Kasper J. Larsen
Oct8-04, 06:20 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 all,\n\n\n\nI happened to stumble upon a tutorial on general relativity authored\nby John Baez (http://math.ucr.edu/home/baez/gr/ricci.weyl.html). In\nthe section called "The Ricci and Weyl Tensors" the author tries to\ngive a geometric interpretation of these tensors. However, I am not at\nall clear about the precise meaning of what he writes:\n\n\n" [...] We consider a bunch of initially comoving coffee grounds near\na point P in spacetime, with the coffee ground that actually goes\nthrough P having velocity v at that instant. (Hence the term "instant\ncoffee".) Working in the local rest frame of the coffee ground that\ngoes through P, we consider a small round ball of comoving coffee\ngrounds centered at P, and see what happens as time passes. Each\ncoffee ground moves along a geodesic, but since spacetime is curved,\nthe ball may shrink, expand, rotate, and/or be deformed into an\nellipsoid.\n\nThe Ricci tensor Rab only keeps track of the change of volume of this\nball. Namely, the second time derivative of the volume of the ball is\n-Rabva vb times the ball\'s original volume."\n\n(Actually, Lee C. Loveridge may also be quoted, writing something\nalong the same lines: "The Ricci tensor governs the changing size of a\nsmall volume propagating through a curved space.")\n\n\nIt is the last paragraph quoted from Baez that I am very unsure about.\nThere are several naturally defined balls associated with a manifold,\nso I am unclear about which exact definition is being referred to.\nFurther, some vector field on the manifold or some family of geodesics\nneeds to be chosen for the "ball" to be uniquely determined.\n\n\nMy questions are as below:\n\n1. How would the claim made by Baez be expressed precisely, as a\ntheorem of differential geometry? I should like very much to see a\nproof also. A reference to some bibliography would be nice.\n\n2. Is there some direct geometric interpretation of the Ricci tensor,\napart from it being defined as "the" contraction of the Riemann\ntensor? In terms of sectional curvature, perhaps?\n\n3. Is there some way of interpreting the Weyl tensor, physical as well\nas geometric? (In Baez\'s tutorial, the Weyl tensor is related to\ngravitational waves, but in my opinion the argument given there\ndoesn\'t seem to pinpoint waves as the only possible interpretation.)\n\n\n\nThanks in advance!\n\nKasper J. Larsen\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 all,
I happened to stumble upon a tutorial on general relativity authored
by John Baez (http://math.ucr.edu/home/baez/gr/ricci.weyl.html). In
the section called "The Ricci and Weyl Tensors" the author tries to
give a geometric interpretation of these tensors. However, I am not at
all clear about the precise meaning of what he writes:
" [...] We consider a bunch of initially comoving coffee grounds near
a point P in spacetime, with the coffee ground that actually goes
through P having velocity v at that instant. (Hence the term "instant
coffee".) Working in the local rest frame of the coffee ground that
goes through P, we consider a small round ball of comoving coffee
grounds centered at P, and see what happens as time passes. Each
coffee ground moves along a geodesic, but since spacetime is curved,
the ball may shrink, expand, rotate, and/or be deformed into an
ellipsoid.
The Ricci tensor Rab only keeps track of the change of volume of this
ball. Namely, the second time derivative of the volume of the ball is
-Rabva vb times the ball's original volume."
(Actually, Lee C. Loveridge may also be quoted, writing something
along the same lines: "The Ricci tensor governs the changing size of a
small volume propagating through a curved space.")
It is the last paragraph quoted from Baez that I am very unsure about.
There are several naturally defined balls associated with a manifold,
so I am unclear about which exact definition is being referred to.
Further, some vector field on the manifold or some family of geodesics
needs to be chosen for the "ball" to be uniquely determined.
My questions are as below:
1. How would the claim made by Baez be expressed precisely, as a
theorem of differential geometry? I should like very much to see a
proof also. A reference to some bibliography would be nice.
2. Is there some direct geometric interpretation of the Ricci tensor,
apart from it being defined as "the" contraction of the Riemann
tensor? In terms of sectional curvature, perhaps?
3. Is there some way of interpreting the Weyl tensor, physical as well
as geometric? (In Baez's tutorial, the Weyl tensor is related to
gravitational waves, but in my opinion the argument given there
doesn't seem to pinpoint waves as the only possible interpretation.)
Thanks in advance!
Kasper J. Larsen
I happened to stumble upon a tutorial on general relativity authored
by John Baez (http://math.ucr.edu/home/baez/gr/ricci.weyl.html). In
the section called "The Ricci and Weyl Tensors" the author tries to
give a geometric interpretation of these tensors. However, I am not at
all clear about the precise meaning of what he writes:
" [...] We consider a bunch of initially comoving coffee grounds near
a point P in spacetime, with the coffee ground that actually goes
through P having velocity v at that instant. (Hence the term "instant
coffee".) Working in the local rest frame of the coffee ground that
goes through P, we consider a small round ball of comoving coffee
grounds centered at P, and see what happens as time passes. Each
coffee ground moves along a geodesic, but since spacetime is curved,
the ball may shrink, expand, rotate, and/or be deformed into an
ellipsoid.
The Ricci tensor Rab only keeps track of the change of volume of this
ball. Namely, the second time derivative of the volume of the ball is
-Rabva vb times the ball's original volume."
(Actually, Lee C. Loveridge may also be quoted, writing something
along the same lines: "The Ricci tensor governs the changing size of a
small volume propagating through a curved space.")
It is the last paragraph quoted from Baez that I am very unsure about.
There are several naturally defined balls associated with a manifold,
so I am unclear about which exact definition is being referred to.
Further, some vector field on the manifold or some family of geodesics
needs to be chosen for the "ball" to be uniquely determined.
My questions are as below:
1. How would the claim made by Baez be expressed precisely, as a
theorem of differential geometry? I should like very much to see a
proof also. A reference to some bibliography would be nice.
2. Is there some direct geometric interpretation of the Ricci tensor,
apart from it being defined as "the" contraction of the Riemann
tensor? In terms of sectional curvature, perhaps?
3. Is there some way of interpreting the Weyl tensor, physical as well
as geometric? (In Baez's tutorial, the Weyl tensor is related to
gravitational waves, but in my opinion the argument given there
doesn't seem to pinpoint waves as the only possible interpretation.)
Thanks in advance!
Kasper J. Larsen