Jon Absoul
Aug26-04, 08:19 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>\n[ This is a follow-up to an older thread, where I could find no "post\nmessage" link... The thread is:\nhttp://groups.google.com/groups?hl=en&lr=&ie=UTF-8&threadm=caesde%242g07%241%40godfrey.mcc.ac.uk&rnum=1&prev=/groups%3Fq%3Dabsoul%2BSchwarzschild%26hl%3Den%26lr %3D%26ie%3DUTF-8%26selm%3Dcaesde%25242g07%25241%2540godfrey.mcc.a c.uk%26rnum%3D1\n-jon]\n\n\nBelated thanks to T Essel and Greg Egan for generous posts!\n\n\n[1] SUMMARY\n\nI was looking for vector and tensor components in Schwarzschild\ncoordinates (I\'m doing some numerical work).\n\nI use the Schwarzschild chart:\nds^2 = A^2 dt^2 - dr^2/A^2 - r^2 (du^2 + sin(u)^2 dv^2),\n-infty < t < infty, 2M < r < infty, 0 < u < pi, -pi < v < pi\nA^2 = (1-2M/r)\n\nI get the contra- and covariant 4-velocities (radial motion):\n\nv^\\mu = \\gamma(1/A, vA, 0, 0)\nv_\\mu = \\gamma( A, -v/A, 0, 0)\n\nand the (contravariant) energy-momentum tensor (radial motion):\n\nT^{\\mu\\nu} = \\rho\\gamma^2 *\n1/A^2 v 0 0\nv (vA)^2 0 0\n0 0 0 0\n0 0 0 0\n\n(Notes:\nv is the radial velocity measured by a static observer at r\n("FIDO").\nFor particles, |v|<1, \\gamma=(1-v^2)^{-1/2}.\nFor photons, |v|=1, \\gamma=1.\n\\rho is the density of moving dust and is too low to affect the\nmetric.\nSanity checks:\nA=1: flat spacetime.\nv=0: particle at rest at r.\nv^\\mu v_\\mu = 1 for particles, = 0 for photons.\n)\n\n\n[2] VAIDYA NULL DUST.\n\nNice solution for the metric for outgoing, spherically symmetric,\nincoherent radiation!\n\nHowever, I could not figure out how to transform the metric to\nSchwarzschild-like coordinates, that is to a diagonal metric with\nthe same angular coordinates as the Schwarzschild metric (so that\nr measures circumference/2pi). The normal transform between\nEddintong-Finkelstein coordinates to Schwarzschild coordinates\ndoesn\'t seem to work when m is not constant. Simply setting m=m(r,t)\nin the Schwarzschild chart doesn\'t work.\nAnyone?\n\nAlso, is there a Vaidya reference for (a) citing in a paper, (b)\nreading more?\n\n\n[3] OTHER\n\nT Essel wrote:\n>> I\'m self-studying from several textbooks.\n> Cool! Which ones?\n\nMy two favorites are Foster&Nightingale ("A Short Course in GR") and\nTaylor&Wheeler ("Exploring Black Holes"). I have also read good chunks\nof many others: Rindler, Stephani, MTW, d\'Inverno, Hawkins&Ellis,\nEllis&etal, Wald, to name a few.\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>[ This is a follow-up to an older thread, where I could find no "post
message" link... The thread is:
http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&threadm=caesde%242g07%241%40godfrey.mcc.ac.uk&rnum=1&prev=/groups%3Fq%3Dabsoul%2BSchwarzschild%26hl%3Den%26lr %3D%26ie%3DUTF-8%26selm%3Dcaesde%25242g07%25241%2540godfrey.mcc.a c.uk%26rnum%3D1
-jon]
Belated thanks to T Essel and Greg Egan for generous posts!
[1] SUMMARY
I was looking for vector and tensor components in Schwarzschild
coordinates (I'm doing some numerical work).
I use the Schwarzschild chart:
ds^2 = A^2 dt^2 - dr^2/A^2 - r^2 (du^2 + sin(u)^2 dv^2),-\infty < t < \infty, 2M < r < \infty,< u < \pi, -\pi < v < \piA^2 = (1-2M/r)
I get the contra- and covariant 4-velocities (radial motion):
v^\mu = \gamma(1/A, vA, 0, 0)v_\mu = \gamma( A, -v/A, 0, 0)
and the (contravariant) energy-momentum tensor (radial motion):
T^{\mu\nu} = \rho\gamma^2 *1/A^2[/itex] v
v [itex](vA)^2
(Notes:
v is the radial velocity measured by a static observer at r
("FIDO").
