Fluid Solution - Einstein's Equation
Einstein’s equation
To start we give an exposition of elementary topics in the geometry of space, and
the spatial tensorial character of physical laws in prerelativity physics. We formulate
general relativity and provide a motivation for Einstein’s equation, which relates the
geometry of spacetime to the distribution of matter in the universe.
1.2.1 General and Special Covariance
Assume that space has the manifold structure of R
3
and the association of points of
space with elements (x
1
, x2
, x3
) of R
3
can be accomplished by construction of a rigid
rectilinear grid of metersticks. We call the coordinates of space derived in this manner
as Cartesian coordinates [24]. The distance, S, between two points, x and ¯x, defined
in terms of Cartesian coordinates by
S
2 = (x
1 − x¯
1
)
2 + (x
2 − x¯
2
)
2 + (x
3 − x¯
3
)
2
(1.1)
This formula is the distance between two points.
Referred to equation (1.1), the distance between two nearby points is
(δS)
2 = (δx1
)
2 + (δx2
)
2 + (δx3
)
2
(1.2)
Therefore, the metric of space is given by
ds
2 = (dx
1
)
2 + (dx
2
)
2 + (dx
3
)
2
(1.3)
In the Cartesian coordinate basis, we derive
ds
2 =
X
a,b
hab(dx
a
)(dx
b
) (1.4)
with hab = diag (1, 1, 1). This definition of hab is independent of choice of Cartesian
coordinate system.
When the components of the metric in the Cartesian coordinate basis are constants,
we get
∂ahbc = 0. (1.5)
The space is the manifold R
3 which possesses a flat Riemann metric. We are able to
use the geodesics of the flat metric to construct a Cartesian coordinate system. We use
the fact that initially parallel geodesics remain parallel because the cuvature vanishes.
Minkowski space-time
We introduce a 4-dimensional continuum called space-time in which an event has
coordinates (t, x, y, z) [19].
Minkowski space-time is defined as a 4-dimensional manifold provided with a flat
metric of signature +2. By the definiton, since the metric is flat, there exists a special
coordinate system covering the whole manifold in which the metric is diagonal, with
diagonal elements equal to ±1. For convenience, we prefer to use the convention that
lower case latin indices run from 0 to 3. The special coordinate system is called a
Minkowski coordinate system and is written
(x
a
) = (x
0
, x1
, x2
, x3
) = (c t, x, y, z). (1.6)
In special relativity [24], it is assumed that spacetime has the manifold structure of
R
4
.
The spacetime interval, S, between two events x and ¯x defined by
S = −(x
0 − x¯
0
)
2 + (x
1 − x¯
1
)
2 + (x
2 − x¯
2
)
2 + (x
3 − x¯
3
)
2
(1.7)
in units where c = 1.
From the equation (1.7), we are able to defind the metric of spacetime ηab by
ds
2 =
X
3
a,b=0
ηab dx
a
dx
b with ηab = diag (-1, 1, 1, 1), where x
a
is any global inertial coordinate system.
Therefore, the ordinary derivative operator, ∂a, of the global inertial coordinates
satisfies
∂aηbc = 0 (1.9)
The curvature of ηab vanishes. In addition, we can parameterize timelike curves by
proper time, τ , defined by
τ =
Z
(−ηab T
a T
b
)
1/2
dt, (1.10)
where t = arbitrary parameterizaton of the curve, and T
a = the tangent to the curve
in this parameterization.
The tangent vector u
a
to a timelike curve parameterized by τ is defined by the
4-velocity of the curve. The square of any 4-vector is an invariant and so
u
aua = −c
2
, (1.11)
where c = 1 so we get,
u
aua = −1. (1.12)
In the absence of external forces, its 4-velocity will satisfy the equation of motion,
u
a
∂au
b = 0, (1.13)
where ∂a is the derivative operator associated with ηab. In addition, when forces are
present, the equation (1.13), u
a∂au
b
is nonzero. Furthermore, all material particles
have a parameter known as “rest mass”, m, which appears as a parameter in the
equations of motion when forces are present. We can define the energy momentum
4-vector, p
a
, of a particle of mass m by
p
a = m ua
. (1.14)
Finally, we can define the energy of particle which is measured by an observer – present
at the site of the particle – as
E = −pa v
a
,where v
a
is the 4-velocity of the observer.
