tessel@tum.bot
May17-04, 05:04 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>Oh dear. I growled (in another thread)\n\n> A general request to all posters who have fallen into bad habits:\n> please be kind to the long-suffering moderators--- and readers!-- and\n> make every attempt to write posts which are\n>\n> (i) concise\n>\n> (ii) clear\n>\n> (iii) correct.\n>\n> This will save time and effort and -greatly- improve the impression you\n> make on others here.\n\nand then immediately proceeded to myself write an example of a post which\nviolated at least two of these demands! I apologize for that, and I\'ll\ntry to correct any damage I may have done in this post.\n\nJon, I see now that I somehow overlooked and/or misread your description\nof what you thought you found concerning the behavior of the Ricci scalar\nR in the FRW dust with S^3 hyperslices:\n\n> I also plotted the spacetime curvature scalar, R(t), (the contraction of\n> the Ricci tensor), which turns out to start and end at positive\n> infinity, but which becomes negative near maximum expansions, psi in\n> ~(2.5,3.8).\n\nI should have said something like this:\n\nIn a dust solution, we can find a suitable frame (like the one I gave for\nthe particular dust solution under discussion) which serves as a "local\nLorentz frame" comoving with the dust. Expanded wrt that frame, the\nEinstein tensor has the form\n\nG^(ab) = diag(mu,0,0,0)\n\nwhere mu >= 0 is the density of the dust. Since G^(ab) is the\n"trace-reverse" of R^(ab), the trace of the Einstein tensor is for a dust\njust 8 pi mu = -R. So you are really asking about the time behavior of mu\n(the negative of the Ricci scalar). In an expansion and recollapse model,\nwe would of course expect mu to be minimal--- but positive--- at the\nmoment of maximum expansion, and to increase as we run time forward or\nbackwards from there, blowing up as we approach the Big Bang or Big\nCrunch. In particular, in the specific dust model we are discussing, wrt\nthe conformal chart I gave, we find\n\nmu = 3/(8 pi A^2) csc(t/2)^6,\n\n0 < t < 2 pi\n\nRemember that "delta t" is not a linear function of the proper time\ninterval "delta s" measured by a dust particle; rather, from the line\nelement we have\n\nds = A sin(t/2)^2 dt\n\nwhich gives the proper time interval delta s, measured between two events\non his world line by an ideal observer riding on a dust particle, as an\nintegral from t1 to t2.\n\nSo the answer to your first question is that if you found a behavior\nqualitatively different from 1/sin^6(w) on 0 < w < pi, either you\nincorrectly computed R (probably because you were working with an\nimplicitly defined solution to the Friedmann equation is it appears for\nproper time rather than the conformal time coordinate I am using) or else\nyou plotted it incorrectly (or had a computer try to plot it using some\ninappropriate numerical method).\n\nI also could and should have been -much- clearer in answering your second\nquestion, asking for the intuitive meaning of the Ricci scalar.\n\nFor the benefit of others, I should start by mentioning an underlying\nquestion which often gets short schrift in the American undergraduate\ncurriculum: what is the meaning of the trace of an ordinary matrix?\n\nWe have discussed this question many times in the past, and I am usually\nreluctant to repeat myself at length, but having munged my first reply\nI\'ll break my rule about this. (But you may also want to search for these\npast threads on sci.physics.* and sci.math.*, since I have noticed that if\nI try to explain something n times, my comments tend to become -less-\nclear as n increases--- boredom with the subject presumably accounting for\ndecreased enthusiasm and thus insufficient energy expended to write an\nintelligently written response.)\n\nThere are actually several valuable ways to understand the meaning of the\ntrace. The "algebraic meaning" is of course given by the fact that we can\nwrite the characteristic equation of the n x n matrix A is the polynomial\n\nchi_A(t) = t^n - tr A t^(n-1) + ...\n\nAs you no doubt know, this controls the "eigenthings" of A (and as Cayley\nnoticed, if you plug in A, you find that A is a matrix solution to its own\ncharacteristic equation.)\n\nThe "dynamical meaning" is given by the formula\n\ndet (exp t A) = exp (tr A).\n\nHere, on the LHS\n\nexp(t A) = I + t A + t^2/2 A^2 + t^3/6 A^3 + ....