The document discusses the properties of static stellar models described by Einstein's equations. It defines key terms like the equation of state, adiabatic index, pressure, and density. It describes the conditions imposed on the metric, pressure, and density functions. The boundary of the star is defined as the surface where pressure goes to zero, and must be a smooth surface. Outside the star, the metric and potential are required to approach the Schwarzschild solution asymptotically.

1. Positive mass theorem and adiabatic
index below 6/5 in a static star
A. K. M. Masood-ul-Alam
abulm@math.tsinghua.edu.cn
Mathematical Sciences Center & Department of Mathematical Sciences
Tsinghua University, Beijing 100084, China
Done with Prosper – p. 1/45

2. Pressure, Density, Adiabatic Index
Equation of state: a function of one variable: ρ = ρ(p)
Done with Prosper – p. 2/45

3. Pressure, Density, Adiabatic Index
Equation of state: a function of one variable: ρ = ρ(p)
ρ + p dp
Adiabatic index: γ =
p dρ
Done with Prosper – p. 2/45

4. Pressure, Density, Adiabatic Index
Equation of state: a function of one variable: ρ = ρ(p)
ρ + p dp
Adiabatic index: γ =
p dρ
p, called pressure, is usually small compared to ρ,
called density or energy density or mass-energy ensity
or total mass-energy density including internal energy.
Done with Prosper – p. 2/45

5. Pressure, Density, Adiabatic Index
Equation of state: a function of one variable: ρ = ρ(p)
ρ + p dp
Adiabatic index: γ =
p dρ
p, called pressure, is usually small compared to ρ,
called density or energy density or mass-energy ensity
or total mass-energy density including internal energy.
p
We use geometrized units c = G = 1. p was 2 where c
c
is the speed of light.
Done with Prosper – p. 2/45

6. Pressure, Density, Adiabatic Index
Equation of state: a function of one variable: ρ = ρ(p)
ρ + p dp
Adiabatic index: γ =
p dρ
p, called pressure, is usually small compared to ρ,
called density or energy density or mass-energy ensity
or total mass-energy density including internal energy.
p
We use geometrized units c = G = 1. p was 2 where c
c
is the speed of light.
Ex. If pressure is 1030 dynes/cm2 corresponding to a
density 3.2 × 1011 gm/cm3 , then with
c = 3 × 1010 cm/sec2 , p = 0.0035. Recall
c2 ρ
1 dynes/cm2 = 1 g/(cm sec2 ) ≈ 0.9872 × 10−6 atm.
Done with Prosper – p. 2/45

7. History of the name, Small p/ρ
If we neglect small p/ρ, then
ρ + p dp p d log p
γ= = (1 + )
p dρ ρ d log ρ
coincides with the adiabatic index of perfect gas
ρ dp
γ= .
p dρ
Done with Prosper – p. 3/45

8. History of the name, Small p/ρ
If we neglect small p/ρ, then
ρ + p dp p d log p
γ= = (1 + )
p dρ ρ d log ρ
coincides with the adiabatic index of perfect gas
ρ dp
γ= .
p dρ
The name originated from Poisson adiabatic eqns
P V γ = constant, P 1−γ T γ = constant, T V γ−1 = constant,
where γ is the ratio of specific heats cp /cv . More
generally polytropic changes (quasi-static process in
which dQ/dT = c = constant during the entire process)
satisfy above equations with γ replaced polytropic
cp − c
exponent γ = .
cv − c Done with Prosper – p. 3/45

9. Conditions on ρ and p
Differentiability Conditions: ρ(p) is a piecewise C 1 on
(0, pc ).
We call pc the central pressure. On (0, pc )
Done with Prosper – p. 4/45

10. Conditions on ρ and p
Differentiability Conditions: ρ(p) is a piecewise C 1 on
(0, pc ).
We call pc the central pressure. On (0, pc )
dp
ρ > 0 and > 0.
dρ
Done with Prosper – p. 4/45

11. Conditions on ρ and p
Differentiability Conditions: ρ(p) is a piecewise C 1 on
(0, pc ).
We call pc the central pressure. On (0, pc )
dp
ρ > 0 and > 0.
dρ
dp
For a polytropic gas is the velocity of sound.
dρ
Done with Prosper – p. 4/45

12. Conditions on ρ and p
Differentiability Conditions: ρ(p) is a piecewise C 1 on
(0, pc ).
We call pc the central pressure. On (0, pc )
dp
ρ > 0 and > 0.
dρ
dp
For a polytropic gas is the velocity of sound.
dρ
However constant density (incompressible fluid!) is
dρ
allowed: = 0. Real motivation: history based on
dp
stability considerations.
Done with Prosper – p. 4/45

13. Conditions on ρ and p
Differentiability Conditions: ρ(p) is a piecewise C 1 on
(0, pc ).
We call pc the central pressure. On (0, pc )
dp
ρ > 0 and > 0.
dρ
dp
For a polytropic gas is the velocity of sound.
dρ
However constant density (incompressible fluid!) is
dρ
allowed: = 0. Real motivation: history based on
dp
stability considerations.
lim inf γ(p) = lim sup γ(p).
p↓0 +
p↓0+
Done with Prosper – p. 4/45

14. log p and γ v. log ρ
(1)
(2)
Bigger
Done with Prosper – p. 5/45

15. Static stellar model
Σ is a manifold diffeomorphic to R3 .
Done with Prosper – p. 6/45

16. Static stellar model
Σ is a manifold diffeomorphic to R3 .
p is now a function on Σ.
Done with Prosper – p. 6/45

17. Static stellar model
Σ is a manifold diffeomorphic to R3 .
p is now a function on Σ.
p vanishes outside a compact set in Σ. This outside is
the vacuum and the inside, where p > 0 is the star or
fluid region.
Done with Prosper – p. 6/45

18. Static stellar model
Σ is a manifold diffeomorphic to R3 .
p is now a function on Σ.
p vanishes outside a compact set in Σ. This outside is
the vacuum and the inside, where p > 0 is the star or
fluid region.
Assume ρ = 0 outside.
Done with Prosper – p. 6/45

19. Static stellar model: Einstein Equations
g is a Riemannian metric on Σ.
Done with Prosper – p. 7/45

20. Static stellar model: Einstein Equations
g is a Riemannian metric on Σ.
V is a positive function on Σ.
Done with Prosper – p. 7/45

21. Static stellar model: Einstein Equations
g is a Riemannian metric on Σ.
V is a positive function on Σ.
4
Space-time is (Σ × R, g), where
Done with Prosper – p. 7/45

22. Static stellar model: Einstein Equations
g is a Riemannian metric on Σ.
V is a positive function on Σ.
4
Space-time is (Σ × R, g), where
4
g = −V 2 dt2 + g
Done with Prosper – p. 7/45

23. Static stellar model: Einstein Equations
g is a Riemannian metric on Σ.
V is a positive function on Σ.
4
Space-time is (Σ × R, g), where
4
g = −V 2 dt2 + g
4
Since V and g are independent of time t, g is static.
Done with Prosper – p. 7/45

