GRAVITATION & DARK ENERGY
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Part One (of 3) : Einstein’s Theory of General Relativity … is one of the greatest intellectual achievements of the human race. Let’s see how easy it was for Albert Einstein to discover General Relativity. Einstein’s truly brilliant idea was that the presence of mass-‐energy (e.g., our massive Sun) somehow produces curvature of spacetime, which results in planets moving on curved paths around the Sun—instead of the planets simply moving at constant speed in straight lines (as Newton said happens if no “force” exists). That is, Einstein postulated that:
Curvature of Spacetime = Mass-‐Energy
Mass-‐Energy ! Tµ! which is the energy-‐momentum tensor, with 4 ! 4 =16 components
Example: for a perfect gas (e.g., the “gas” of galaxies in the universe)
Tµ! =
p 0 0 00 p 0 00 0 p 00 0 0 !"c2
"
#
$$$$$
%
&
'''''
Now, what do we place on the “curvature of spacetime” side of our equation? Fortunately, Einstein had available the work of the brilliant mathematicians Riemann and Ricci. Riemann characterized geometrical curvature as R!
"# $ which has 4 ! 4 ! 4 ! 4 = 256 components! We can’t set it equal to Tµ
! because the latter has only 4 ! 4 =16 components. But! Ricci to the rescue: the purely mathematical operation of contraction gives us: R!
"!# = R"# where the right-‐hand side is named the Ricci tensor. It has only TWO indices, so we are in business! Writing it in mixed covariant-‐contravariant form (like Tµ
! ) gives us
Einstein’s First Guess: Rµ! = G
c4
p 0 0 00 p 0 00 0 p 00 0 0 !"c2
"
#
$$$$$
%
&
'''''
(G is Newton’s G, the c4 fixes units)
Looks good, but there is a problem: mass-‐energy is conserved, but the Ricci tensor is not. “Conserved” means that its derivative is zero. So, Einstein needed to patch the left-‐hand side so that its derivative would also be zero. Hey, it looks like a kluge, but here it is:
where R = R!! is the contraction of Ricci’s tensor, and
gµ! is the metric tensor (just 1’s on the diagonal—
trivial). Since we are requiring derivatives to be zero, naturally we can toss in an arbitrary constant, ! (the famous Cosmological Constant): and we have done that!
If we also stick in the 8! —we’ve done that, too—our equation easily reduces, in the first approximation (and ignoring Λ) to Newton’s Law of gravity! We have derived the…
Field Equations of General Relativity!
Rµ! " 12R gµ
!#$%
&'( + )gµ
! = 8* Gc4Tµ
!#$%
&'(
where p is the pressure (for galaxies, ~ zero), and ρ is the density. (Isaac Newton did not know Special Relativity, and thus did not know that p, tiny as it is, has to be included. So, Newton’s law of gravity is only one equation (dealing with the –ρc2 ) and not the 16 equations (well, in this simple example, two independent equations) of General Relativity!)
GRAVITATION & DARK ENERGY
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Part Two (of 3) : The Cosmological Constant: Dark Energy , and the cosmological constant !gµ
!must be structured like
Tµ! =
p 0 0 00 p 0 00 0 p 00 0 0 !"c2
"
#
$$$$$
%
&
'''''
Well, we will do what Einstein did, and consider the possibility that the vacuum is not vacuous, as we’d always thought, but rather has some constant mass density !V (subscript V means vacuum). A simple thought experiment then gives us the pressure pVfor the vacuum:
Suppose the above were a conventional piston-‐and-‐cylinder, with gas in it, not vacuum, and we added heat !Q . There would be an increase in the temperature (that is, in the energy E) of the gas, and the piston would move. Conservation of energy, of course, applies:
!Q = !E + p!V , where V is the volume. This is also called the first law of thermodynamics, but it is just conservation of energy. p!V is the work done as the piston moves (high school physics). Now apply the equation to our thought experiment above, for the case !Q = 0 , that is, no heat added. Then we have
0 = !Vc2!V + pV!V
Any outward motion of the piston (“expansion of the universe”) spreads further apart any galaxies that are in the cylinder (thus reducing their mutual gravitational attraction) but (by Einstein’s remarkable hypothesis) does not diminish the vacuum’s mass density !V at any location. !V!V is the change in the mass contained in the cylinder (remember that density is mass divided by volume V). OK! Cancel the !V 's and we get our answer:
pV = !!Vc2 (which is negative; and which is enormous, because of the c2 ) and
!gµ! = ""Vc
2
1 0 0 00 1 0 00 0 1 00 0 0 1
#
$
%%%%
&
'
((((
= ""Vc2gµ
!
