Murat Ozer
Oct12-06, 04:14 AM
Consider a reference system, call it S', uniformly rotating at
angular velocity w with respect to a stationary system, call it S.
Suppose that the coordinates of S' are related to those of S by the
following "Lorentz transformations" in cylindrical coordinates:
dt = gamma(dt' +wr^2/c^2 dt'), (1)
dphi = gamma(dphi' + wdt'), dr = dr', dz = dz',
where gamma = 1/Sqrt(1 - w^2r^2/c^2).
Under these Lorentz transformations the infinitesimal distance squared
ds^2 in S
ds^2 = c^2dt^2 - dr^2 - r^2dphi^2 - dz^2, (2)
is transformed into
ds'^2 = c^2dt'^2 - dr'^2 - r^2dphi'^2 - dz'^2, (3)
which is the infinitesimal distance squared in S'. Now, S is an
inertial reference frame, and ds' has the form required of an
inertial frame.
Even though there is an acceleration, the centripetal acceleration, the
angular acceleration dw/dt = 0 and the net torque acting on the system
is zero. So, is this system, S', an inertial system? To me the answer
seems to be affirmative. After all, a friction free rotating system
would continue its state of rotation indefinitely until a torque acts
on it. I have not seen any extension of Newton's laws of motion to
rotating systems.
For sure, somebody must have treated this extension.
My second question is the following: How would a person in a uniformly
rotating system determine the type of transformations relating his/her
coordintas to those of S? This question seems very relevant to me,
because instead of the Lorentz transformations above, the two systems
can also be related through the relativistic Galilean transformations,
given by
dt = dt' (4)
dphi = dphi' + wdt',
dr = dr', dz = dz'
under which the metric ds is transformed into
ds'^2 = (1 - w^2r^2/c^2)c^2dt'^2 - dr'^2 -
r^2dphi'^2 - 2wr^2dt'dphi' - dz'^2. (5)
So, how would a person determine whether the metric in his/her frame is
that given by (3) or (5)?
Comments please...
Regards,
Murat Ozer
angular velocity w with respect to a stationary system, call it S.
Suppose that the coordinates of S' are related to those of S by the
following "Lorentz transformations" in cylindrical coordinates:
dt = gamma(dt' +wr^2/c^2 dt'), (1)
dphi = gamma(dphi' + wdt'), dr = dr', dz = dz',
where gamma = 1/Sqrt(1 - w^2r^2/c^2).
Under these Lorentz transformations the infinitesimal distance squared
ds^2 in S
ds^2 = c^2dt^2 - dr^2 - r^2dphi^2 - dz^2, (2)
is transformed into
ds'^2 = c^2dt'^2 - dr'^2 - r^2dphi'^2 - dz'^2, (3)
which is the infinitesimal distance squared in S'. Now, S is an
inertial reference frame, and ds' has the form required of an
inertial frame.
Even though there is an acceleration, the centripetal acceleration, the
angular acceleration dw/dt = 0 and the net torque acting on the system
is zero. So, is this system, S', an inertial system? To me the answer
seems to be affirmative. After all, a friction free rotating system
would continue its state of rotation indefinitely until a torque acts
on it. I have not seen any extension of Newton's laws of motion to
rotating systems.
For sure, somebody must have treated this extension.
My second question is the following: How would a person in a uniformly
rotating system determine the type of transformations relating his/her
coordintas to those of S? This question seems very relevant to me,
because instead of the Lorentz transformations above, the two systems
can also be related through the relativistic Galilean transformations,
given by
dt = dt' (4)
dphi = dphi' + wdt',
dr = dr', dz = dz'
under which the metric ds is transformed into
ds'^2 = (1 - w^2r^2/c^2)c^2dt'^2 - dr'^2 -
r^2dphi'^2 - 2wr^2dt'dphi' - dz'^2. (5)
So, how would a person determine whether the metric in his/her frame is
that given by (3) or (5)?
Comments please...
Regards,
Murat Ozer