\n\nis the matrix exponential, which you may have encountered in a course on\nODEs; the exp on the RHS is the ordinary exponential. If so, in that\ncourse you were actually doing a bit of Lie theory! In this context, the\ntrace can be understood as the "logarithmic derivative" of the volume of a\nparallelepiped under a "flow". For example, if\n\n[ 0 -1 ]\nA = [ 1 0 ]\n\nthen you can verify using trig identities that\n\n[ cos(t) -sin(t) ]\nexp(t A) = [ sin(t) cos(t) ]\n\ni.e. a rotation in E^2. Here, the area of parallepipeds is of course\npreserved by rotation; similarly for "boosts", but not for "dilations".\n\nYou can look for past threads called "What is a Vector Field?" and "What\nis Lie Theory?", which explains how A is related to the system of linear\nODEs\n\n[ x* ] [ 0 -1 ] [ x ]\n[ y* ] = [ 1 0 ] [ y ]\n\nwhere * = d/ds (s is the independent variable here, not parallelepipeds\ninterpreted as proper time, of course), and also to the first order linear\npartial differential operator\n\n-y @/@x + x @/@y\n\nwhere @/@x is partial differentiation wrt x. The point is that this works\nfor any A! Also, note that our formula relating det and tr immediately\nimplies that the exponential of a traceless matrix is a unimodular matrix.\nThis simple observation is very important in Lie theory and in the\nzillions of subjects (like gtr) which lean heavily upon it.\n\nHere is a slightly different question: can we find any additional\nintuition for the trace of a -symmetric- matrix S? Indeed, we can!---\nobserve that we can consider\n\n[ a p q ] [ x ]\n[ p b r ] [ y ]\n[x y z] [ q r c ] [ z ] = Q(X)\n\nas a positive definite quadratic form on E^3. Now -average- the values of\nQ(X) over the unit sphere, i.e. evaluate Q for all -unit vectors- in E^3\nand integrate. Plugging in\n\nx = sin(u) cos(v)\n\ny = sin(u) sin(v)\n\nz = cos(u)\n\nwe find that the "average value" of Q(X) on unit vectors is\n\nint_(-pi)^pi int_0^pi Q(X) sin(u) du dv\n--------------------------------------- = a + b + c = tr S\nint_(-pi)^pi int_0^pi sin(u) du dv\n\nThis obviously generalizes to E^n, but not to E^(1,n), so you may think\nthis is irrelevant to gtr. (To get compact "unit spheres", we need the\nquadratic form we use to define "unit spheres" to be positive definite).\nBut wait!\n\nLet me rephrase your second question yet again: can we give any intuition\nfor the role of the Ricci tensor itself (not just its trace) in gtr? The\nanswer is: certainly! In fact, we can find several closely related\nintuitive interpretations. The simplest is explained on John Baez\'s\nwebpages\n\nhttp://math.ucr.edu/home/baez/einstein/einstein.html\n\nNext, the Ricci tensor happens to be symmetric, so we might try to employ\nthe "averaging" interpretation of the trace of a symmetric matrix\n(thinking of R(X,X), X some vector field, as, roughly speaking, a\nquadratic form varying smoothly from event to event, i.e. a symmetric\nsecond rank tensor field). But we have an indefinite form, so we appear\nto be stuck.\n\nHowever, in the ADM reformulation of the EFE, we choose a spatial\nhyperslice of our Lorentzian manifold which satisfies the "constraint\nequations". This has intrinsic geometry as a three dimensional Riemannian\nmanifold, including a Ricci tensor R. Then we try to evolve the geometry\nof the slice according to the "evolution equations". There are many\nsubtleties here which I am avoiding, but the point is that the trace of\nthe Ricci tensor on each hyperslice, i.e. the Ricci scalar of the\nhyperslice (-not- the same thing at all as the Ricci scalar of the\nspacetime itself) can be interpreted as an average over a very small unit\nsphere.\n\nThis is closely related to the reformulation of the EFE which I mentioned\nin my previous response, to wit: if we choose a "totally geodesic spatial\nhyperslice" S through an event P, and evaluate the Ricci curvature scalar\nof S at P, then\n\nR = 16 pi mu\n\nwhere mu is the matter density measured at P by an observer whose world\nline is orthogonal there to the hyperslice S. And as we\'ve seen, here R\ncan be interpreted as an average over a small unit sphere around P of the\nquadratic form\n\nQ(N) = R_(ab) N^a N^b,\n\nwhere N is a unit vector in the tangent space to S at P.\n\nThis is completely equivalent to G^(ab) = 8 pi T^(ab), but unfortunately\napparently not very useful. But I might mention here that in this\ncontext, we have another intuitive interpretation of R, in terms of a kind\nof second order deviation of small spheres in S from the E^3 area formula\nA = 4 pi r^2.\n\nA more important way in which the Ricci tensor--- rather than its "trace\nreverse", the Einstein curvature tensor--- enters directly into gtr is via\nsome of the most important equations in this subject: the "Raychaudhuri\nequation", and its close relatives, the "optical equations".