Murat Ozer
Sep12-04, 02:23 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hello!\nI would like to bring to your attention an issue that has been\nbothering me for some time. It concerns the definition of proper time\nin a rotating system, like a turntable. In Part 1 of the following, I\nwill go over how proper time is defined in Special Relativity. In Part\n2, I will treat the relativistic rotating frame and point out that\nproper time for this case is not quite defined as in Part 1.\nPart 1: Special Relativity\nAs usual, consider two reference frames S and S\', S\' moving linearly\nin the x direction away from S with a velocity v. The infinitesimal\nLorentz transformations between S\' and S are:\n\ndt\'= (gamma)(dt - vdx), (1)\n\ndx\' = (gamma)(dx - vdt), (2)\n\ndy\' = dy, dz\' = dz, (3)\n\nwhere (gamma) = 1/sqrt(1-v^2) and c has been set to 1.\n\nThe invariant interval in S and S\' are\n\nds^2 = dt^2 - dx^2-dy^2 - dz^2, (4)\n\nds\'^2 = dt\'^2 - dx\'^2 -dy\'^2 - dz\'^2. (5)\n\nFactoring out dt^2 , equation (4) can be cast into\n\nds =sqrt(1 - v^2) dt (6)\n\nBy definition, the proper time dT of a clock at rest in S\' is found by\nsetting\ndx\' = dy\' = dz\' =0, and dt‘ = dT which results in\n\nds\' = dT (T = Tau). (7)\n\nComparing equations (6) and (7) gives, due to the invariance of the\ninterval,\n\ndT = sqrt(1 - v^2) dt. (8)\n\nIncidentially, the time dilation equation (8) can be obtained directly\nfrom the Lorentz transformations (1,(2), and (3). From (2),\n\n(gamma)dx = dx\' + (gamma)vdt.\n\nInserting this in (1) yields\n\ndt\'=sqrt(1- v^2)dt - vdx\'.\n\nNow setting dx\' =0 (along with dy\' = dz\' =0) and dt\' = dT gives (8).No\nsurprise…\n(End of part 1)\nPart2: Motion in a Rotating Relativistic Frame\nLet us now consider a frame at rest, S, and a second frame S\' rotating\nabout the z axis - a turntable - with a constant angular velocity w.\nIn cylindrical coordinates the interval in S is\n\nds^2 = dt^2 - dr^2 - r^2d(phi)^2 - dz^2. (9)\n\nThe coordinates of S\' are related to those of S by the\ntransformations\n\ndt\' = dt, (10)\n\nd(phi)\' = d(phi) - wdt, (11)\n\ndr\' = dr, dz\' = dz, (12)\n\nin terms of which the interval in S\' becomes\n\nds\'^2 = (1 - w^2r‘^2)dt‘^2 - dr‘^2 - r\'^2d(phi)\'^2\n- 2wr\'^2d(phi)‘dt\' - dz\'^2, (13)\n\nwhere S and S\' have the same common origin so that r = r\' and z = z\'.\n\nEquation (9) can be cast into\n\nds^2 = (1 - dr^2/dt^2 - r^2d(phi)^2/dt^2 - dz^2/dt^2) dt^2,\n= (1 - 0 - r^2w^2 - 0) dt^2 = (1 - w^2r^2) dt^2.\nOr,\n\nds = sqrt(1- w^2r^2)dt, (14)\n\nwhich corresponds to equation (6) above.\n\nNow, for a clock at rest on the turntable dr\' = d(phi)\' = dz\' =0 so\nthat\n\nds\'^2 = (1 - w^2r^2) dt\'^2, (15)\n\nand\n\nds\' = sqrt(1 - w^2r^2) dt\' (16)\n\nIn the literature the proper time of the clock at rest on the\nturntable is defined as\n\nds = ds\' = dT (17)\n\nso that (14) becomes\n\ndT = sqrt(1 - w^2r^2) dt (18)\n\nand (16) becomes\n\ndT =sqrt(1 - w^2r^2) dt\' (19)\n\nEquation (18) is the one quoted in the literature. Defining the proper\ntime this way, dt\' looses its meaning that for a clock at rest on the\nturntable, dt\' is its proper time! Indeed, equation (19) reveals that\nfor a clock at rest on the turntable dt\' is not the proper time. Both\ndT and dt‘ refer to the same clock at rest, and they have different\nmeanings. IS NOT THIS IN CONTRADICTION WITH THE DEFINITION OF PROPER\nTIME?