Rµ! " 12R gµ
! + #gµ! = 8$G
c4Tµ
!
nothing
(not even vacuum)
vacuum
V p
V
except that, since we want to represent a cosmological constant, all four entries on the diagonal must be identical, and must be constant. What does that take?
Dark Energy! We’ll see that it repels, instead of attracting!
GRAVITATION & DARK ENERGY
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Part Three (of 3) : Finally, let us Compare with What Newton Thought We want to compare our result with what Newton would have asserted, using his
F = ma ! m !!R( ) and his Law of Gravity, F = !GMmR2
(the minus sign indicating an attractive
force). Our galaxy, having mass m, is attracted by everything in an arbitrary huge sphere, of radius R, containing total mass M. The density for that vast sphere is, of course
Combining our 3 “Newton equations” gives us:
It is negative! The expansion of the Universe is decelerating! But, what does Einstein say?
The above is a compact way of writing 16 separate equations (μ and ν each go from 1 to 4, since there are 4 dimensions: 3 space, and 1 time). You saw that for a gas of galaxies, only the 4 diagonal elements of the energy-‐momentum tensor were non-‐zero (and also that 3 elements were the same, p). All 4 diagonal elements are 1’s for the metric tensor. As for the Ricci tensor, if you assume the simplest possible isotropic geometry (called Robertson-‐Walker) for the expanding universe (such simplicity is strongly supported by observations of the famous 3K background radiation), the first three elements are again the same. We’ll use RW for a flat universe, since observations indicate that our universe is flat, to a high degree. So, instead of the fancy equation above, with its raised and lowered indices (the reason for “raised/lowered” is too boring to discuss, and we don’t need it anyway), our fancy equation reduces to just two equations, both at high-‐school-‐level:
!!R2
R2+ 2!!RR
= 8!Gc2
p and !R2
R2= 8!G
3"
The first equation, the one with p on the right, occurs 3 times—but of course, we don’t need 3 copies! We have omitted Λ—we will see it appear as dark energy, from Einstein’s ρV idea (and his deduction that pV = !!Vc
2 ), so that ! and p each have two contributors: 1) matter M (including dark matter), and 2) dark energy (subscript V). So ! = !M + !V and p = pM + pV ! pV (since the pressure of the galaxies, as well as of the only-‐weakly-‐interacting dark matter, is ~ zero, in drastic contrast to the enormous pressure of the dark energy). In our equations, R is an arbitrary distance proportional to the size of the universe at any time, R with a dot on it (Isaac Newton’s notation) is the universe’s expansion speed at that time, and R with two dots on it is the acceleration of the expansion of the universe—which is what interests us most. Now eliminate !R from our two equations (easy). Result:
If we ignore the dark energy, the first part gives our negative Newtonian result! But, if the universe has expanded enough (diminishing ρM to the point that it is negligible, as in our present universe) we get our final expression above for the acceleration of the universe:
—which is positive! So, instead of living in a universe that is decelerating (negative !!R ), we find ourselves in a universe that is accelerating its expansion! So, Dark Energy rules!
acceleration = !!R = ! 4"
3G#M R
Rµ! " 12R gµ
! + #gµ! = 8$G
c4Tµ
!
acceleration = !!R = ! 4"
3c2G #Mc
2 + 3pV + #Vc2( )R = ! 4"
3c2G #Mc
2 ! 2#Vc2( )R $ + 8"
3G#V R = !!R
!M = M4"3R3.
R M
m
GRAVITATION & DARK ENERGY
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… but, in the vacuum, and with cosmological constant zero, these
Field Equations of General Relativity reduce to
The photo shows Albert Einstein writing this equation (in covariant form; that is, with two subscripts). Einstein followed his equation with a question mark!
Anyone reading my four-‐page account might wonder why, General Relativity being so simple and so apparently arbitrary (especially regarding dark energy), it is held in such high esteem. It is because this theory makes quantitative predictions, for example of the evolution of the orbits of cosmic binary neutron stars, that have always turned out to be highly accurate! The Global Positioning System (GPS) could not function without our use of General Relativity; Special Relativity (has time as the 4th dimension, but: with a MINUS sign in the Pythagorean Theorem) led directly to atomic power, and to the hydrogen bomb.
Rµ! " 12R gµ
! + #gµ! = 8$G
c4Tµ
!
Rµ! = 0