24. Static stellar model: Einstein Equations
g is a Riemannian metric on Σ.
V is a positive function on Σ.
4
Space-time is (Σ × R, g), where
4
g = −V 2 dt2 + g
4
Since V and g are independent of time t, g is static.
4
Space-time metric g satisfies Einstein equations
4 1 4 4
Ric(g)αβ − Scalar(g)g αβ = 8πTαβ
2
4
with energy-momentum tensor Tαβ = (ρ + p)uα uβ + pg αβ
Done with Prosper – p. 7/45

25. Static stellar model – cont.
uα four dimensional velocity of the fluid.
Done with Prosper – p. 8/45

26. Static stellar model – cont.
uα four dimensional velocity of the fluid.
∂
Static fluid: uα = V −1 unit timelike vector.
∂t
Done with Prosper – p. 8/45

27. Static stellar model – cont.
uα four dimensional velocity of the fluid.
∂
Static fluid: uα =V −1 unit timelike vector.
∂t
Σ, 3-space at any one time, is the rest frame of the fluid.
Done with Prosper – p. 8/45

28. On (Σ, g)
V and g then satisfy the equations
Rij = V −1 i j V + 4π(ρ − p)gij
∆V = 4πV (ρ + 3p)
Done with Prosper – p. 9/45

29. On (Σ, g)
V and g then satisfy the equations
Rij = V −1 i j V + 4π(ρ − p)gij
∆V = 4πV (ρ + 3p)
Rij is the Ricci curvature of g .
Done with Prosper – p. 9/45

30. On (Σ, g)
V and g then satisfy the equations
Rij = V −1 i j V + 4π(ρ − p)gij
∆V = 4πV (ρ + 3p)
Rij is the Ricci curvature of g .
idenotes covariant derivative and
i ij ∂2 k ∂ denotes the Laplacian
∆= i =g i ∂xj
−Γ
∂x ∂xk
relative to g .
Done with Prosper – p. 9/45

31. Surface of the Star
p = 0 on the surface of the star.
Done with Prosper – p. 10/45

32. Surface of the Star
p = 0 on the surface of the star.
Assume that the surface is smooth.
Done with Prosper – p. 10/45

33. Surface of the Star
p = 0 on the surface of the star.
Assume that the surface is smooth.
V and g are C 3 except at the surface of the star (and
possibly at several smooth V = const surfaces inside
the star).
Done with Prosper – p. 10/45

34. Surface of the Star
p = 0 on the surface of the star.
Assume that the surface is smooth.
V and g are C 3 except at the surface of the star (and
possibly at several smooth V = const surfaces inside
the star).
V and g are globally C 1,1 .
Done with Prosper – p. 10/45

35. Surface of the Star
The contracted Bianchi identity for g implies that p is a
Lipschitz function on Σ and (almost everywhere)
ip = −V −1 (ρ + p) iV
Done with Prosper – p. 11/45

36. Surface of the Star
The contracted Bianchi identity for g implies that p is a
Lipschitz function on Σ and (almost everywhere)
ip = −V −1 (ρ + p) iV
If ρ(p) is a Lipschitz function of p this implies
dρ i
∆p + 2 + V −1 V ip = −4π(ρ + p)(ρ + 3p) ≤ 0
dp
Done with Prosper – p. 11/45

37. Surface of the Star
The contracted Bianchi identity for g implies that p is a
Lipschitz function on Σ and (almost everywhere)
ip = −V −1 (ρ + p) iV
If ρ(p) is a Lipschitz function of p this implies
dρ i
∆p + 2 + V −1 V ip = −4π(ρ + p)(ρ + 3p) ≤ 0
dp
Thus if ρ(p) is a Lipschitz function of p then p vanishes
identically.
Done with Prosper – p. 11/45

38. Surface of the Star
The contracted Bianchi identity for g implies that p is a
Lipschitz function on Σ and (almost everywhere)
ip = −V −1 (ρ + p) iV
If ρ(p) is a Lipschitz function of p this implies
dρ i
∆p + 2 + V −1 V ip = −4π(ρ + p)(ρ + 3p) ≤ 0
dp
Thus if ρ(p) is a Lipschitz function of p then p vanishes
identically.
So for such an equation of state ρ has a jump
discontinuity across the surface of the star and V and g
can only be C 1,1 across the surface of the star.
Done with Prosper – p. 11/45

39. p is a Function of V
We write i p = −V −1 (ρ + p) iV in the form
dp
= −V −1 (ρ + p)
dV
Done with Prosper – p. 12/45

40. p is a Function of V
We write i p = −V −1 (ρ + p) iV in the form
dp
= −V −1 (ρ + p)
dV
Thus the boundary of the star is a level set of V
henceforth dentoted by V = VS .
Done with Prosper – p. 12/45

41. p is a Function of V
We write i p = −V −1 (ρ + p) iV in the form
dp
= −V −1 (ρ + p)
dV
Thus the boundary of the star is a level set of V
henceforth dentoted by V = VS .
Since we assumed ρ to be a piecewise C 1 function of p
for p > 0,
Done with Prosper – p. 12/45

42. p is a Function of V
We write i p = −V −1 (ρ + p) iV in the form
dp
= −V −1 (ρ + p)
dV
Thus the boundary of the star is a level set of V
henceforth dentoted by V = VS .
Since we assumed ρ to be a piecewise C 1 function of p
for p > 0,
ρ = ρ(V ) is also a piecewise C 1 function of V on [Vc , VS ].
Vc is the minimum value of V on Σ.
Done with Prosper – p. 12/45

43. Asymptotic Conditions
Outside a compact set Σ is diffeomorphic to a R3 minus
a ball, and w.r.t. to the standard coordinate system in
R3 we have
M
V =1− + O(r−2 )
r
2M
gab = 1+ δab + O(r−2 )
r
Done with Prosper – p. 13/45

44. Asymptotic Conditions
Outside a compact set Σ is diffeomorphic to a R3 minus
a ball, and w.r.t. to the standard coordinate system in
R3 we have
M
V =1− + O(r−2 )
r
2M
gab = 1+ δab + O(r−2 )
r
δab is the Euclidean metric, and r is the spherical
coordinate associated with δab .
Done with Prosper – p. 13/45

45. Asymptotic Conditions
Outside a compact set Σ is diffeomorphic to a R3 minus
a ball, and w.r.t. to the standard coordinate system in
R3 we have
M
V =1− + O(r−2 )
r
2M
gab = 1+ δab + O(r−2 )
r
δab is the Euclidean metric, and r is the spherical
coordinate associated with δab .
The constant M is the mass of the 3-metric g.
Done with Prosper – p. 13/45

46. Asymptotic Conditions
Outside a compact set Σ is diffeomorphic to a R3 minus
a ball, and w.r.t. to the standard coordinate system in
R3 we have
M
V =1− + O(r−2 )
r
2M
gab = 1+ δab + O(r−2 )
r
δab is the Euclidean metric, and r is the spherical
coordinate associated with δab .
The constant M is the mass of the 3-metric g.
Above decay conditions follow from weaker conditions
on V and g. Physical motivation: finite energy static star.
Done with Prosper – p. 13/45