In special relativity, the energy is the “time component” of the 4-vector, p
a
. At the rest
frame, a particle with respect to the observer, equation (1.15) reduces to the familiar
formula E = m c2
. When the spacetime metric, ηab is flat, and the parallel transport
is path independent, we are able to define the energy of a particle as measured by an
observer who is not present at the site of the particle and has 4-velocity parallel to
that of the distant observer.
1.3.2 The Stress-Energy Tensor
In special relavity, we define the energy-momentum 4-vector of a particle of mass m
as in equation (1.14).
From [19], the Minkowski line element takes the form
ds
2 = −dt
2 + dx
2 + dy
2 + dz
2
(1.16)
We can write this in tensorial form as
ds
2 = ηab dx
a
dx
b
, (1.17)
We take ηab to denote the Minkowski metric
ηab ≡
−1 0 0 0
0 +1 0 0
0 0 +1 0
0 0 0 +1
= diag (−1, +1, +1, +1) (1.18)
In relativistic units the equation for the proper time satisfies
dτ
2 = −ds
2
. (1.19)
Now we present proper time τ relates to coordinate time t for any observer whose
velocity at time t is v, where
v =
dx
dt
,
dy
dt
,
dz
dt
. So we have,
dτ
2 = −ds
2 = −(−c
2
dt
2 + dx
2 + dy
2 + dz
2
)
= −dt
2
"
−c
2 +
dx
dt
2
+
dy
dt
2
+
dz
dt
2
#
= −c
2
dt
2
(
−1 +
1
c
2
"
dx
dt
2
+
dy
dt
2
+
dz
dt
2
#)
= c
2
dt
2
1 −
v
2
c
2
dτ =
1 −
v
2
c
2
1/2
c dt (1.21)
The time-component of the energy-momentum vector does represent the energy of the
particle
p
0 =
E
c
, γ =
1 −
v
2
c
2
−1/2
, and E = mc2
γ. (1.22)
The space-components are the components of the three-dimensional momentum
p = mγv. (1.23)
A perfect fluid is defined to be a continuous distribution of matter with stress
energy tensor Tab of the form
Tab = ρuaub + p(ηab + uaub), (1.24)
where u
a
is the 4-velocity of the fluid, ρ is the mass-energy density in the rest-frame
of fluid, and p is the the pressure in the rest-frame of the fluid.
When there is no external forces, the equation of motion of a perfect fluid is simply
∂
a Tab = 0.
Consider ∂
aTab, we can write this in terms of ρ, p, and u
a as
∂
aTab = ∂
a
[ρuaub + p(ηab + uaub)]
= (∂
a
ρ)uaub + (∂
a
p)(ηab + uaub)
+(ρ + p)(∂
aua)ub + (ρ + p)ua(∂
aub)
= [(u
a
∂aρ) + (ρ + p)∂
aua] ub
+ [(ρ + p)u
a
∂aub + ∂
a
p(ηab + uaub)] (1.26)
For equation (1.26) we can project the resulting equation parallel and perpendicular
to u
b
, we find:
[(u
a
∂aρ) + (ρ + p)∂
a
ua] = 0, (1.27)
[(ρ + p)u
a
∂aub + ∂
a
p(ηab + uaub)] = 0. (1.28)
In the non-relativistic limit, when p ≪ ρ , u
µ = (1,
−→v ), and v
dp
dt
≪ |−→∇p|, equation
(1.27) becomes,
u
a
∂aρ + ρ ∂a ua = 0
⇒ ∂tρ +
−→v ·
−→∇ ρ + ρ
−→∇ · −→v = 0
⇒ ∂tρ +
−→∇ · (ρ
−→v ) = 0 (1.29)
and equation (1.28) becomes
ρ
∂
−→v
∂t + (−→v ·
−→∇)
−→v
= −
−→∇p (1.30)
1.3.3
Relativistic hydrodynamics
Ordinary hydrodynamics is dealing with two basic equations [22]:
The equation of continuity
∂tρ + ∇ · (ρ
−→v ) = 0 (1.31)
This relates to the density and velocity of the fluid. It is equivalent to the conservation
of mass.