\n\nOn the other hand, if we stick to the definition of proper time and\ndefine it for a clock at rest on the turntable as dt\' = dT, equation\n(16) becomes\n\nds\'= sqrt(1 - w^2r^2)dT , (20)\n\nwhich when compared with (14) gives\n\ndT = dt (21)\n\nIn other words, there is no time dilation! This is consistent with the\ntransformation equation (10) above, which would directly give dT = dt.\n\nJohn B. Kogut, in his book "Introduction to Relativity" treats motion\nin a rotating, relativistic frame in Section 7.2, p.100. Having\neliminated the d(phi)\'dt\' cross term in equation (13) above he writes\n\nds^2 = (c^2 - w^2r^2)dt\'^2-(dr^2 + c^2r^2d(phi)\'^2/(c^2-w^2r^2)+\ndz^2),\n\nhis equation (7.2.6). He then says\n\n" In order to interpret Eq.(7.2.6), we can compare it to the invariant\ninterval of an inertial frame of reference chosen to approximate the\ntransverse velocity wr locally. For example, taking a clock at rest in\nthe rotating frame,\n\nds^2 = (c^2 - w^2r^2)dt\'^2,\n\nand comparing it to a clock at rest in a locally inertial frame,\n\nds^2 = c^2dT^2,\nwe have time dilation again,\n\ndT=sqrt(1-w^2r^2/c^2)dt\'. "\n\nIs this a bad phrasing or a bad mistake?\nNote that even though the last expression in Kogut\'s statement has the\nsame form as my equation (19) above, the meaning of dT in the two\nequations are quite different.\nComments on Kogut\'s statement and my thinking above will be\nappreciated very much.\n\nRegards,\n\nMurat Ozer\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello!
I would like to bring to your attention an issue that has been
bothering me for some time. It concerns the definition of proper time
in a rotating system, like a turntable. In Part 1 of the following, I
will go over how proper time is defined in Special Relativity. In Part
2, I will treat the relativistic rotating frame and point out that
proper time for this case is not quite defined as in Part 1.
Part 1: Special Relativity
As usual, consider two reference frames S and S', S' moving linearly
in the x direction away from S with a velocity v. The infinitesimal
Lorentz transformations between S' and S are:
dt'= (\gamma)(dt -[/itex] vdx), (1)
dx' = (\gamma)(dx - vdt), (2)
dy' = dy, dz' = dz, (3)
where (\gamma) = 1/\sqrt(1-v^2) and c has been set to 1.
The invariant interval in S and S' are
ds^2 = dt^2 - dx^2-dy^2 - dz^2, (4)
ds'^2 = dt'^2 - dx'^2 -dy'^2 - dz'^2. (5)
Factoring out dt^2 , equation (4) can be cast into
ds =\sqrt(1 - v^2) dt (6)
By definition, the proper time dT of a clock at rest in S' is found by
setting
dx' = dy' = dz' =0, and dt‘ = dT which results in
ds' = dT (T = \Tau). (7)
Comparing equations (6) and (7) gives, due to the invariance of the
interval,
dT = \sqrt(1 - v^2) dt. (8)
Incidentially, the time dilation equation (8) can be obtained directly
from the Lorentz transformations (1,(2), and (3). From (2),
(\gamma)dx = dx' + (\gamma)vdt.
Inserting this in (1) yields
dt'=\sqrt(1- v^2)dt - vdx'.
Now setting dx' =0 (along with dy' = dz' =0) and dt' = dT gives (8).No
surprise…
(End of part 1)
Part2: Motion in a Rotating Relativistic Frame
Let us now consider a frame at rest, S, and a second frame S' rotating
about the z axis - a turntable - with a constant angular velocity w.
In cylindrical coordinates the interval in S is
ds^2 = dt^2 - dr^2 - r^{2d}(\phi)^2 - dz^2. (9)
The coordinates of S' are related to those of S by the
transformations
dt' = dt, (10)
d(\phi)' = d(\phi) - wdt, (11)
dr' = dr, dz' = dz, (12)
in terms of which the interval in S' becomes
ds'^2 = (1 - w^{2r}‘^2)dt‘^2 - dr‘^2 - r'^2d(\phi)'^2- 2wr'^2d(\phi)‘dt' - dz'^2, (13)
where S and S' have the same common origin so that r = r' and z = z'.