47. Impossibilty of γ ≤ 6/5 Everywhere
1
Theorem [Lindblom, Masood-ul-Alam] Assume is
γ(p)
bounded as p → 0+ . Then γ must satisfy the inequality
2
6 p 6
γ> 1+ ≥
5 ρ 5
at some point inside a finite star.
Done with Prosper – p. 14/45

48. Impossibilty of γ ≤ 6/5 Everywhere
1
Theorem [Lindblom, Masood-ul-Alam] Assume is
γ(p)
bounded as p → 0+ . Then γ must satisfy the inequality
2
6 p 6
γ> 1+ ≥
5 ρ 5
at some point inside a finite star.
Limits on the adiabatic index in static stellar models, in
B.L. Hu and T. Jacobson ed. Directions in General Relativity,
Cambridge University Press (1993).
Done with Prosper – p. 14/45

49. 1/γ(p) is bounded
1
is bounded if we exclude indefinite oscillations of γ
γ(p)
near p = 0. We also assume that ρ = ρ(p) is a
non-decreasing C 1 function in some 0 < p < .
Done with Prosper – p. 15/45

50. 1/γ(p) is bounded
1
is bounded if we exclude indefinite oscillations of γ
γ(p)
near p = 0. We also assume that ρ = ρ(p) is a
non-decreasing C 1 function in some 0 < p < .
Theorem [Lindblom, Masood-ul-Alam] γ(p) > 1 at some
point in every open interval (0, ).
Done with Prosper – p. 15/45

51. 1/γ(p) is bounded
1
is bounded if we exclude indefinite oscillations of γ
γ(p)
near p = 0. We also assume that ρ = ρ(p) is a
non-decreasing C 1 function in some 0 < p < .
Theorem [Lindblom, Masood-ul-Alam] γ(p) > 1 at some
point in every open interval (0, ).
In particular if the equation of state is nice enough such
1
that lim inf γ(p) = lim sup γ(p) then is bounded on
p↓0 +
p↓0+ γ(p)
(0, pc ].
Done with Prosper – p. 15/45

52. History
2
6 p
The "γ > 1+ somewhere theorem" was
5 ρ
motivated by Beig and Simon’s observation that the
2
6 p
equality everywhere γ = 1+ gives
5 ρ
1 1/5
p(ρ) = (ρ0 − ρ1/5 ) and leads to the uniqueness of
6
Buchdahl’s solutions. In these solutions and in
corresponding Newtonian theory of polytropic fluids
ρ = Apn/n+1 , n = constant ≥ 5 the fluid extends to
infinity.
Done with Prosper – p. 16/45

53. History
2
6 p
The "γ > 1+ somewhere theorem" was
5 ρ
motivated by Beig and Simon’s observation that the
2
6 p
equality everywhere γ = 1+ gives
5 ρ
1 1/5
p(ρ) = (ρ0 − ρ1/5 ) and leads to the uniqueness of
6
Buchdahl’s solutions. In these solutions and in
corresponding Newtonian theory of polytropic fluids
ρ = Apn/n+1 , n = constant ≥ 5 the fluid extends to
infinity.
Beig, R., Simon, W.: On the Spherical Symmetry of
Static Perfect Fluids in General Relativity, Lett. Math.
Phys. 21: 245-250, 1991
Done with Prosper – p. 16/45

54. History cont.
Beig and Simon was in turn inspired by my use of the
2
6 p
condition γ ≥ 1+ for deriving spherical
5 ρ
symmetry of static stars using the positive mass
theorem.
Done with Prosper – p. 17/45

55. History cont.
Beig and Simon was in turn inspired by my use of the
2
6 p
condition γ ≥ 1+ for deriving spherical
5 ρ
symmetry of static stars using the positive mass
theorem.
Masood-ul-Alam: A proof of the uniqueness of static
stellar models with small dρ/dp, Classical Quantum
Gravity 5, 409-491, 1988
Done with Prosper – p. 17/45

56. History cont.
Beig and Simon was in turn inspired by my use of the
2
6 p
condition γ ≥ 1+ for deriving spherical
5 ρ
symmetry of static stars using the positive mass
theorem.
Masood-ul-Alam: A proof of the uniqueness of static
stellar models with small dρ/dp, Classical Quantum
Gravity 5, 409-491, 1988
However the condition was known earlier. It occurred in
a series expansion near the center of the star in the
study of stellar stability.
Done with Prosper – p. 17/45

57. History cont.
In relation to the positive mass theorem the condition
arose when we demanded
Done with Prosper – p. 18/45

58. History cont.
In relation to the positive mass theorem the condition
arose when we demanded
d2 ψ
2
≥0
dV
Done with Prosper – p. 18/45

59. History cont.
In relation to the positive mass theorem the condition
arose when we demanded
d2 ψ
2
≥0
dV
so that
˜ d2 ψ
Scalar(ψ 4 g) = 8ψ −5 (W − W ) 2 ≥ 0
dV
Done with Prosper – p. 18/45

60. History cont.
In relation to the positive mass theorem the condition
arose when we demanded
d2 ψ
2
≥0
dV
so that
˜ d2 ψ
Scalar(ψ 4 g) = 8ψ −5 (W − W ) 2 ≥ 0
dV
˜
if W − W ≥ 0.
Done with Prosper – p. 18/45

61. History cont.
In relation to the positive mass theorem the condition
arose when we demanded
d2 ψ
2
≥0
dV
so that
˜ d2 ψ
Scalar(ψ 4 g) = 8ψ −5 (W − W ) 2 ≥ 0
dV
˜
if W − W ≥ 0.
˜ ˜
Here W = | V |2 and W = W (V ) is related to the
conformal function (more later).
Done with Prosper – p. 18/45

62. 2
6
Proof of γ>
5
1+
p
ρ
somewhere Thm
Consider the conformal metric ψ 4 g where

 1 VS
 (1 + V )exp − VS
 ρ(v)dv
S
ψ(V ) = 2 1 + VS V v(ρ(v) + 3p(v))
 1

 (1 + V ) in vacuum
2
Done with Prosper – p. 19/45

63. 2
6
Proof of γ>
5
1+
p
ρ
somewhere Thm
Consider the conformal metric ψ 4 g where

 1 VS
 (1 + V )exp − VS
 ρ(v)dv
S
ψ(V ) = 2 1 + VS V v(ρ(v) + 3p(v))
 1

 (1 + V ) in vacuum
2
Scalar curvature R of Ψ4 g vanishes in the vacuum.
Done with Prosper – p. 19/45

64. 2
6
Proof of γ>
5
1+
p
ρ
somewhere Thm
Consider the conformal metric ψ 4 g where

 1 VS
 (1 + V )exp − VS
 ρ(v)dv
S
ψ(V ) = 2 1 + VS V v(ρ(v) + 3p(v))
 1

 (1 + V ) in vacuum
2
Scalar curvature R of Ψ4 g vanishes in the vacuum.
Inside the star
Done with Prosper – p. 19/45

