Re: TomTom's stupidity (re: was always TomTom's stupidity)
- From: stevendaryl3016@xxxxxxxxx (Daryl McCullough)
- Date: 4 Oct 2005 20:37:27 -0700
schoenfeld1@xxxxxxxxx says...
>Daryl McCullough wrote:
>> schoenfeld1@xxxxxxxxx says...
>>
>> >ds^2 = d(ict)^2 + dx^2 + dy^2 + dz^2 is a *EUCLIDEAN* metric.
>>
>> No, it's not. There is no "i" in Euclidean geometry.
>>
>> The presence of that "i" makes the metric non-positive-definite.
>> For sufficiently large values of dt, ds^2 is negative.
>
>The time coordinate is imaginary
There are no imaginary distances in Euclidean geometry.
>and the metric is a EUCLIDEAN.
This is a silly argument over terminology. The important
distinction is whether ds^2 is positive definite or not.
When x,y,z, and t are all real, then ds^2 can be positive,
negative, or 0. That's very different from Euclidean geometry.
>> >you cannot, with any system of equations, formulate a theory
>> >over Minkowski spacetime which gives the same predictions as GR.
>>
>> Nobody said otherwise. You are confusing two different things:
>> (1) Is it possible to use SR with noninertial coordinates? Yes,
>> it certainly is possible.
>
>You cannot do this without additional postulate(s).
Maybe you can't, but I certainly can.
>What justification have you, from the SR postulates,
>to assume that accelerating frames preserve their proper
>lengths?
Nothing physical depends on a choice of coordinates. So if
you have a description of physics that works in one set of
coordinates, then you can figure out the description in
any other set of coordinates. It's just calculus.
Do you think that any additional physical assumptions are
needed to use spherical coordinates to do Newtonian physics?
It's just calculus.
>Consider a 1 unit ruler initially at rest in frame A.
>At time t = 0 the ruler is accelerated instanteously
>to velocity v = 0.866c.
That's ambiguous. Do you mean that, as measured in frame
A, both ends accelerate at the same time? (Perhaps by attaching
rockets to both ends, and setting them off at a designated time)
Or do you mean that at t=0, one end of the ruler accelerates
instantaneously, and the other end moves when the shock wave
hits it? In both cases, the proper length of the ruler will
change. In the first case, the proper length will increase,
and in the second case, it will either increase or decrease,
depending on whether the ruler is accelerated by pushing one
end or by pulling the other end.
>Scenario 1:
>>From stationary frame A, if you let the left end change velocity at
>t = 0 then you would expect that the right end to accelerate at
>t = 0 as well.
Why would you expect that? Think about it: Suppose you push on
one end of a spring. Does the other end move instantly? No, the
spring compresses. The same thing will happen if you push on one
end of a rigid ruler; it will compress.
>At time t > 0 the ruler would end up with the same
>length in A's frame.
That's true, assuming that you coordinate the acceleration
of the two ends. If you pull on one end, and push on the
other, then it is possible to maintain a constant length,
as measured in frame A.
>However, in the rulers frame, it's length at
>time t > 0 would now be doubled (length contraction) and this
>would have a physical effect on the ruler (the ruler would
>break for example).
Yes, if you accelerate it rapidly enough, the ruler will break.
>How do the postulates SR prevent this scenario?
SR doesn't prevent you from breaking rulers. If you push too hard
on one end, the ruler be crushed. If you pull too hard on the
other end, the ruler will be stretched or broken. SR places
absolute limits on the rigidity of any material object.
What does this have to do with using noninertial rest frames?
>Scenario 2:
>If you delay* the acceleration of one of the ends (say the right end)
>such that it accelerates at t > 0 in A's frame then you can arrange it
>such that the proper lengths of the ruler remain preserved (i.e. in
>rulers frame) but change in the stationary frame (ruler length would
>halve in A's frame).
>
>How do the postulates of SR dictate this scenario? (this is the
>scenario you subtely invoke)
This is the case that noninertial coordinates are most useful.
If every point along the ruler undergoes constant proper acceleration,
then the situation can be described in terms of coordinates X and T
related to the coordinates x and t of frame A as follows:
x = X cosh(gT/c)
t = X/c sinh(gT/c)
In terms of the coordinates X and T, the ruler will maintain a constant
length as it accelerates.
>> (2) Is the result a relativistic theory
>> of gravity? No, it's not. It's the theory of an accelerated observer
>> in *flat* spacetime, while gravity is (in Einstein's GR) the
>> effect of *curved* spacetime.
>
>Sure you can construct a mathematical model (with implicit assumption)
>of how accelerated observers work in SR, but this model ends up have
>little relation to the physics of accelerated observers (assuming
>strong EP is true).
The physics is the *same* whether you are using inertial coordinates
or noninertial coordinates. Physical effects are not affected by a
change of coordinates, only how those effects are *described*. The
EP has *no* bearing on this issue. You are misusing it.
>Also, I suspect the SR postulates are unable to rigorously define how
>accelerated observers should work anyway (read my two scenarios and
>justify why one is false and why one is true from SR postulates).
That's completely wrong. SR is all that you need to be able to
figure out what things look like from the point of view of an
accelerated observer.
>Einstein probably realized this
It's false. So Einstein didn't "realize" it.
>and invented strong EP which leads to predictions your SR
>accelerated frames can't possible make anyway.
You completely misunderstand the EP. You don't need the EP
to do any problem in flat spacetime. It doesn't matter whether
observers are accelerating or not. Everything about the physics
of flat spacetime is described by SR (together with laws for
how non-gravitational forces behave). What the EP allows you
to do is to make a prediction about how clocks behave in
a gravitational field. It works like this:
1. Using Special Relativity alone, you can predict that clocks
aboard an accelerated rocket will undergo position-dependent
time dilation; clocks in the front of the rocket will run
faster than clocks in the rear of the rocket.
2. Now, assume that locally gravity is indistinguishable from
acceleration. Then it follows that clocks at rest in a gravitational
field will experience the same position-dependent time dilation:
clocks higher in the gravitational field will run faster than
clocks lower in the gravitational field.
The equivalence principle allows you to use SR to solve (approximately)
some simple problems involving gravity. That's why Einstein could predict
gravitational time dilation several years before he actually developed
General Relativity. Gravitational time dilation follows from SR together
with the equivalence principle.
--
Daryl McCullough
Ithaca, NY
.
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