Re: Are SR effects real or not? Simplified case.

On Jul 18, 9:23 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
PD says...



On Jul 17, 10:53=A0am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
"The height of Daryl McCullough as measured in a frame in
which Daryl McCullough is at rest" is perfectly well-defined
invariant quantity.

Well, I see what you're saying, I really do. But doesn't it seem a
little empty to you that if you have an object being looked at in
frames F and F', to say while standing in frame F' that "the length of
the object as measured in F is an invariant quantity in this frame"?

A rule of thumb for whether a quantity is an invariant or not is
whether it can be expressed in manifestly covariant form. What I
mean by that is that it should be expressed in terms of
scalars, 4-vectors and tensors and any indices should occur
twice: once lowered and once raised (implying the summation
convention). (This is not foolproof because something can
appear to be a 4-vector or tensor when it really is not.
For example, if A^u is a vector field, then @/@x^v A^u
seems like a tensor of the form T^u_v, but it isn't really,
unless you use the covariant derivative instead of the
partial derivative).

So the energy of particle X is not an invariant, because
energy is the time-component of a 4-vector, and there is
no corresponding lowered index. But the energy of particle
X as measured in the frame of observer Y *is* an invariant,
because it can be written in the manifestly covariant form:

P^u U_u

where P^u is the 4-momentum of particle X, and U^u is
the 4-velocity of the observer Y.

This form is obviously an invariant.

Yes, I see your point here.


Other sorts of measurements can be reexpressed in terms
of invariants, but as my example with "proper length"
shows, it's sometimes complicated.



It's a little complicated to come up with a covariant
expression for proper length, but here's an attempt.
It's necessary to assume that it is moving smoothly
enough (if the two ends are moving erratically, proper
length is not really well-defined).

Let e1 be some event taking place at one end of a stick.
Let X1^u be the position of this event. Let U1^u be the
4-velocity of that end of the stick at that moment.

Let e2 be some event taking place at the other end of
the stick. Let X2^u be the position of thise event, and
let U2^u be the 4-velocity of the other end at that
moment.

To compute the proper length, we can't just compute
the invariant interval between e1 and e2, because
e2 may not be simultaneous with e1 in the frame of
the stick. Instead, we need to find some event e3
such that (1) e3 takes place at the same end of
the stick as e2, and (2) e3 is simultaneous with
e1 in the frame of the stick.

OK, so what this does is choose to measure the length of the stick not
by the events in this frame, but by the events in a different frame.

I don't know what you mean. An event doesn't belong to a frame.
An event is a point in spacetime. Its *coordinates* depend on
the frame, but the event isn't in a frame.

You are simply choosing events by which you are measuring the length
in the stick's frame.

I'm just carrying out the definition of "proper length" as
"the length of the object as measured in a frame in which
the object is at rest".

Yes.


In basic physical terms, this is the moral equivalent of what Henri
Wilson has suggested. He says it is foolish to measure the length of a
moving object, using the same prescription you use for a stationary
object. He says you have to stop the object and then measure its
length.
What you are doing is the mathematical equivalent of stopping the
object and then measuring it.

No, it's the mathematical equivalent of accelerating *myself*
so that I'm at rest relative to the object, and then measuring
the object.

Yes, I agree again. We agree that proper length is only *measurable*
in the frame in which the object is at rest.

To stop the object means to exert forces on the
object, which could possibly change its length.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Are SR effects real or not? Simplified case.
    ... teaches that length is relative to a frame of reference." ... One way of thinking about relativity is in terms of coordinate-dependent ... "The proper length of the object" which is the same in any ... length" as the name for the invariant quantity, ...
    (sci.physics.relativity)
  • Re: Are SR effects real or not? Simplified case.
    ... teaches that length is relative to a frame of reference." ... One way of thinking about relativity is in terms of coordinate-dependent ... "The proper length of the object" which is the same in any ... length" as the name for the invariant quantity, ...
    (sci.physics.relativity)
  • Re: Are SR effects real or not? Simplified case.
    ... the object as measured in F is an invariant quantity in this frame"? ... expression for proper length, but here's an attempt. ... object and then measuring it. ...
    (sci.physics.relativity)
  • Re: So... Lerentz Contractions are *physical* not observered?
    ... the invariant length of an ... object is its length in its own rest frame). ... but does that mean that Lorentz contraction is somehow not ...
    (sci.physics.relativity)
  • Re: Simple four velocity question
    ... frame A, then that measurement *in frame A* is invariant, ... This is why I referred to conservation of mechanical energy. ... conservation law is a *projection* onto that reference frame. ...
    (sci.physics.relativity)
Quantcast