65. 4
Proof cont.: Scalar curvature of Ψ g
The scalar curvature R of Ψ4 g satisfies
Done with Prosper – p. 20/45

66. 4
Proof cont.: Scalar curvature of Ψ g
The scalar curvature R of Ψ4 g satisfies
Ψ4 (1 + VS )R = 16πρ(1 − VS )+
2
8ρ2 V S| V |2 p 2 + 3VS
3 1+ −γ
γV 2 (ρ + 3p)2 ρ 1 + VS
which is non-negative whenever
Done with Prosper – p. 20/45

67. 4
Proof cont.: Scalar curvature of Ψ g
The scalar curvature R of Ψ4 g satisfies
Ψ4 (1 + VS )R = 16πρ(1 − VS )+
2
8ρ2 V S| V |2 p 2 + 3VS
3 1+ −γ
γV 2 (ρ + 3p)2 ρ 1 + VS
which is non-negative whenever
2
p 1 + VS
γ ≤3 1+
ρ 2 + 3VS
Done with Prosper – p. 20/45

68. 4
Proof cont.: Scalar curvature of Ψ g
The scalar curvature R of Ψ4 g satisfies
Ψ4 (1 + VS )R = 16πρ(1 − VS )+
2
8ρ2 V S| V |2 p 2 + 3VS
3 1+ −γ
γV 2 (ρ + 3p)2 ρ 1 + VS
which is non-negative whenever
2
p 1 + VS
γ ≤3 1+
ρ 2 + 3VS
2
1 + VS 2 6 p
For VS < 1, > . So R ≥ 0 if γ ≤ 1+
2 + 3VS 5 5 ρ
Done with Prosper – p. 20/45

69. Proof cont.
2
6 p
The mass of Ψ4 gis zero. Thus if γ ≤ 1+
5 ρ
everywhere inside the star then Ψ4 g is Euclidean by the
positive mass theorem:
Done with Prosper – p. 21/45

70. Proof cont.
2
6 p
The mass of Ψ4 gis zero. Thus if γ ≤ 1+
5 ρ
everywhere inside the star then Ψ4 g is Euclidean by the
positive mass theorem:
Theorem [Schoen, Yau] Let (N, η) be a complete oriented
3-dimensional Riemannian manifold. Suppose (N, η) is
asymptotically flat and has non-negative scalar
curvature. Then the mass of (N.η) is non-negative and
if the mass is zero then (N, η) is isometric to R3 with the
standard euclidean metric.
Done with Prosper – p. 21/45

71. Investigating where γ is small
There has been one more paper related to the
2
6 p
"γ > 1+ somewhere theorem."
5 ρ
Done with Prosper – p. 22/45

72. Investigating where γ is small
There has been one more paper related to the
2
6 p
"γ > 1+ somewhere theorem."
5 ρ
Stricter constraint on γ involving Vc and VS has been
derived using the positive mass theorem:
Done with Prosper – p. 22/45

73. Investigating where γ is small
There has been one more paper related to the
2
6 p
"γ > 1+ somewhere theorem."
5 ρ
Stricter constraint on γ involving Vc and VS has been
derived using the positive mass theorem:
Shiromizu, S., Yoshino, H.: Positive Energy Theorem
Implies Constraints on Static Stellar Models,, Prog.
Theor. Phys. 116, No. 6 (2006) 1159-1164.
Done with Prosper – p. 22/45

74. Investigating where γ is small
There has been one more paper related to the
2
6 p
"γ > 1+ somewhere theorem."
5 ρ
Stricter constraint on γ involving Vc and VS has been
derived using the positive mass theorem:
Shiromizu, S., Yoshino, H.: Positive Energy Theorem
Implies Constraints on Static Stellar Models,, Prog.
Theor. Phys. 116, No. 6 (2006) 1159-1164.
However so far nobody investigated the region where γ
is small from the viewpoint of positive mass theorem.
Done with Prosper – p. 22/45

75. Conformal function
To investigate the extent of the region where γ is small
and the limits on it imposed by the positive mass
theorem we need detail analysis of spherically
symmetric equations,
Done with Prosper – p. 23/45

76. Conformal function
To investigate the extent of the region where γ is small
and the limits on it imposed by the positive mass
theorem we need detail analysis of spherically
symmetric equations,
in particular analysis of ψ such that ψ 4 g is Euclidean.
Done with Prosper – p. 23/45

77. Conformal function
To investigate the extent of the region where γ is small
and the limits on it imposed by the positive mass
theorem we need detail analysis of spherically
symmetric equations,
in particular analysis of ψ such that ψ 4 g is Euclidean.
d ˜ ˜
Let W = | V |2 and W = W (V ). Suppose ψ satisfies
d2 ψ 2π dψ
= ρψ − 2V (ρ + 3p)
dV 2
W˜ dV
Done with Prosper – p. 23/45

78. Conformal function
To investigate the extent of the region where γ is small
and the limits on it imposed by the positive mass
theorem we need detail analysis of spherically
symmetric equations,
in particular analysis of ψ such that ψ 4 g is Euclidean.
d ˜ ˜
Let W = | V |2 and W = W (V ). Suppose ψ satisfies
d2 ψ 2π dψ
= ρψ − 2V (ρ + 3p)
dV 2
W˜ dV
Then the scalar curvature R of ψ 4 g satisfies
˜ d2 ψ
R = 8ψ −5 (W − W ) 2
dV
Done with Prosper – p. 23/45

79. Can assume spherical symmetry
˜
Then W = W (V ). Can take W = W to make R zero.
Done with Prosper – p. 24/45

80. Can assume spherical symmetry
˜
Then W = W (V ). Can take W = W to make R zero.
d2 ψ
Knowledge of the sign of 2
at V = k is crucial.
dV
Done with Prosper – p. 24/45

81. Can assume spherical symmetry
˜
Then W = W (V ). Can take W = W to make R zero.
d2 ψ
Knowledge of the sign of 2
at V = k is crucial.
dV
d2 ψ
Lemma At V < VS where 2
= 0 and ψ(V ) is thrice
dV
differentiable, we have
d3 ψ 5πρ2 ψ 6 p 2
= [γ − (1 + ) ]
dV 3
γW˜ V (ρ + 3p) 5 ρ
Done with Prosper – p. 24/45

82. Notes regarding the Lemma
d2 ψ dψ
When 2
= 0, 2V (ρ + 3p) = ρψ
dV dV
Done with Prosper – p. 25/45

83. Notes regarding the Lemma
d2 ψ dψ
When 2
= 0, 2V (ρ + 3p) = ρψ
dV dV
6 p 2 d2 ψ
If γ > (1 + ) near the center then 2
> 0 in a
5 ρ dV
deleted neighborhood of the center. This is because
d2 ψ
2
= 0 at the center of the star. Roughly and
dV
physically speaking "soft center" is not good for the
stability of the star.
Done with Prosper – p. 25/45

