Murat
Oct12-06, 04:11 AM
I would like to point out,once again,that the definition of proper
time in General Relativity needs to be modified/corrected. Even
though the present definition dTau = ds/c for timelike geodesics
(Tau = proper time, ds = infinitesimal interval), gives correct
answers for most cases,there is at least one case for which it
gives the wrong answer. Let me elaborate:
Let us consider a stationary frame, frame S, whose metric is given by
ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2, (1)
and another frame,frame S',moving in the x direction with respect to
S with velocity v whose coordinates are related to those of S by the
Lorentz Transformations,with gamma = 1/Sqrt(1- v^2/c^2):
dt = gamma(dt'+ v/c^2dx'), (2)
and
dx = gamma(dx'+ vdt'), dy = dy', dz = dz'.
The metric in S' is
ds'^2 = c^2dt'^2 -dx'^2 - dy'^2 - dz'^2 (3)
By factoring out c^2dt^2 eq.(1) can be written as,
ds^2 = (1-v^2/c^2)c^2dt^2. (4)
Now, proper time dTau is the time kept by a clock at rest. Hence ,
dt' =dTau, when dx' = dy' = dz' = 0, (5)
which yields
ds'^2 = c^2 dTau^2. (6)
The invariance of the infinitesimal intervals, ds = ds', gives
ds/c = ds'/c = dTau = Sqrt(1- v^2/c^2) dt. (7)
In arriving at Eq. (7) we did not refer to the Lorentz
taransformations.
>From eq. (2a), using the fact that dt' = dTau when dx' = 0=dy'=dz'
we get
dt = gamma dTau, or dTau = Sqrt(1-v^2/c^2)dt. (8)
Obviously, the expressions (7) and (8) for dTau obtained from
the invariant intervals ds and ds', and from the Lorentz
transformations
are identical, as they must be.
One may then conclude from this exercise that
ds = ds' = cdTau. (9)
As for General Relativity, the Special Relativistic definition (9)
of proper time is carried over without paying attention that eq. (9)
may not always be correct. It gives the correct answer for most cases
As I will show, the definition (9) leads to the wrong expression for
dTau for a rotating turntable:
Let us consider again two systems (S) and (S') with common origins and
z axes such that (S')is rotating about the z axis with constant angular
velocity w.In cylindrical coordinates the "relativistic"
transformations are given by
dt = dt' (10)
dphi = dphi' + wdt = dphi' + wdt'
dr = dr', dz = dz'.
The metric
ds^2 = c^2dt^2 - dr^2 - r^2dphi^2 - dz^2 in (S) (11)
is transformed into
ds'^2 = (1 - w^2r^2/c^2)c^2dt'^2 - 2wr^2dphi'dt' - dr'^2
- r^2dphi'^2 -dz'^2 (in S'). (12)
Let us obtain dTau from (11) and (12) by following the same reasoning
above: Again, by factoring out c^2dt^2 eq. (11) gives
ds = Sqrt(1-w2 r^2/c^2)cdt. (13)
Setting dt' = dTau when dr' = dphi' =dz' =0 in eq. (12) yields
ds' = Sqrt(1-w^2r^2/c^2) cdTau. (14)
And ds = ds' gives
dTau = dt. (15)
On the other hand, the transformations (10), upon using dt' = dTau
when dr' = dphi' =dz' =0 give
dt = dTau. (16)
The expressions (15) and (16) obtained in different ways are identical.
However, if one uses the definition ds = cdTau, one gets from eq. (12)
dTau = Sqrt(1-w^2r^2/c^2)dt' = Sqrt(1-w^2 r^2/c^2)dt. (17)
This expression does not agree with
dTau = dt (18)
obtained directly from the transformation equation dt = dt'.
The reason for this conflict is the definition ds = cdTau; it gives
the correct expression for dTau only if the metric coefficient
g_t't' = 1. In the rotating turntable example above
g_t't' = (1 - w^2r^2/c^2) and hence ds = cdTau fails.
I do not see any logical inconsistency in this treatment. Your remarks
,especially from the general relativity experts, will be most welcomed.