For particles, |v|<1, \gamma=(1-v^2)^{-1/2}.
For photons, |v|=1, \gamma=1.\rho is the density of moving dust and is too low to affect the
metric.
Sanity checks:
A=1: flat spacetime.
v=0: particle at rest at r.
v^\mu v_\mu = 1 for particles, = for photons.
)
[2] VAIDYA NULL DUST.
Nice solution for the metric for outgoing, spherically symmetric,
incoherent radiation!
However, I could not figure out how to transform the metric to
Schwarzschild-like coordinates, that is to a diagonal metric with
the same angular coordinates as the Schwarzschild metric (so that
r measures circumference/2pi). The normal transform between
Eddintong-Finkelstein coordinates to Schwarzschild coordinates
doesn't seem to work when m is not constant. Simply setting m=m(r,t)
in the Schwarzschild chart doesn't work.
Anyone?
Also, is there a Vaidya reference for (a) citing in a paper, (b)
reading more?
[3] OTHER
T Essel wrote:
>> I'm self-studying from several textbooks.
> Cool! Which ones?
My two favorites are Foster&Nightingale ("A Short Course in GR") and
Taylor&Wheeler ("Exploring Black Holes"). I have also read good chunks
of many others: Rindler, Stephani, MTW, d'Inverno, Hawkins&Ellis,
Ellis&etal, Wald, to name a few.
message" link... The thread is:
http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&threadm=caesde%242g07%241%40godfrey.mcc.ac.uk&rnum=1&prev=/groups%3Fq%3Dabsoul%2BSchwarzschild%26hl%3Den%26lr %3D%26ie%3DUTF-8%26selm%3Dcaesde%25242g07%25241%2540godfrey.mcc.a c.uk%26rnum%3D1
-jon]
Belated thanks to T Essel and Greg Egan for generous posts!
[1] SUMMARY
I was looking for vector and tensor components in Schwarzschild
coordinates (I'm doing some numerical work).
I use the Schwarzschild chart:
ds^2 = A^2 dt^2 - dr^2/A^2 - r^2 (du^2 + sin(u)^2 dv^2),-\infty < t < \infty, 2M < r < \infty,< u < \pi, -\pi < v < \piA^2 = (1-2M/r)
I get the contra- and covariant 4-velocities (radial motion):
v^\mu = \gamma(1/A, vA, 0, 0)v_\mu = \gamma( A, -v/A, 0, 0)
and the (contravariant) energy-momentum tensor (radial motion):
T^{\mu\nu} = \rho\gamma^2 *1/A^2[/itex] v
v [itex](vA)^2
(Notes:
v is the radial velocity measured by a static observer at r
("FIDO").
For particles, |v|<1, \gamma=(1-v^2)^{-1/2}.
For photons, |v|=1, \gamma=1.\rho is the density of moving dust and is too low to affect the
metric.
Sanity checks:
A=1: flat spacetime.
v=0: particle at rest at r.
v^\mu v_\mu = 1 for particles, = for photons.
)
[2] VAIDYA NULL DUST.
Nice solution for the metric for outgoing, spherically symmetric,
incoherent radiation!
However, I could not figure out how to transform the metric to
Schwarzschild-like coordinates, that is to a diagonal metric with
the same angular coordinates as the Schwarzschild metric (so that
r measures circumference/2pi). The normal transform between
Eddintong-Finkelstein coordinates to Schwarzschild coordinates
doesn't seem to work when m is not constant. Simply setting m=m(r,t)
in the Schwarzschild chart doesn't work.
Anyone?
Also, is there a Vaidya reference for (a) citing in a paper, (b)
reading more?
[3] OTHER
T Essel wrote:
>> I'm self-studying from several textbooks.
> Cool! Which ones?
My two favorites are Foster&Nightingale ("A Short Course in GR") and
Taylor&Wheeler ("Exploring Black Holes"). I have also read good chunks
of many others: Rindler, Stephani, MTW, d'Inverno, Hawkins&Ellis,
Ellis&etal, Wald, to name a few.