84. Notes regarding the Lemma
d2 ψ dψ
When 2
= 0, 2V (ρ + 3p) = ρψ
dV dV
6 p 2 d2 ψ
If γ > (1 + ) near the center then 2
> 0 in a
5 ρ dV
deleted neighborhood of the center. This is because
d2 ψ
2
= 0 at the center of the star. Roughly and
dV
physically speaking "soft center" is not good for the
stability of the star.
d2 ψ
Near the surface of the star 2
> 0. This follows
dV
p
because → 0 as p ↓ 0.
ρ
Done with Prosper – p. 25/45

85. Extent of the region where γ is small
Theorem [Masood-ul-Alam] There cannot be an interval
2
6 p
j < V < k inside a finite star where γ < 1+
5 ρ
Done with Prosper – p. 26/45

86. Extent of the region where γ is small
Theorem [Masood-ul-Alam] There cannot be an interval
2
6 p
j < V < k inside a finite star where γ < 1+
5 ρ
d2 ψ
if 2
≥ 0 at V = k− .
dV
Done with Prosper – p. 26/45

87. Extent of the region where γ is small
Theorem [Masood-ul-Alam] There cannot be an interval
2
6 p
j < V < k inside a finite star where γ < 1+
5 ρ
d2 ψ
if 2
≥ 0 at V = k− .
dV
d2 ψ
if 2
< 0 at V = k− and ... ? ...
dV
Done with Prosper – p. 26/45

88. Extent of the region where γ is small
Theorem [Masood-ul-Alam] There cannot be an interval
2
6 p
j < V < k inside a finite star where γ < 1+
5 ρ
d2 ψ
if 2
≥ 0 at V = k− .
dV
d2 ψ
if 2
< 0 at V = k− and ... ? ...
dV
k− allows ρ to be discontinuous at k.
Done with Prosper – p. 26/45

89. Proof
Choose a conformal function u(V ) on [j, k] that matches
at V = k in a C 1,1 fashion and gives positive scalar
curvature on [j, k] as before. Then if
Done with Prosper – p. 27/45

90. Proof
Choose a conformal function u(V ) on [j, k] that matches
at V = k in a C 1,1 fashion and gives positive scalar
curvature on [j, k] as before. Then if
d ln u
0 < 2j < 1,
dV V =j+
we can construct portion of a new star solving the
spherically symmetric equations with appopriate initial
˜
conditions at V = j. The new pair (W , ψ) makes the
metric ψ 4 g scalar flat in this portion.
Done with Prosper – p. 27/45

91. Proof
Choose a conformal function u(V ) on [j, k] that matches
at V = k in a C 1,1 fashion and gives positive scalar
curvature on [j, k] as before. Then if
d ln u
0 < 2j < 1,
dV V =j+
we can construct portion of a new star solving the
spherically symmetric equations with appopriate initial
˜
conditions at V = j. The new pair (W , ψ) makes the
metric ψ 4 g scalar flat in this portion.
d2 ψ
If 2
≥ 0 at V = k− , the above condition is satisfied.
dV
Done with Prosper – p. 27/45

92. Proof
Choose a conformal function u(V ) on [j, k] that matches
at V = k in a C 1,1 fashion and gives positive scalar
curvature on [j, k] as before. Then if
d ln u
0 < 2j < 1,
dV V =j+
we can construct portion of a new star solving the
spherically symmetric equations with appopriate initial
˜
conditions at V = j. The new pair (W , ψ) makes the
metric ψ 4 g scalar flat in this portion.
d2 ψ
If 2
≥ 0 at V = k− , the above condition is satisfied.
dV
Done with Prosper – p. 27/45

93. Proof
Choose a conformal function u(V ) on [j, k] that matches
at V = k in a C 1,1 fashion and gives positive scalar
curvature on [j, k] as before. Then if
d ln u
0 < 2j < 1,
dV V =j+
we can construct portion of a new star solving the
spherically symmetric equations with appopriate initial
˜
conditions at V = j. The new pair (W , ψ) makes the
metric ψ 4 g scalar flat in this portion.
d2 ψ
If 2
≥ 0 at V = k− , the above condition is satisfied.
dV
Done with Prosper – p. 27/45

94. Proof
Choose a conformal function u(V ) on [j, k] that matches
at V = k in a C 1,1 fashion and gives positive scalar
curvature on [j, k] as before. Then if
d ln u
0 < 2j < 1,
dV V =j+
we can construct portion of a new star solving the
spherically symmetric equations with appopriate initial
˜
conditions at V = j. The new pair (W , ψ) makes the
metric ψ 4 g scalar flat in this portion.
d2 ψ
If 2
≥ 0 at V = k− , the above condition is satisfied.
dV
Done with Prosper – p. 27/45

95. Proof cont.: Conformal functions on [j, k]
k ρ(s)ds
−α
u(V ) = ψ(k)e V 2s(ρ(s) + 3p(s))
Done with Prosper – p. 28/45

96. Proof cont.: Conformal functions on [j, k]
k ρ(s)ds
−α
u(V ) = ψ(k)e V 2s(ρ(s) + 3p(s))
3p(k) d ln ψ
with constant α = 2k 1 +
ρ(k− ) dV
V =k
Done with Prosper – p. 28/45

97. Proof cont.: Conformal functions on [j, k]
k ρ(s)ds
−α
u(V ) = ψ(k)e V 2s(ρ(s) + 3p(s))
3p(k) d ln ψ
with constant α = 2k 1 +
ρ(k− ) dV
V =k
u satisfies
d ln u αρ
2V =
dV ρ + 3p
Done with Prosper – p. 28/45

98. Proof cont.: Conformal functions on [j, k]
k ρ(s)ds
−α
u(V ) = ψ(k)e V 2s(ρ(s) + 3p(s))
3p(k) d ln ψ
with constant α = 2k 1 +
ρ(k− ) dV
V =k
u satisfies
d ln u αρ
2V =
dV ρ + 3p
u matches with ψ at V = k in a C 1,1 fashion.
Done with Prosper – p. 28/45

99. Proof cont.: Conformal functions on [j, k]
k ρ(s)ds
−α
u(V ) = ψ(k)e V 2s(ρ(s) + 3p(s))
3p(k) d ln ψ
with constant α = 2k 1 +
ρ(k− ) dV
V =k
u satisfies
d ln u αρ
2V =
dV ρ + 3p
u matches with ψ at V = k in a C 1,1 fashion.
d ln u
α ≤ 1 =⇒ 0 < 2j <1
dV V =j+
Done with Prosper – p. 28/45

100. Proof cont.:Conformal functions on [j, k]
d2 ψ 2π dψ
Recall, = ρψ − 2V (ρ + 3p)
dV 2
W˜ dV
2πρψ 3p d ln ψ
= 1 − 2V 1+
W˜ ρ dV
Done with Prosper – p. 29/45

101. Proof cont.:Conformal functions on [j, k]
d2 ψ 2π dψ
Recall, = ρψ − 2V (ρ + 3p)
dV 2
W˜ dV
2πρψ 3p d ln ψ
= 1 − 2V 1+
W˜ ρ dV
d2 ψ 2π ψρ−
= (1 − α)
dV 2 W˜
V =k−
Done with Prosper – p. 29/45