Regards,
Murat Ozer
time in General Relativity needs to be modified/corrected. Even
though the present definition dTau = ds/c for timelike geodesics
(Tau = proper time, ds = infinitesimal interval), gives correct
answers for most cases,there is at least one case for which it
gives the wrong answer. Let me elaborate:
Let us consider a stationary frame, frame S, whose metric is given by
ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2, (1)
and another frame,frame S',moving in the x direction with respect to
S with velocity v whose coordinates are related to those of S by the
Lorentz Transformations,with gamma = 1/Sqrt(1- v^2/c^2):
dt = gamma(dt'+ v/c^2dx'), (2)
and
dx = gamma(dx'+ vdt'), dy = dy', dz = dz'.
The metric in S' is
ds'^2 = c^2dt'^2 -dx'^2 - dy'^2 - dz'^2 (3)
By factoring out c^2dt^2 eq.(1) can be written as,
ds^2 = (1-v^2/c^2)c^2dt^2. (4)
Now, proper time dTau is the time kept by a clock at rest. Hence ,
dt' =dTau, when dx' = dy' = dz' = 0, (5)
which yields
ds'^2 = c^2 dTau^2. (6)
The invariance of the infinitesimal intervals, ds = ds', gives
ds/c = ds'/c = dTau = Sqrt(1- v^2/c^2) dt. (7)
In arriving at Eq. (7) we did not refer to the Lorentz
taransformations.
>From eq. (2a), using the fact that dt' = dTau when dx' = 0=dy'=dz'
we get
dt = gamma dTau, or dTau = Sqrt(1-v^2/c^2)dt. (8)
Obviously, the expressions (7) and (8) for dTau obtained from
the invariant intervals ds and ds', and from the Lorentz
transformations
are identical, as they must be.
One may then conclude from this exercise that
ds = ds' = cdTau. (9)
As for General Relativity, the Special Relativistic definition (9)
of proper time is carried over without paying attention that eq. (9)
may not always be correct. It gives the correct answer for most cases
As I will show, the definition (9) leads to the wrong expression for
dTau for a rotating turntable:
Let us consider again two systems (S) and (S') with common origins and
z axes such that (S')is rotating about the z axis with constant angular
velocity w.In cylindrical coordinates the "relativistic"
transformations are given by
dt = dt' (10)
dphi = dphi' + wdt = dphi' + wdt'
dr = dr', dz = dz'.
The metric
ds^2 = c^2dt^2 - dr^2 - r^2dphi^2 - dz^2 in (S) (11)
is transformed into
ds'^2 = (1 - w^2r^2/c^2)c^2dt'^2 - 2wr^2dphi'dt' - dr'^2
- r^2dphi'^2 -dz'^2 (in S'). (12)
Let us obtain dTau from (11) and (12) by following the same reasoning
above: Again, by factoring out c^2dt^2 eq. (11) gives
ds = Sqrt(1-w2 r^2/c^2)cdt. (13)
Setting dt' = dTau when dr' = dphi' =dz' =0 in eq. (12) yields
ds' = Sqrt(1-w^2r^2/c^2) cdTau. (14)
And ds = ds' gives
dTau = dt. (15)
On the other hand, the transformations (10), upon using dt' = dTau
when dr' = dphi' =dz' =0 give
dt = dTau. (16)
The expressions (15) and (16) obtained in different ways are identical.
However, if one uses the definition ds = cdTau, one gets from eq. (12)
dTau = Sqrt(1-w^2r^2/c^2)dt' = Sqrt(1-w^2 r^2/c^2)dt. (17)
This expression does not agree with
dTau = dt (18)
obtained directly from the transformation equation dt = dt'.
The reason for this conflict is the definition ds = cdTau; it gives
the correct expression for dTau only if the metric coefficient
g_t't' = 1. In the rotating turntable example above
g_t't' = (1 - w^2r^2/c^2) and hence ds = cdTau fails.
I do not see any logical inconsistency in this treatment. Your remarks
,especially from the general relativity experts, will be most welcomed.
Regards,
Murat Ozer