\nUnfortunately, despite their importance for many purposes, these are not\neven mentioned in most textbooks! But if you read John\'s essay, you\'ve\nalready encountered the basic idea: the quadratic form R_(ab) U^a U^b,\nwhere now U is a unit timelike vector in the spacetime itself, determines\nthe volume decrease of a small sphere of initially static test particles\ndue to the presence of mass-energy inside the sphere. The Raychaudhuri\nequation generalizes this to allow for the possibility that the cloud of\ntest particles is, at time zero, rotating or shearing or already\nexpanding/contracting.\n\nThe optical equations use R_(ab) K^a K^b, where now K is a -null vector-.\nHere the question is: how does the area and shape of a narrow pencil of\nlight rays evolve over time, due if you will to "light bending" effects\ncaused by the curvature of spacetime? Here too, the Ricci tensor plays\nthe role of determining a "contraction", which here effects (all other\nthings being equal) a kind of "focusing". This may help explain why some\nof the "energy conditions" involve R_(ab) U^a U^b or R_(ab) K^a K^b.\n\nThere\'s more, for example the Ricci tensor appears in an important general\n"decomposition" of the Riemann tensor into "conformal", "scalar" and\n"Ricci" parts. But this post is already too long!\n\nAt this point, I could append the computation of the curvature scalar via\nthe method of Cartan. However, I\'m running out of time and space, and\nI\'ve cranked through very similar computations in this group in the past,\nso I\'ll just suggest that interested readers search for those posts, or\nconsult the following references:\n\nFor intrinsic/extrinsic curvature of spatial hyperslices, see at least the\npictures in the classic\n\nauthor = {Charles W. Misner and Kip S. Thorne and John Archibald Wheeler},\ntitle = {Gravitation},\npublisher = {W. H. Freeman},\nyear = 1970}\n\nand then try\n\nauthor = {Hans Stephani},\ntitle = {General Relativity: An Introduction of the Theory of the\nGravitational Field},\npublisher = {Cambridge University Press},\nedition = {Second},\nnote = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},\nyear = 1990}\n\nMTW also discuss Cartan\'s slick way of computing the curvature tensors,\nwhich employs his "exterior calculus", via connection one-forms and\ncurvature two-forms.\n\nFor Raychaudhuri, see Stephani or the classic monograph\n\nauthor = {S. W. Hawking and G. F. R. Ellis},\ntitle = {The Large Scale Structure of Space-Time},\npublisher = {Cambridge University Press},\nyear = 1973}\n\nHE also discuss the optical equations, but I prefer the discussion in\n\nauthor = {P.J.E. Peebles},\ntitle = {Principles of physical cosmology},\npublisher = {Princeton University Press},\nyear = 1993}\n\nFor totally geodesic hyperslices and more on intrinsic/extrinsic\ncurvatures, see\n\nauthor = {Theodore Frankel},\ntitle = {Gravitational Curvature: an Introduction to {E}instein\'s Theory},\npublisher = {W. H. Freeman},\nyear = 1979}\n\nFor frames and coframes, try\n\nauthor = {F. de Felice and C.J.S. Clarke},\ntitle = {Relativity on Curved Manifolds},\npublisher = {Cambridge University Press},\nyear = 1990}\n\nLast but not least, the undergraduate textbook\n\nauthor = {Antal Fekete},\ntitle = {Real Linear Algebra},\npublisher = {Marcel Dekker},\nyear = 1985}\n\noffers much more valuable intuition into the basic notions of linear\nalgebra, including shears, dilations, and many other things I and others\noften refer to here, and I highly recommend it. For more intuition for\nthe coefficients of the characteristic polynomial which appear between\ntrace and determinant, see the discussion of slices of "energy ellipsoids"\nin\n\nauthor = {Diestel, Reinhard},\ntitle = {Graph Theory},\npublisher = {Springer},\nseries = {Graduate texts in mathematics},\nvolume = 173,\nyear = 2000}\n\nauthor = {Bollob\\\'as, B\\\'ela},\ntitle = {Modern Graph Theory},\nseries = {Graduate texts in mathematics},\nvolume = 184,\npublisher = {Springer-Verlag},\nyear = 1998}\n\n"T. Essel" (hiding somewhere in cyberspace)\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>Oh dear. I growled (in another thread)
> A general request to all posters who have fallen into bad habits:
> please be kind to the long-suffering moderators--- and readers!-- and
> make every attempt to write posts which are
>
> (i) concise
>
> (ii) clear
>
> (iii) correct.
>
> This will save time and effort and -greatly- improve the impression you
> make on others here.
and then immediately proceeded to myself write an example of a post which
violated at least two of these demands! I apologize for that, and I'll
try to correct any damage I may have done in this post.