102. Proof cont.:Conformal functions on [j, k]
d2 ψ 2π dψ
Recall, = ρψ − 2V (ρ + 3p)
dV 2
W˜ dV
2πρψ 3p d ln ψ
= 1 − 2V 1+
W˜ ρ dV
d2 ψ 2π ψρ−
= (1 − α)
dV 2 W˜
V =k−
d2 ψ
Thus if ≥ 0, α ≤ 1 and hence
dV 2
V =k−
Done with Prosper – p. 29/45

103. Proof cont.:Conformal functions on [j, k]
d2 ψ 2π dψ
Recall, = ρψ − 2V (ρ + 3p)
dV 2
W˜ dV
2πρψ 3p d ln ψ
= 1 − 2V 1+
W˜ ρ dV
d2 ψ 2π ψρ−
= (1 − α)
dV 2 W˜
V =k−
d2 ψ
Thus if ≥ 0, α ≤ 1 and hence
dV 2
V =k−
d ln u
0< < 1 which allows us to continue inward.
dV V =j+
Done with Prosper – p. 29/45

104. Proof cont.:Scalar curvature of u g4
u4 scalar(u4 g) =
2W α p 2
6(1 + ) − 5γ + γ(1 − α) + 16πρ(1 − α)
3p 2 ρ
γV 2 (1 + )
ρ
Done with Prosper – p. 30/45

105. Proof cont.
A detailed analysis of the spherically symmetric
equations are necessary to ensure
Proof goes through even if for new star W becomes
zero early?
ψ matches at V = j in C 1,1 fashion.
ψ > 0.
Done with Prosper – p. 31/45

106. Proof cont.
A detailed analysis of the spherically symmetric
equations are necessary to ensure
Proof goes through even if for new star W becomes
zero early?
ψ matches at V = j in C 1,1 fashion.
ψ > 0.
On an interval [a, b] we shall denote by r = r(V ),
2
m = m(V ), W ˜ = W (V ) ≡ 1 − 2m
˜ dr
, and
r dV
ψ = ψ(V ) to be the solutions of the following equations
with initial values of the functions specified at j :
Done with Prosper – p. 31/45

107. Proof cont.
A detailed analysis of the spherically symmetric
equations are necessary to ensure
Proof goes through even if for new star W becomes
zero early?
ψ matches at V = j in C 1,1 fashion.
ψ > 0.
On an interval [a, b] we shall denote by r = r(V ),
2
m = m(V ), W ˜ = W (V ) ≡ 1 − 2m
˜ dr
, and
r dV
ψ = ψ(V ) to be the solutions of the following equations
with initial values of the functions specified at j :
Done with Prosper – p. 31/45

108. Proof cont.
A detailed analysis of the spherically symmetric
equations are necessary to ensure
Proof goes through even if for new star W becomes
zero early?
ψ matches at V = j in C 1,1 fashion.
ψ > 0.
On an interval [a, b] we shall denote by r = r(V ),
2
m = m(V ), W ˜ = W (V ) ≡ 1 − 2m
˜ dr
, and
r dV
ψ = ψ(V ) to be the solutions of the following equations
with initial values of the functions specified at j :
Done with Prosper – p. 31/45

109. Proof cont.
A detailed analysis of the spherically symmetric
equations are necessary to ensure
Proof goes through even if for new star W becomes
zero early?
ψ matches at V = j in C 1,1 fashion.
ψ > 0.
On an interval [a, b] we shall denote by r = r(V ),
2
m = m(V ), W ˜ = W (V ) ≡ 1 − 2m
˜ dr
, and
r dV
ψ = ψ(V ) to be the solutions of the following equations
with initial values of the functions specified at j :
Done with Prosper – p. 31/45

110. Proof cont. Spherically Symmetric Equations
dr r(r − 2m)
= r-equation
dV V (m + 4πr3 p)
Done with Prosper – p. 32/45

111. Proof cont. Spherically Symmetric Equations
dr r(r − 2m)
= r-equation
dV V (m + 4πr3 p)
dm 4πr3 (r − 2m)ρ
= m-equation
dV V (m + 4πr3 p)
Done with Prosper – p. 32/45

112. Proof cont. Spherically Symmetric Equations
dr r(r − 2m)
= r-equation
dV V (m + 4πr3 p)
dm 4πr3 (r − 2m)ρ
= m-equation
dV V (m + 4πr3 p)
˜ V 2 (m + 4πr3 p)2 ˜
W (V ) = W -equation
r3 (r − 2m)
Done with Prosper – p. 32/45

113. Proof cont. Spherically Symmetric Equations
dr r(r − 2m)
= r-equation
dV V (m + 4πr3 p)
dm 4πr3 (r − 2m)ρ
= m-equation
dV V (m + 4πr3 p)
˜ V 2 (m + 4πr3 p)2 ˜
W (V ) = W -equation
r3 (r − 2m)
dψ ψ 2m
= 1− 1− 1st order ψ -equation
dV 2r W˜ r
Done with Prosper – p. 32/45

114. Proof cont. Spherically Symmetric Equations
Einstein equations yield first two equations. With the
help of i p = −V −1 (ρ + p) i V they can be written as
TOV (Tolman-Oppenheimer-Volkoff) eqns.
Done with Prosper – p. 33/45

115. Proof cont. Spherically Symmetric Equations
Einstein equations yield first two equations. With the
help of i p = −V −1 (ρ + p) i V they can be written as
TOV (Tolman-Oppenheimer-Volkoff) eqns.
The solution ψ of the last equation makes ψ 4 g
Euclidean when g is spherically symmetric.
Done with Prosper – p. 33/45

116. Proof cont. Spherically Symmetric Equations
Einstein equations yield first two equations. With the
help of i p = −V −1 (ρ + p) i V they can be written as
TOV (Tolman-Oppenheimer-Volkoff) eqns.
The solution ψ of the last equation makes ψ 4 g
Euclidean when g is spherically symmetric.
Differentiating this equation wherever possible and
using the other three equations we get the second
derivative of ψ(V ) we met before:
d2 ψ 2π dψ
= ρψ − 2V (ρ + 3p) 2nd order ψ -equation
dV 2
W˜ dV
Done with Prosper – p. 33/45

117. Proof cont. A new system of ODEs
Theorem [Lindblom,Masood-ul-Alam] ˜
If r(V ), m(V ), W (V )
˜
is a solution of the following system, r > 2m > 0, W > 0,
and initially the values of the three functions are related
˜ ˜
by W -equation, then W -equation continues to hold
subsequently .
1/2
dr 2m ˜ −1/2
= 1− W
dV r
1/2
dm 2m ˜
2
= 4πρr 1 − W −1/2
dV r
dW˜
= 8πV (ρ + p) − 4V mr−3
dV
Done with Prosper – p. 34/45

