Re: Length contraction formula applied in presence of masses

Yes, I've been reading Taylor/Wheeler's Exploring black holes.
-Karan

p.s. The aim of this post is to keep the thread alive. People, I need answers.


karan@xxxxxxxxxx (karan) wrote in message news:<547d92f1.0504020605.6c312a39@xxxxxxxxxxxxxxxxxx>...
> How did you get the expression for dT and dR in terms of Schwarzchild
> coordinates without using Lorentz? The other two for dT' and dR' are
> easy to get.
>
> What I really want to know is that how can one justify the use of
> Lorentz formulae between one inertial(freefall) and another non
> inertial frame. You would agree that a particle at rest would not
> remain at rest in the shell frame. Could you explicitly show that in
> local spacetime it can be considered as inertial.
>
>
>
> Secondly,inside the event horizon r and t switch roles. But, isnt time
> still measured by the freefaller with his watch . And isnt distance
> still measured with a meter stick.And if the ratio of distance
> measured upon time taken is greater than 1 , we interpret that as
> velocity being greater than 1.(units such that c=1). Still pretty
> confused.
>
> Another, we cant 'go back in r' inside the event horizon.fine. We can
> go back in time? Time is space like..... r,phi,theta ? What sense does
> all this make , if any.
>
>
> I've made another post please see if you can reply to that too.
>
> Thanks,
> Karan
>
>
>
>
>
>
>
> George Jones <george_llew_jones@xxxxxxxxx> wrote in message news:<FN6dnSD8p9IN5tDfRVn-iA@xxxxxxx>...
> > karan wrote:
> > > What would be the relation between distances as measured by an
> > > observer freely falling towards a centre of attraction of some mass
> > > and another observer placed at a constant distance from the centre of
> > > attraction(say, on a shell)?
> > > In the book I was reading the relation used is that of length
> > > contraction. But the 'shell' frame is NOT an inertial frame of
> > > reference. Then how is the use of length contraction formula
> > > justified?
> >
> > By any chance, is the book "Exploring Black Holes" by Taylor and
> > Wheeler?
> >
> > Any observer (not just inertial one) in special or general relativity
> > can locally set up an orthonormal coordinate system. Usually an event on
> > the observer's worldline is chosen as the origin of these local
> > spacetime coordinates. By local, I mean that the coordinate system
> > covers only a small patch of spacetime.
> >
> > If 2 observers set up local orthonormal coordinate systems that overlap,
> > all of the results relating inertial coordinates are valid on the
> > overlap region - stuff like time, dilation, Lorentz contraction, and
> > Lorentz transformations.
> >
> > Now suppose we take the specific case of Schwarzschild spacetime with
> > one observer freely from rest at inifinity and and the other observer
> > standing on a spherical shell. Suppose further that the 2 observers are
> > coincident at event O, and that each observer sets up a local
> > orthonormal coordinate system with O as the origin.
> >
> > For simplicity, restrict attention to displacements in the r and t
> > Schwarzschild coordinates, so that the local coordinate systems are
> > 2-dimensional. Let (T, X) be the local coordinates set up by the freely
> > falling observer and (T', X') the shell observer's local coordinates.
> > So, when Schwarzschild coordinates (t, r) are taken into account, any
> > event in a small region of spacetime around O can be labeled by 3 sets
> > of coordinates.
> >
> > To emphasize that everthing is local, I'm going to use infinitesimal
> > coordinate difference from now on. Now conssider the coordinate
> > differences between event O and an event A such dT' = 0 and dX' =/= 0.
> > The shell observer thinks that event A happen simultaneously with O, but
> > a spatial distance dX' away from 0. By Lorentz contraction,
> > dX' = dX/gamma, where gamma = (1 - v^2)^(-1/2) and v is the relative
> > speed between the 2 observers.
> >
> > I think what you want though, is an explicit construction of the (T, X)
> > and (T', X') coordinate systems that demonstrates all this. I have
> > worked through such a construction, but I used spacetime 4-vectors to
> > derive the coordinates. If you're reading Taylor and Wheeler, this
> > method may be unfamiliar to you.
> >
> > I will, however, write down the results of my construction.
> >
> > dT = (1 - 2M/r)dt - (2M/r)^(1/2)dr
> > dX = -(2M/r)^(1/2)(1 - 2M/r)^(-1)dt + dr
> >
> > dT' = (1 - 2M/r)^(-1/2)dt
> > dX' = (1 - 2M/r)^(1/2)dr
> >
> > Exercise: Express the coordinates dT and dX in terms of dT', dX', and v,
> > where outside the event horizon v = -(2M/r)^(1/2), as you give below.
> > You should arrive at the standard Lorentz transformation between
> > coordinates.
> >
> > > Another problem,
> > > for a body in free fall starting at rest far from the entre of a black
> > > hole we have,dr_freefall/dt_freefall = -(2M/r)^.5
> > >
> > > dr_freefall = distances measured by the observer in free fall.
> > > M = mass of the black hole.
> > > r= Schwarzschild reduced coordinate.
> > >
> > >
> > > inside the event horizon of a black hole this velocity seems to exceed
> > > that of light. Whats wrong?
> >
> > Inside the event horizon, -r is a time coordinate. Why? Hint: look
> > carefully at the Schwarzschild metric. So, the correct interpretation
> > of the above formula is that inside the event horizon, the rate of
> > change of the coordinate time -r with respect to the free faller's
> > proper time is greater than 1.
> >
> > Regards,
> > George
.

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