Equation (9) can be cast into
ds^2 = (1 - dr^2/dt^2 - r^{2d}(\phi)^2/dt^2 - dz^2/dt^2) dt^2,= (1 -- r^{2w}^2 - 0) dt^2 = (1 - w^{2r}^2) dt^2.
Or,
ds = \sqrt(1- w^{2r}^2)dt, (14)
which corresponds to equation (6) above.
Now, for a clock at rest on the turntable dr' = d(\phi)' = dz' =0 so
that
ds'^2 = (1 - w^{2r}^2) dt'^2, (15)
and
ds' = \sqrt(1 - w^{2r}^2) dt' (16)
In the literature the proper time of the clock at rest on the
turntable is defined as
ds = ds' = dT (17)
so that (14) becomes
dT = \sqrt(1 - w^{2r}^2) dt (18)
and (16) becomes
dT =\sqrt(1 - w^{2r}^2) dt' (19)
Equation (18) is the one quoted in the literature. Defining the proper
time this way, dt' looses its meaning that for a clock at rest on the
turntable, dt' is its proper time! Indeed, equation (19) reveals that
for a clock at rest on the turntable dt' is not the proper time. Both
dT and dt‘ refer to the same clock at rest, and they have different
meanings. IS NOT THIS IN CONTRADICTION WITH THE DEFINITION OF PROPER
TIME?
On the other hand, if we stick to the definition of proper time and
define it for a clock at rest on the turntable as dt' = dT, equation
(16) becomes
ds'= \sqrt(1 - w^{2r}^2)dT , (20)
which when compared with (14) gives
dT = dt (21)
In other words, there is no time dilation! This is consistent with the
transformation equation (10) above, which would directly give dT = dt.
John B. Kogut, in his book "Introduction to Relativity" treats motion
in a rotating, relativistic frame in Section 7.2, p.100. Having
eliminated the d(\phi)'dt' cross term in equation (13) above he writes
[itex]ds^2 = (c^2 - w^{2r}^2)dt'^2-(dr^2 + c^{2r}^2d(\phi)'^2/(c^2-w^{2r}^2)+dz^2),
his equation (7.2.6). He then says
" In order to interpret Eq.(7.2.6), we can compare it to the invariant
interval of an inertial frame of reference chosen to approximate the
transverse velocity wr locally. For example, taking a clock at rest in
the rotating frame,
ds^2 = (c^2 - w^{2r}^2)dt'^2,
and comparing it to a clock at rest in a locally inertial frame,
ds^2 = c^{2dT}^2,
we have time dilation again,
dT=\sqrt(1-w^{2r}^2/c^2)dt'. "
Is this a bad phrasing or a bad mistake?
Note that even though the last expression in Kogut's statement has the
same form as my equation (19) above, the meaning of dT in the two
equations are quite different.
Comments on Kogut's statement and my thinking above will be
appreciated very much.
Regards,
Murat Ozer
I would like to bring to your attention an issue that has been
bothering me for some time. It concerns the definition of proper time
in a rotating system, like a turntable. In Part 1 of the following, I
will go over how proper time is defined in Special Relativity. In Part
2, I will treat the relativistic rotating frame and point out that
proper time for this case is not quite defined as in Part 1.
Part 1: Special Relativity
As usual, consider two reference frames S and S', S' moving linearly
in the x direction away from S with a velocity v. The infinitesimal
Lorentz transformations between S' and S are:
dt'= (\gamma)(dt -[/itex] vdx), (1)
dx' = (\gamma)(dx - vdt), (2)
dy' = dy, dz' = dz, (3)
where (\gamma) = 1/\sqrt(1-v^2) and c has been set to 1.
The invariant interval in S and S' are
ds^2 = dt^2 - dx^2-dy^2 - dz^2, (4)
ds'^2 = dt'^2 - dx'^2 -dy'^2 - dz'^2. (5)
Factoring out dt^2 , equation (4) can be cast into
ds =\sqrt(1 - v^2) dt (6)
By definition, the proper time dT of a clock at rest in S' is found by
setting
dx' = dy' = dz' =0, and dt‘ = dT which results in
ds' = dT (T = \Tau). (7)
Comparing equations (6) and (7) gives, due to the invariance of the
interval,
dT = \sqrt(1 - v^2) dt. (8)
Incidentially, the time dilation equation (8) can be obtained directly
from the Lorentz transformations (1,(2), and (3). From (2),
(\gamma)dx = dx' + (\gamma)vdt.