118. Proof cont. A new system of ODEs
Theorem [Lindblom,Masood-ul-Alam] ˜
If r(V ), m(V ), W (V )
˜
is a solution of the following system, r > 2m > 0, W > 0,
and initially the values of the three functions are related
˜ ˜
by W -equation, then W -equation continues to hold
subsequently .
1/2
dr 2m ˜ −1/2
= 1− W
dV r
1/2
dm 2m ˜
2
= 4πρr 1 − W −1/2
dV r
dW˜
= 8πV (ρ + p) − 4V mr−3
dV
Done with Prosper – p. 34/45

119. Proof cont. A new system of ODEs
Theorem [Lindblom,Masood-ul-Alam] ˜
If r(V ), m(V ), W (V )
˜
is a solution of the following system, r > 2m > 0, W > 0,
and initially the values of the three functions are related
˜ ˜
by W -equation, then W -equation continues to hold
subsequently .
1/2
dr 2m ˜ −1/2
= 1− W
dV r
1/2
dm 2m ˜
2
= 4πρr 1 − W −1/2
dV r
dW˜
= 8πV (ρ + p) − 4V mr−3
dV
Done with Prosper – p. 34/45

120. Proof cont. Why/when two systems are equivalent
The old system yields the new system. The new system
satisfies
1/2
d 2m ˜
1− W 1/2
− (m + 4πr3 p)V r−2 = 0
dV r
Done with Prosper – p. 35/45

121. Proof cont. Why/when two systems are equivalent
The old system yields the new system. The new system
satisfies
1/2
d 2m ˜
1− W 1/2
− (m + 4πr3 p)V r−2 = 0
dV r
˜
Thus if W -equation holds at V = j, then it continues to
hold for V < j until Vc or a new center before Vc is
reached.
Done with Prosper – p. 35/45

122. Proof cont. Why/when two systems are equivalent
The old system yields the new system. The new system
satisfies
1/2
d 2m ˜
1− W 1/2
− (m + 4πr3 p)V r−2 = 0
dV r
˜
Thus if W -equation holds at V = j, then it continues to
hold for V < j until Vc or a new center before Vc is
reached.
Only two possible types of new center can occur. For
the center and the precise meaning of subsequently :
Done with Prosper – p. 35/45

123. Proof cont. Why/when two systems are equivalent
The old system yields the new system. The new system
satisfies
1/2
d 2m ˜
1− W 1/2
− (m + 4πr3 p)V r−2 = 0
dV r
˜
Thus if W -equation holds at V = j, then it continues to
hold for V < j until Vc or a new center before Vc is
reached.
Only two possible types of new center can occur. For
the center and the precise meaning of subsequently :
Lindblom, L., Masood-ul-Alam, A.K.M.: On the
Spherical Symmetry of Static Stellar Models Commun.
Math. Phys. 162, 123-145 (1994)
Done with Prosper – p. 35/45

124. Proof cont.
Integrate the system for V ≤ j with initial conditions at
V =j
˜ j 2 (m + 4πr 3 p(j))2
W (j) =
r 3 (r − 2m)
j 2m d ln ψ
1− 1− = j
2r ˜
W r dV
V =j +
Done with Prosper – p. 36/45

125. Proof cont.
Integrate the system for V ≤ j with initial conditions at
V =j
˜ j 2 (m + 4πr 3 p(j))2
W (j) =
r 3 (r − 2m)
j 2m d ln ψ
1− 1− = j
2r ˜
W r dV
V =j +
d ln ψ
Provided 0 < 2 j <1 it is possible to find
dV
V =j +
˜
positive numbers r(j), m(j), W (j) such that they satisfy
the above initial conditions. For details:
Done with Prosper – p. 36/45

126. Proof cont.
Integrate the system for V ≤ j with initial conditions at
V =j
˜ j 2 (m + 4πr 3 p(j))2
W (j) =
r 3 (r − 2m)
j 2m d ln ψ
1− 1− = j
2r ˜
W r dV
V =j +
d ln ψ
Provided 0 < 2 j <1 it is possible to find
dV
V =j +
˜
positive numbers r(j), m(j), W (j) such that they satisfy
the above initial conditions. For details:
Done with Prosper – p. 36/45

127. Proof cont.
Masood-ul-Alam, A.K.M.: Proof that static stellar
models are spherical Gen Relativ Gravit 39,55-85
(2007)
Done with Prosper – p. 37/45

128. Proof cont.
Masood-ul-Alam, A.K.M.: Proof that static stellar
models are spherical Gen Relativ Gravit 39,55-85
(2007)
Yaohua Wang (a student of Prof. X. Zhang) exactly
solved the algebraic system of initial conditions.
Done with Prosper – p. 37/45

129. Neutrino Connection?
Can ψ be related to neutrino?
Done with Prosper – p. 38/45

130. Neutrino Connection?
Can ψ be related to neutrino?
(A stupid analogy) Think the conformal function as a
globally compensating function in the sense that it
makes g curvature-free. Neutrino was first introduced
"... as a desperate remedy to save the principle of
energy conservation ...."
Done with Prosper – p. 38/45

131. Neutrino Connection?
Can ψ be related to neutrino?
(A stupid analogy) Think the conformal function as a
globally compensating function in the sense that it
makes g curvature-free. Neutrino was first introduced
"... as a desperate remedy to save the principle of
energy conservation ...."
(A profound analogy?) ψ 4 = ||ξ||2 where ξ is a fixed
spinor satisfying Dirac equation (actually neutrino
equation) relative to the nonnegative scalar curvature
metric g that produces the mass in the Witten-Bartnik
mass formula.
Done with Prosper – p. 38/45

132. Neutrino Connection?
Can ψ be related to neutrino?
(A stupid analogy) Think the conformal function as a
globally compensating function in the sense that it
makes g curvature-free. Neutrino was first introduced
"... as a desperate remedy to save the principle of
energy conservation ...."
(A profound analogy?) ψ 4 = ||ξ||2 where ξ is a fixed
spinor satisfying Dirac equation (actually neutrino
equation) relative to the nonnegative scalar curvature
metric g that produces the mass in the Witten-Bartnik
mass formula.
Inverse β−decay:
electron + proton → neutron + neutrino
Done with Prosper – p. 38/45

133. Open Problems
Investigate situations at higher densities from
geometrical viewpoint. Pion condensation.
Accompanied neutrino (including massive νµ ) emission.
Done with Prosper – p. 39/45

134. Open Problems
Investigate situations at higher densities from
geometrical viewpoint. Pion condensation.
Accompanied neutrino (including massive νµ ) emission.
Forget adiabatic index. Consider when p/ρ is not
negligible. Ex. Pressure is 1037 dynes/cm2
corresponding to a density 10 16 gm/cm3 p ≈ 3.3.
c2 ρ
Done with Prosper – p. 39/45

135. Open Problems
Investigate situations at higher densities from
geometrical viewpoint. Pion condensation.
Accompanied neutrino (including massive νµ ) emission.
Forget adiabatic index. Consider when p/ρ is not
negligible. Ex. Pressure is 1037 dynes/cm2
corresponding to a density 10 16 gm/cm3 p ≈ 3.3.
c2 ρ
2
6 6
Generalize the inequality γ ≥ 1+ to include
5 5
rotation:
Done with Prosper – p. 39/45

