Quote by Austin0
Well you still have that v in the rhs of your equation. What does it represent??
stevendaryl;4215692 1) said:
It doesn't represent anything--2) it's just a number that is used to describe3) the relationship between two coordinate systems, and also happens to be 4) the speed of Achilles in one of the coordinate systems. .
Well i asked a perfectly cogent and relevant question. 1)you deny it is a valid question . then 2) you immediately contradict yourself and present two different possible reasonable answers 3),and 4) but both your answers seem questionable.
Working from the information defined by pervect it is not possible to derive a velocity for the Achilles frame in the Zeno frame as far as i can see,
Likewise it is not possible to define a velocity for Achilles himself in the Zeno frame.
So again I ask what is the velocity referring to that could be a valid part of the Zeno metric?.
And how do you arrive at it??
stevendaryl said:
I think that you are having trouble grasping the idea of an arbitrary, noninertial, curvilinear coordinate system (as opposed to an inertial, Cartesian coordinate system).
I have no trouble with the idea of an arbitrary non-linear coordinate system.
In fact, back at my second post I brought up this possibility
Quote by Austin0
Having done so it appears that it was not explicitly stated that the intervals were equivalent. And in fact they would not correspond to time on any normal clock with a constant rate.
So are you talking about an arbitrary clock that speeds up over time ??
DaleSpam denied this back then but it appears that that is exactly the case here,,,, DO you now agree??
stevendaryl said:
An inertial Cartesian coordinate system is set up in some standard way (for example, using light signals to measure distances and using a standard clock to measure time, and using the Einstein synchronization convention for synchronizing distant clocks). But you can use any convention you like to set up a coordinate system. Let be an inertial Cartesian coordinate system for some region of spacetime.
Let (x,t) be an inertial Cartesian coordinate system for some region R of spacetime. Let T'(x,t), X'(x,t), X(x',t'), T(x',t') be any four differentiable functions such that for any pair (x,t) describing a point in R,
X(X'(x,t), T'(x,t)) = x
T(X'(x,t), T'(x,t)) = t
Then within region R, you can use coordinates x', t' defined by
x' = X'(x,t)
t' = T'(x,t)
As far as GR is concerned, (x',t') can be used just as well as (x,t).
this appears to me to be a generalization of the concept of transformation between relative frames. Is this correct??
if this is so i don't see the relevance.
This particular case is not about setting up a system from the ground but working within the constraints of defined relationships and partial definitions without a completely defined system for Zeno .We can assume a standard inertial system for Achilles but we have only some data from observations in Zeno frame to go by.
DaleSpam said:
Consider the inertial frame where Achilles is at rest. In this frame the turtle's worldline is given by (t,100-vt) where v is the relative velocity between Achilles and the turtle. So in this frame Achilles is a distance d=100-vt behind the turtle. The definition of Zeno time, n, given is d=100/2^n. Substituting in and simplifying we get the following transform between the inertial frame and Zeno coordinates:
n=log_2 \left( \frac{100}{100-vt} \right)
Don't you agree that to assert an equivalence between coordinates or values between two frames in relative motion you need to transform the values from one frame to the other.
If in fact you do not
already have the correct transform functions, the T,X,T' and X' in your generalization you cannot simply assume the equivalence between some values in both frames and derive a valid transform from that . There has to be some relevant basis for the equivalence from first principles to justify such an identity and substitution.
Wouldn't you agree??
SO in this case we are given : d=100-vt in the A frame and d=100/2^n in the Z frame.
Is the 100 in the A frame equivalent to the 100 in the Z frame?
Assuming that at A time t =0 Achilles is at x=0 and the tortoise is at x=100 and at Z time n= 0 Achilles is at x'=0 and the tortoise is at x'=100. isn't it axiomatic that if these events are simultaneous in the A frame that they cannot be simultaneous in the Z frame?? It follows that the distances , the spatial intervals in the two frames cannot be congruent also Yes??
So if the intervals dx,t=0 and dx', t'=0 are not equivalent, even initially when you can assign coordinates to the positions in the Z frame, how do you justify the equivalence 100-vt=100/2^n over time when the systems are not only in relative motion but one of them is non-linear??
Where you do not have a basis to even determine coordinate positions in the Z frame for A and the tortoise or relate times in that frame to the A frame??
It appears to me that to make this assumption of equivalence is unfounded and circular. I.e.,,to
determine if these are equivalent
requires a
valid transformation so to use them to derive a transformation then
makes them equivalent circularly.
stevendaryl said:
In the case DaleSpam is talking about,
X'(x,t) = x
T'(x,t) = log_2(\dfrac{100}{100-vt})
stevendaryl said:
You are asking what the physical interpretation of the noninertial coordinates are--coordinates don't HAVE a physical interpretation, or they don't need to, anyway. They're just a way of identifying points in spacetime. They're just names, but names chosen in a "smooth" way, so that you know that nearby points will have names that are close together as numbers. .
in another thread you stated that gravitational time dilation could be eliminated by a coordinate choice remember??
I asked you if you were talking about an arbitrary scaling of clock periodicity and you agreed, correct?
So then we are talking about a physical interpretation of clock rates. AN artificial mechanical adjustment to the workings of the mechanism. What could be clearer than that??
In this case this means a mechanistic device that exponentially increases the rate at which the hands spin or the LED increments or whatever means that is used to actually indicate the measure of time,,,, CORRECT?
Such artificial scaling is in fact used in the GPS system right?? Those clocks physically increment at a different rate yes??