Inserting this in (1) yields
dt'=\sqrt(1- v^2)dt - vdx'.
Now setting dx' =0 (along with dy' = dz' =0) and dt' = dT gives (8).No
surprise…
(End of part 1)
Part2: Motion in a Rotating Relativistic Frame
Let us now consider a frame at rest, S, and a second frame S' rotating
about the z axis - a turntable - with a constant angular velocity w.
In cylindrical coordinates the interval in S is
ds^2 = dt^2 - dr^2 - r^{2d}(\phi)^2 - dz^2. (9)
The coordinates of S' are related to those of S by the
transformations
dt' = dt, (10)
d(\phi)' = d(\phi) - wdt, (11)
dr' = dr, dz' = dz, (12)
in terms of which the interval in S' becomes
ds'^2 = (1 - w^{2r}‘^2)dt‘^2 - dr‘^2 - r'^2d(\phi)'^2- 2wr'^2d(\phi)‘dt' - dz'^2, (13)
where S and S' have the same common origin so that r = r' and z = z'.
Equation (9) can be cast into
ds^2 = (1 - dr^2/dt^2 - r^{2d}(\phi)^2/dt^2 - dz^2/dt^2) dt^2,= (1 -- r^{2w}^2 - 0) dt^2 = (1 - w^{2r}^2) dt^2.
Or,
ds = \sqrt(1- w^{2r}^2)dt, (14)
which corresponds to equation (6) above.
Now, for a clock at rest on the turntable dr' = d(\phi)' = dz' =0 so
that
ds'^2 = (1 - w^{2r}^2) dt'^2, (15)
and
ds' = \sqrt(1 - w^{2r}^2) dt' (16)
In the literature the proper time of the clock at rest on the
turntable is defined as
ds = ds' = dT (17)
so that (14) becomes
dT = \sqrt(1 - w^{2r}^2) dt (18)
and (16) becomes
dT =\sqrt(1 - w^{2r}^2) dt' (19)
Equation (18) is the one quoted in the literature. Defining the proper
time this way, dt' looses its meaning that for a clock at rest on the
turntable, dt' is its proper time! Indeed, equation (19) reveals that
for a clock at rest on the turntable dt' is not the proper time. Both
dT and dt‘ refer to the same clock at rest, and they have different
meanings. IS NOT THIS IN CONTRADICTION WITH THE DEFINITION OF PROPER
TIME?
On the other hand, if we stick to the definition of proper time and
define it for a clock at rest on the turntable as dt' = dT, equation
(16) becomes
ds'= \sqrt(1 - w^{2r}^2)dT , (20)
which when compared with (14) gives
dT = dt (21)
In other words, there is no time dilation! This is consistent with the
transformation equation (10) above, which would directly give dT = dt.
John B. Kogut, in his book "Introduction to Relativity" treats motion
in a rotating, relativistic frame in Section 7.2, p.100. Having
eliminated the d(\phi)'dt' cross term in equation (13) above he writes
[itex]ds^2 = (c^2 - w^{2r}^2)dt'^2-(dr^2 + c^{2r}^2d(\phi)'^2/(c^2-w^{2r}^2)+dz^2),
his equation (7.2.6). He then says
" In order to interpret Eq.(7.2.6), we can compare it to the invariant
interval of an inertial frame of reference chosen to approximate the
transverse velocity wr locally. For example, taking a clock at rest in
the rotating frame,
ds^2 = (c^2 - w^{2r}^2)dt'^2,
and comparing it to a clock at rest in a locally inertial frame,
ds^2 = c^{2dT}^2,
we have time dilation again,
dT=\sqrt(1-w^{2r}^2/c^2)dt'. "
Is this a bad phrasing or a bad mistake?
Note that even though the last expression in Kogut's statement has the
same form as my equation (19) above, the meaning of dT in the two
equations are quite different.
Comments on Kogut's statement and my thinking above will be
appreciated very much.
Regards,
Murat Ozer