136. Open Problems
Investigate situations at higher densities from
geometrical viewpoint. Pion condensation.
Accompanied neutrino (including massive νµ ) emission.
Forget adiabatic index. Consider when p/ρ is not
negligible. Ex. Pressure is 1037 dynes/cm2
corresponding to a density 10 16 gm/cm3 p ≈ 3.3.
c2 ρ
2
6 6
Generalize the inequality γ ≥ 1+ to include
5 5
rotation:
layer by layer.
Done with Prosper – p. 39/45

137. Open Problems
Investigate situations at higher densities from
geometrical viewpoint. Pion condensation.
Accompanied neutrino (including massive νµ ) emission.
Forget adiabatic index. Consider when p/ρ is not
negligible. Ex. Pressure is 1037 dynes/cm2
corresponding to a density 10 16 gm/cm3 p ≈ 3.3.
c2 ρ
2
6 6
Generalize the inequality γ ≥ 1+ to include
5 5
rotation:
layer by layer.
angular velocity depending on radius.
Done with Prosper – p. 39/45

138. Open Problems cont.
Relax staticity and isentropy conditions: Isentropic
equation of state p = p(ρ). entropy depending on radius.
In the zero entropy case thermal contributions to the
pressure and density are neglected. Examples: Cold
stars such as Neutron stars, White dwarfs. Constant
entropy per baryon independent of the radius of the star
(Supermassive stars).
Done with Prosper – p. 40/45

139. Open Problems cont.
Relax staticity and isentropy conditions: Isentropic
equation of state p = p(ρ). entropy depending on radius.
In the zero entropy case thermal contributions to the
pressure and density are neglected. Examples: Cold
stars such as Neutron stars, White dwarfs. Constant
entropy per baryon independent of the radius of the star
(Supermassive stars).
For our simple situation one escape from no-go result
was that the star extends to infinity. What are possible
escapes when rotation, local non-spherical symmetry
are important?
Done with Prosper – p. 40/45

140. Wild Guessing: Gravitation Rules.
Gravitation and Thermodynamics are the foundation of
Physics.
Done with Prosper – p. 41/45

141. Wild Guessing: Gravitation Rules.
Gravitation and Thermodynamics are the foundation of
Physics.
Quantization, principle of least/stationary action are
mathematical properties of differential equations.
Done with Prosper – p. 41/45

142. Wild Guessing: Gravitation Rules.
Gravitation and Thermodynamics are the foundation of
Physics.
Quantization, principle of least/stationary action are
mathematical properties of differential equations.
Space-time (Einstein equations but here we have no
"real sense" of time because of the staticity
assumption) determines the energy-momentum tensor.
Solving for the space-time metric given the
energy-momentum tensor is backward thinking.
Done with Prosper – p. 41/45

143. Wild Guessing: Gravitation Rules.
Gravitation and Thermodynamics are the foundation of
Physics.
Quantization, principle of least/stationary action are
mathematical properties of differential equations.
Space-time (Einstein equations but here we have no
"real sense" of time because of the staticity
assumption) determines the energy-momentum tensor.
Solving for the space-time metric given the
energy-momentum tensor is backward thinking.
Possible causes of the "patterns" we see in the
energy-momentum tensor: its tensorial nature, coupled
with the approximate local and asymptotic symmetry of
Minkowski space-time? Stability?
Done with Prosper – p. 41/45

144. Non-staticity and time
Non-staticity and "real sense" of time.
Done with Prosper – p. 42/45

145. Non-staticity and time
Non-staticity and "real sense" of time.
Introduce time "minimally." Manage non-conformally flat
Riemannian 3-metric with non-zero Cotton tensor and
avoid giving importance of extrinsic curvature of the
embedding in 3+1 dimension as far as possible.
Done with Prosper – p. 42/45

146. Non-staticity and time
Non-staticity and "real sense" of time.
Introduce time "minimally." Manage non-conformally flat
Riemannian 3-metric with non-zero Cotton tensor and
avoid giving importance of extrinsic curvature of the
embedding in 3+1 dimension as far as possible.
Can this time be "maya" (illusion)? Real time is energy.
In what sense?
Done with Prosper – p. 42/45

147. Non-staticity and time
Non-staticity and "real sense" of time.
Introduce time "minimally." Manage non-conformally flat
Riemannian 3-metric with non-zero Cotton tensor and
avoid giving importance of extrinsic curvature of the
embedding in 3+1 dimension as far as possible.
Can this time be "maya" (illusion)? Real time is energy.
In what sense?
Is any one "more basic?" Which one: linear or angular
momentum? Rotation or translation?
Done with Prosper – p. 42/45

148. Non-staticity and time
Non-staticity and "real sense" of time.
Introduce time "minimally." Manage non-conformally flat
Riemannian 3-metric with non-zero Cotton tensor and
avoid giving importance of extrinsic curvature of the
embedding in 3+1 dimension as far as possible.
Can this time be "maya" (illusion)? Real time is energy.
In what sense?
Is any one "more basic?" Which one: linear or angular
momentum? Rotation or translation?
Dual space of 3-space.
Done with Prosper – p. 42/45

149. Suggested Readings on Gravity
Hu, B. L.: General Relativity as
Geometro-Hydrodynamics, arXiv:gr-qc/9607070v1
(1996).
Barut, A. O., Cruz, M. G., Sobouti, Y.: Localized
solutions of the linearized gravitational field equations in
free space, Class. Quantum Grav. II, (1994) 2537-2543.
Finster, F., Smoller, J., Yau, S.-T.: The coupling of
gravity to spin and electromagnetism, Mod. Phys. Lett.
A 14, (1999) 1053-1057.
Penrose, R.: Chapter 30 in The Road to Reality, (2004).
Done with Prosper – p. 43/45

150. References
Lindblom, L., Masood-ul-Alam, A.: Limits on the adiabatic index in
static stellar models, in B.L. Hu and T. Jacobson ed. Directions in
General Relativity, Cambridge University Press (1993).
Beig, R., Simon, W.: On the spherical symmetry of static perfect
fluids in general relativity, Lett. Math. Phys. 21, 245-250 (1991).
Shiromizu, S., Yoshino, H.: Positive Energy Theorem Implies
Constraints on Static Stellar Models,, Prog. Theor. Phys. 116, No. 6
(2006) 1159-1164.
Shiromizu, T., Yamada, S., Yoshino, H.: On existence of matter
outside a static black hole, Journal of Mathematical Physics 47,
112502(1-8) (2006).
Done with Prosper – p. 44/45

151. References
Carrasco, A., Mars, M., Simon, W.: On perfect fluids and black holes
in static equilibrium, Journal of Physics: Conference Series 66,
012012(1-6) (2007).
Beig, R., Simon, W.: On the uniqueness of static perfect fluid
solutions in general relativity, Commun. Math. Phys. 144, 373-390
(1992).
Done with Prosper – p. 45/45