Jon, I see now that I somehow overlooked and/or misread your description
of what you thought you found concerning the behavior of the Ricci scalar
R in the FRW dust with S^3 hyperslices:
> I also plotted the spacetime curvature scalar, R(t), (the contraction of
> the Ricci tensor), which turns out to start and end at positive
> infinity, but which becomes negative near maximum expansions, \psi in
> ~(2.5,3.8).
I should have said something like this:
In a dust solution, we can find a suitable frame (like the one I gave for
the particular dust solution under discussion) which serves as a "local
Lorentz frame" comoving with the dust. Expanded wrt that frame, the
Einstein tensor has the form
G^(ab) = diag(\mu,0,0,0)
where \mu >= is the density of the dust. Since G^(ab) is the
"trace-reverse" of R^(ab), the trace of the Einstein tensor is for a dust
just 8 \pi \mu = -R. So you are really asking about the time behavior of \mu
(the negative of the Ricci scalar). In an expansion and recollapse model,
we would of course expect \mu to be minimal--- but positive--- at the
moment of maximum expansion, and to increase as we run time forward or
backwards from there, blowing up as we approach the Big Bang or Big
Crunch. In particular, in the specific dust model we are discussing, wrt
the conformal chart I gave, we find
\mu = 3/(8 \pi A^2) csc(t/2)^6,
< t < 2 \pi
Remember that "\delta t" is not a linear function of the proper time
interval "\delta s" measured by a dust particle; rather, from the line
element we have
ds = A sin(t/2)^2 dt
which gives the proper time interval \delta s, measured between two events
on his world line by an ideal observer riding on a dust particle, as an
integral from t1 to t2.
So the answer to your first question is that if you found a behavior
qualitatively different from 1/sin^6(w) on < w < \pi, either you
incorrectly computed R (probably because you were working with an
implicitly defined solution to the Friedmann equation is it appears for
proper time rather than the conformal time coordinate I am using) or else
you plotted it incorrectly (or had a computer try to plot it using some
inappropriate numerical method).
I also could and should have been -much- clearer in answering your second
question, asking for the intuitive meaning of the Ricci scalar.
For the benefit of others, I should start by mentioning an underlying
question which often gets short schrift in the American undergraduate
curriculum: what is the meaning of the trace of an ordinary matrix?
We have discussed this question many times in the past, and I am usually
reluctant to repeat myself at length, but having munged my first reply
I'll break my rule about this. (But you may also want to search for these
past threads on sci.physics.* and sci.math.*, since I have noticed that if
I try to explain something n times, my comments tend to become -less-
clear as n increases--- boredom with the subject presumably accounting for
decreased enthusiasm and thus insufficient energy expended to write an
intelligently written response.)
There are actually several valuable ways to understand the meaning of the
trace. The "algebraic meaning" is of course given by the fact that we can
write the characteristic equation of the n x n matrix A is the polynomial
\chi_A(t) = t^n - tr A t^(n-1) + .[/itex]..
As you no doubt know, this controls the "eigenthings" of A (and as Cayley
noticed, if you plug in A, you find that A is a matrix solution to its own
characteristic equation.)
The "dynamical meaning" is given by the formula
det (\exp t A) = \exp (tr A).
Here, on the LHS
\exp(t A) = I + t A + t^2/2 A^2 + t^3/6 A^3 + ....
is the matrix exponential, which you may have encountered in a course on
ODEs; the \exp on the RHS is the ordinary exponential. If so, in that
course you were actually doing a bit of Lie theory! In this context, the
trace can be understood as the "logarithmic derivative" of the volume of a
parallelepiped under a "flow". For example, if
[ -1 ]A = [ 1 ]
then you can verify using trig identities that
[ cos(t) -sin(t) ]\exp(t A) = [ sin(t) cos(t) ]
i.e. a rotation in E^2. Here, the area of parallepipeds is of course
preserved by rotation; similarly for "boosts", but not for "dilations".
You can look for past threads called "What is a Vector Field?" and "What
is Lie Theory?", which explains how A is related to the system of linear
ODEs
[ x* ] [ -1 ] [ x ][ y* ] = [ 1 ] [ y ]
where * = d/ds (s is the independent variable here, not parallelepipeds
interpreted as proper time, of course), and also to the first order linear
partial differential operator
-y @/@x + x @/@y
where @/@x is partial differentiation wrt x. The point is that this works
for any A! Also, note that our formula relating det and tr immediately
implies that the exponential of a traceless matrix is a unimodular matrix.
This simple observation is very important in Lie theory and in the
zillions of subjects (like gtr) which lean heavily upon it.
