in the X,T
coordinate system, he must accelerate *less* than g.
If his acceleration is greater than this, then his X
position will change. The distance between the two
observers, as measured in the X-T coordinate system
will *increase.

So, from the point of view of the "accelerated coordinate
system" of the left observer, the distance between the
two observers will increase, which is in agreement with
what is computed from the moving frame.

If the gravitational force felt by one observer
is different to that felt by the other,

I didn't say that. I said that *if* the two rockets
are to maintain a constant distance between each
other (as measured in their own, accelerated coordinate
system), *then* the "higher" rocket must accelerate
less than the "lower" rocket.

If they maintain *identical* accelerations, then
the distance between them *increases* (as measured
in their accelerating coordinate system).

despite the fact that the experiments are identical,
then the laws of physics are dependent on location.

Yes, in an *accelerated* coordinate system, physical
phenomena depend on location.

--
Daryl McCullough
Ithaca, NY

.

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