The Euler equation is the fluid mechanics equivalent to Newton’s second law, it
relates the acceleration of a particle following the flow
−→a =
dv
dt
=
∂
−→v
∂t + (−→v ·
−→∇)
−→v (1.32)
We can write it in term of force density −→f and mass density ρ:
−→a =
∂
−→v
∂t + (−→v ·
−→∇)
−→v =
−→f
ρ
(1.33)
These laws have relativistic generalizations which are:
∇a(ρ va
) = 0 (1.34)
A
a = V
b ∇b V
a
(1.35)
where Aa
is now a 4-vector field of 4-accelerations.
The relativistic continuity equation yields
∂(ρ γ)
∂t = ∇ · (ρ γ −→v ) (1.36)
We interpret ρ as proportional to the number density of particles as measured by an
observer moving with the fluid. Indeed, the γ factor is corresponding to the fact that
Lorentz contraction “squashes” in the direction of motion, therefore, as seen by an
observer moving with respect to the fluid the number density of particles is ρ γ.
Now for the 4-acceleration
A
i = (γ ∂t + γ [
−→v ·
−→∇]) [γ vi
] (1.37)
and
A
0 = (γ ∂t + γ [
−→v ·
−→∇]) [γ] (1.38)
The standard Newtonian results is reproduced at the low velocity, where γ → 1.
The special theory of relativity only deals with flat spacetime and the motion of
objects is usually treated in terms of Lorentz transformations and translations from
inertial frame to another. The general theory of relativity extends it to deal with
non-inertial frames, and via Einstein’s equations with curved spacetimes as well. We
need to understand the basic concepts of special relativity before turning to general relativity—as otherwise the mathematical constructions used in general relativity
would appear rather unmotivated.
8
1.4 Conclusion
1.4.1 Postulational formulation of special relativity
There are two sets of postulates which are useful to generalize the general theory [19].
Postulate I. Space and time are represented by a 4-dimensional manifold provided with a
symmetric affine connection, Γa
bc, and a metric tensor, gab, which is satisfied as
follows:
(i) gab is non-singular with signature − + ++;
(ii) ∇c gab = 0;
(iii) Ra
bcd = 0.
The Postulate states that Γa
bc is the metric connection and that the metric is flat.
Postulate II. There exist privileged classes of curves in the manifold singled out as follows:
(i) ideal clocks travel along timelike curves and measure the parameter τ defined by
dτ
2 = −gab dx
a dx
b
;
(ii) free particles travel along timelike geodesics;
(iii) light rays travel along null geodesics.
The first part of the second postulate makes physical the distinction between space
and time in the manifold. In Minkowski coordinates, it distinguishes the coordinate x
0
from the other three as the “time” coordinate. Furthermore, it states that the proper
time τ which a clock measures is in accordance with the clock hypothesis. The rest of
Postulate II singles out the privileged curves that free particles and light rays travel
along.
The correspondence principle
Any new theory is consistent with any acceptable earlier theories within their range of
validity. General relativity must agree on the one hand with special relativity in the
absence of gravitation and on the other hand with Newtonian gravitational theory in
the limit of weak gravitational fields and low velocities by comparing with the speed
of light. This gives rise to a correspondence principle, as in figure (1.1), where arrows
indicate directions of increased specialization.
Figure 1.1: This structure shows the correspondence principle for general relativity.
© Karam OUHAROU. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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