Here is a slightly different question: can we find any additional
intuition for the trace of a -symmetric- matrix S? Indeed, we can!---
observe that we can consider
[ a p q ] [ x ]
[ p b r ] [ y ]
[x y z] [ q r c ] [ z ] = Q(X)
as a positive definite quadratic form on E^3. Now -average- the values of
Q(X) over the unit sphere, i.e. evaluate Q for all -unit vectors- in E^3
and integrate. Plugging in
x = sin(u) cos(v)y = sin(u) sin(v)z = cos(u)
we find that the "average value" of Q(X) on unit vectors is
\int_(-\pi)^\pi \int_0^\pi Q(X) sin(u) du dv
--------------------------------------- [itex]= a + b + c = tr S\int_(-\pi)^\pi \int_0^\pi sin(u) du dv
This obviously generalizes to E^n, but not to E^(1,n), so you may think
this is irrelevant to gtr. (To get compact "unit spheres", we need the
quadratic form we use to define "unit spheres" to be positive definite).
But wait!
Let me rephrase your second question yet again: can we give any intuition
for the role of the Ricci tensor itself (not just its trace) in gtr? The
answer is: certainly! In fact, we can find several closely related
intuitive interpretations. The simplest is explained on John Baez's
webpages
http://math.ucr.edu/home/baez/einstein/einstein.html
Next, the Ricci tensor happens to be symmetric, so we might try to employ
the "averaging" interpretation of the trace of a symmetric matrix
(thinking of R(X,X), X some vector field, as, roughly speaking, a
quadratic form varying smoothly from event to event, i.e. a symmetric
second rank tensor field). But we have an indefinite form, so we appear
to be stuck.
However, in the ADM reformulation of the EFE, we choose a spatial
hyperslice of our Lorentzian manifold which satisfies the "constraint
equations". This has intrinsic geometry as a three dimensional Riemannian
manifold, including a Ricci tensor R. Then we try to evolve the geometry
of the slice according to the "evolution equations". There are many
subtleties here which I am avoiding, but the point is that the trace of
the Ricci tensor on each hyperslice, i.e. the Ricci scalar of the
hyperslice (-not- the same thing at all as the Ricci scalar of the
spacetime itself) can be interpreted as an average over a very small unit
sphere.
This is closely related to the reformulation of the EFE which I mentioned
in my previous response, to wit: if we choose a "totally geodesic spatial
hyperslice" S through an event P, and evaluate the Ricci curvature scalar
of S at P, then
R = 16 \pi \mu
where \mu is the matter density measured at P by an observer whose world
line is orthogonal there to the hyperslice S. And as we've seen, here R
can be interpreted as an average over a small unit sphere around P of the
quadratic form
Q(N) = R_(ab) N^a N^b,
where N is a unit vector in the tangent space to S at P.
This is completely equivalent to G^(ab) = 8 \pi T^(ab), but unfortunately
apparently not very useful. But I might mention here that in this
context, we have another intuitive interpretation of R, in terms of a kind
of second order deviation of small spheres in S from the E^3 area formula
A = 4 \pi r^2.
A more important way in which the Ricci tensor--- rather than its "trace
reverse", the Einstein curvature tensor--- enters directly into gtr is via
some of the most important equations in this subject: the "Raychaudhuri
equation", and its close relatives, the "optical equations".
Unfortunately, despite their importance for many purposes, these are not
even mentioned in most textbooks! But if you read John's essay, you've
already encountered the basic idea: the quadratic form R_(ab) U^a U^b,
where now U is a unit timelike vector in the spacetime itself, determines
the volume decrease of a small sphere of initially static test particles
due to the presence of mass-energy inside the sphere. The Raychaudhuri
equation generalizes this to allow for the possibility that the cloud of
test particles is, at time zero, rotating or shearing or already
expanding/contracting.
The optical equations use R_(ab) K^a K^b, where now K is a -null vector-.
Here the question is: how does the area and shape of a narrow pencil of
light rays evolve over time, due if you will to "light bending" effects
caused by the curvature of spacetime? Here too, the Ricci tensor plays
the role of determining a "contraction", which here effects (all other
things being equal) a kind of "focusing". This may help explain why some
of the "energy conditions" involve R_(ab) U^a U^b or R_(ab) K^a K^b.
There's more, for example the Ricci tensor appears in an important general
"decomposition" of the Riemann tensor into "conformal", "scalar" and
"Ricci" parts. But this post is already too long!
At this point, I could append the computation of the curvature scalar via
the method of Cartan. However, I'm running out of time and space, and
I've cranked through very similar computations in this group in the past,
so I'll just suggest that interested readers search for those posts, or
consult the following references:
For intrinsic/extrinsic curvature of spatial hyperslices, see at least the
pictures in the classic
author = {Charles W. Misner and Kip S. Thorne and John Archibald Wheeler},
title = {Gravitation},
publisher = {W. H. Freeman},
year = 1970}
and then try
author = {Hans Stephani},
title = {General Relativity: An Introduction of the Theory of the
Gravitational Field},
publisher = {Cambridge University Press},
edition = {Second},
note = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},
year = 1990}
MTW also discuss Cartan's slick way of computing the curvature tensors,
which employs his "exterior calculus", via connection one-forms and
curvature two-forms.
For Raychaudhuri, see Stephani or the classic monograph
author = {S. W. Hawking and G. F. R. Ellis},
title = {The Large Scale Structure of Space-Time},
publisher = {Cambridge University Press},
year = 1973}
HE also discuss the optical equations, but I prefer the discussion in
author = {P.J.E. Peebles},
title = {Principles of physical cosmology},
publisher = {Princeton University Press},
year = 1993}
For totally geodesic hyperslices and more on intrinsic/extrinsic
curvatures, see
author = {Theodore Frankel},
title = {Gravitational Curvature: an Introduction to {E}instein's Theory},
publisher = {W. H. Freeman},
year = 1979}
For frames and coframes, try
author = {F. de Felice and C.J.S. Clarke},
title = {Relativity on Curved Manifolds},
publisher = {Cambridge University Press},
year = 1990}
Last but not least, the undergraduate textbook
author = {Antal Fekete},
title = {Real Linear Algebra},
publisher = {Marcel Dekker},
year = 1985}
offers much more valuable intuition into the basic notions of linear
algebra, including shears, dilations, and many other things I and others
often refer to here, and I highly recommend it. For more intuition for
the coefficients of the characteristic polynomial which appear between
trace and determinant, see the discussion of slices of "energy ellipsoids"
in
author = {Diestel, Reinhard},
title = {Graph Theory},
publisher = {Springer},
series = {Graduate texts in mathematics},
volume = 173,
year = 2000}
author = {Bollob\'as, B\'ela},
title = {Modern Graph Theory},
series = {Graduate texts in mathematics},
volume = 184,
publisher = {Springer-Verlag},
year = 1998}
"T. Essel" (hiding somewhere in cyberspace)
> A general request to all posters who have fallen into bad habits:
> please be kind to the long-suffering moderators--- and readers!-- and
> make every attempt to write posts which are
>
> (i) concise
>
> (ii) clear
>
> (iii) correct.
>
> This will save time and effort and -greatly- improve the impression you
> make on others here.
and then immediately proceeded to myself write an example of a post which
violated at least two of these demands! I apologize for that, and I'll
try to correct any damage I may have done in this post.
Jon, I see now that I somehow overlooked and/or misread your description
of what you thought you found concerning the behavior of the Ricci scalar
R in the FRW dust with S^3 hyperslices:
> I also plotted the spacetime curvature scalar, R(t), (the contraction of
> the Ricci tensor), which turns out to start and end at positive
> infinity, but which becomes negative near maximum expansions, \psi in
> ~(2.5,3.8).
I should have said something like this:
In a dust solution, we can find a suitable frame (like the one I gave for
the particular dust solution under discussion) which serves as a "local
Lorentz frame" comoving with the dust. Expanded wrt that frame, the
Einstein tensor has the form
G^(ab) = diag(\mu,0,0,0)
where \mu >= is the density of the dust. Since G^(ab) is the
"trace-reverse" of R^(ab), the trace of the Einstein tensor is for a dust
just 8 \pi \mu = -R. So you are really asking about the time behavior of \mu
(the negative of the Ricci scalar). In an expansion and recollapse model,
we would of course expect \mu to be minimal--- but positive--- at the
moment of maximum expansion, and to increase as we run time forward or
backwards from there, blowing up as we approach the Big Bang or Big
Crunch. In particular, in the specific dust model we are discussing, wrt
the conformal chart I gave, we find
\mu = 3/(8 \pi A^2) csc(t/2)^6,
< t < 2 \pi
Remember that "\delta t" is not a linear function of the proper time
interval "\delta s" measured by a dust particle; rather, from the line
element we have
ds = A sin(t/2)^2 dt
which gives the proper time interval \delta s, measured between two events
on his world line by an ideal observer riding on a dust particle, as an
integral from t1 to t2.
So the answer to your first question is that if you found a behavior
qualitatively different from 1/sin^6(w) on < w < \pi, either you
incorrectly computed R (probably because you were working with an
implicitly defined solution to the Friedmann equation is it appears for
proper time rather than the conformal time coordinate I am using) or else
you plotted it incorrectly (or had a computer try to plot it using some
inappropriate numerical method).
I also could and should have been -much- clearer in answering your second
question, asking for the intuitive meaning of the Ricci scalar.
For the benefit of others, I should start by mentioning an underlying
question which often gets short schrift in the American undergraduate
curriculum: what is the meaning of the trace of an ordinary matrix?
We have discussed this question many times in the past, and I am usually
reluctant to repeat myself at length, but having munged my first reply
I'll break my rule about this. (But you may also want to search for these
past threads on sci.physics.* and sci.math.*, since I have noticed that if
I try to explain something n times, my comments tend to become -less-
clear as n increases--- boredom with the subject presumably accounting for
decreased enthusiasm and thus insufficient energy expended to write an
intelligently written response.)
There are actually several valuable ways to understand the meaning of the
trace. The "algebraic meaning" is of course given by the fact that we can
write the characteristic equation of the n x n matrix A is the polynomial
\chi_A(t) = t^n - tr A t^(n-1) + .[/itex]..
As you no doubt know, this controls the "eigenthings" of A (and as Cayley
noticed, if you plug in A, you find that A is a matrix solution to its own
characteristic equation.)
The "dynamical meaning" is given by the formula
det (\exp t A) = \exp (tr A).
Here, on the LHS
\exp(t A) = I + t A + t^2/2 A^2 + t^3/6 A^3 + ....
is the matrix exponential, which you may have encountered in a course on
ODEs; the \exp on the RHS is the ordinary exponential. If so, in that
course you were actually doing a bit of Lie theory! In this context, the
trace can be understood as the "logarithmic derivative" of the volume of a
parallelepiped under a "flow". For example, if
[ -1 ]A = [ 1 ]
then you can verify using trig identities that
[ cos(t) -sin(t) ]\exp(t A) = [ sin(t) cos(t) ]
i.e. a rotation in E^2. Here, the area of parallepipeds is of course
preserved by rotation; similarly for "boosts", but not for "dilations".
You can look for past threads called "What is a Vector Field?" and "What
is Lie Theory?", which explains how A is related to the system of linear
ODEs
[ x* ] [ -1 ] [ x ][ y* ] = [ 1 ] [ y ]
where * = d/ds (s is the independent variable here, not parallelepipeds
interpreted as proper time, of course), and also to the first order linear
partial differential operator
-y @/@x + x @/@y
where @/@x is partial differentiation wrt x. The point is that this works
for any A! Also, note that our formula relating det and tr immediately
implies that the exponential of a traceless matrix is a unimodular matrix.
This simple observation is very important in Lie theory and in the
zillions of subjects (like gtr) which lean heavily upon it.
Here is a slightly different question: can we find any additional
intuition for the trace of a -symmetric- matrix S? Indeed, we can!---
observe that we can consider
[ a p q ] [ x ]
[ p b r ] [ y ]
[x y z] [ q r c ] [ z ] = Q(X)
as a positive definite quadratic form on E^3. Now -average- the values of
Q(X) over the unit sphere, i.e. evaluate Q for all -unit vectors- in E^3
and integrate. Plugging in
x = sin(u) cos(v)y = sin(u) sin(v)z = cos(u)
we find that the "average value" of Q(X) on unit vectors is
\int_(-\pi)^\pi \int_0^\pi Q(X) sin(u) du dv
--------------------------------------- [itex]= a + b + c = tr S\int_(-\pi)^\pi \int_0^\pi sin(u) du dv
This obviously generalizes to E^n, but not to E^(1,n), so you may think
this is irrelevant to gtr. (To get compact "unit spheres", we need the
quadratic form we use to define "unit spheres" to be positive definite).
But wait!
Let me rephrase your second question yet again: can we give any intuition
for the role of the Ricci tensor itself (not just its trace) in gtr? The
answer is: certainly! In fact, we can find several closely related
intuitive interpretations. The simplest is explained on John Baez's
webpages
http://math.ucr.edu/home/baez/einstein/einstein.html
Next, the Ricci tensor happens to be symmetric, so we might try to employ
the "averaging" interpretation of the trace of a symmetric matrix
(thinking of R(X,X), X some vector field, as, roughly speaking, a
quadratic form varying smoothly from event to event, i.e. a symmetric
second rank tensor field). But we have an indefinite form, so we appear
to be stuck.
However, in the ADM reformulation of the EFE, we choose a spatial
hyperslice of our Lorentzian manifold which satisfies the "constraint
equations". This has intrinsic geometry as a three dimensional Riemannian
manifold, including a Ricci tensor R. Then we try to evolve the geometry
of the slice according to the "evolution equations". There are many
subtleties here which I am avoiding, but the point is that the trace of
the Ricci tensor on each hyperslice, i.e. the Ricci scalar of the
hyperslice (-not- the same thing at all as the Ricci scalar of the
spacetime itself) can be interpreted as an average over a very small unit
sphere.
This is closely related to the reformulation of the EFE which I mentioned
in my previous response, to wit: if we choose a "totally geodesic spatial
hyperslice" S through an event P, and evaluate the Ricci curvature scalar
of S at P, then
R = 16 \pi \mu
where \mu is the matter density measured at P by an observer whose world
line is orthogonal there to the hyperslice S. And as we've seen, here R
can be interpreted as an average over a small unit sphere around P of the
quadratic form
Q(N) = R_(ab) N^a N^b,
where N is a unit vector in the tangent space to S at P.
This is completely equivalent to G^(ab) = 8 \pi T^(ab), but unfortunately
apparently not very useful. But I might mention here that in this
context, we have another intuitive interpretation of R, in terms of a kind
of second order deviation of small spheres in S from the E^3 area formula
A = 4 \pi r^2.
A more important way in which the Ricci tensor--- rather than its "trace
reverse", the Einstein curvature tensor--- enters directly into gtr is via
some of the most important equations in this subject: the "Raychaudhuri
equation", and its close relatives, the "optical equations".
Unfortunately, despite their importance for many purposes, these are not
even mentioned in most textbooks! But if you read John's essay, you've
already encountered the basic idea: the quadratic form R_(ab) U^a U^b,
where now U is a unit timelike vector in the spacetime itself, determines
the volume decrease of a small sphere of initially static test particles
due to the presence of mass-energy inside the sphere. The Raychaudhuri
equation generalizes this to allow for the possibility that the cloud of
test particles is, at time zero, rotating or shearing or already
expanding/contracting.
The optical equations use R_(ab) K^a K^b, where now K is a -null vector-.
Here the question is: how does the area and shape of a narrow pencil of
light rays evolve over time, due if you will to "light bending" effects
caused by the curvature of spacetime? Here too, the Ricci tensor plays
the role of determining a "contraction", which here effects (all other
things being equal) a kind of "focusing". This may help explain why some
of the "energy conditions" involve R_(ab) U^a U^b or R_(ab) K^a K^b.
There's more, for example the Ricci tensor appears in an important general
"decomposition" of the Riemann tensor into "conformal", "scalar" and
"Ricci" parts. But this post is already too long!
At this point, I could append the computation of the curvature scalar via
the method of Cartan. However, I'm running out of time and space, and
I've cranked through very similar computations in this group in the past,
so I'll just suggest that interested readers search for those posts, or
consult the following references:
For intrinsic/extrinsic curvature of spatial hyperslices, see at least the
pictures in the classic
author = {Charles W. Misner and Kip S. Thorne and John Archibald Wheeler},
title = {Gravitation},
publisher = {W. H. Freeman},
year = 1970}
and then try
author = {Hans Stephani},
title = {General Relativity: An Introduction of the Theory of the
Gravitational Field},
publisher = {Cambridge University Press},
edition = {Second},
note = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},
year = 1990}
MTW also discuss Cartan's slick way of computing the curvature tensors,
which employs his "exterior calculus", via connection one-forms and
curvature two-forms.
For Raychaudhuri, see Stephani or the classic monograph
author = {S. W. Hawking and G. F. R. Ellis},
title = {The Large Scale Structure of Space-Time},
publisher = {Cambridge University Press},
year = 1973}
HE also discuss the optical equations, but I prefer the discussion in
author = {P.J.E. Peebles},
title = {Principles of physical cosmology},
publisher = {Princeton University Press},
year = 1993}
For totally geodesic hyperslices and more on intrinsic/extrinsic
curvatures, see
author = {Theodore Frankel},
title = {Gravitational Curvature: an Introduction to {E}instein's Theory},
publisher = {W. H. Freeman},
year = 1979}
For frames and coframes, try
author = {F. de Felice and C.J.S. Clarke},
title = {Relativity on Curved Manifolds},
publisher = {Cambridge University Press},
year = 1990}
Last but not least, the undergraduate textbook
author = {Antal Fekete},
title = {Real Linear Algebra},
publisher = {Marcel Dekker},
year = 1985}
offers much more valuable intuition into the basic notions of linear
algebra, including shears, dilations, and many other things I and others
often refer to here, and I highly recommend it. For more intuition for
the coefficients of the characteristic polynomial which appear between
trace and determinant, see the discussion of slices of "energy ellipsoids"
in
author = {Diestel, Reinhard},
title = {Graph Theory},
publisher = {Springer},
series = {Graduate texts in mathematics},
volume = 173,
year = 2000}
author = {Bollob\'as, B\'ela},
title = {Modern Graph Theory},
series = {Graduate texts in mathematics},
volume = 184,
publisher = {Springer-Verlag},
year = 1998}
"T. Essel" (hiding somewhere in cyberspace)