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Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.1
A brief introduction to general relativity
The concept of manifold: A manifold is a topological space that can be continuously parameterized (i.e.
differentiable), whose dimension is equal to the number of parameters required
for specifying a point in this manifold.
Locally, this manifold looks like the Euclidean four-space R4.
Examples:
1- The circle S1 = {(x,y)∈ IR2 | x2+y2 = 1} is a manifold of dimension one.
2- The circle Sn = {(x1, x2, x3, …xn,)∈ IRn+1 | x12+ x2
2+ x32+ …xn+1
2 = 1} is a
manifold of dimension n.
3- The position coordinates of a particle (ξ 1, ξ 2, ξ 3) together with its three
momentum coordinates (p1,p2,p3) build up a 6-dimensional manifold.
The geometry of a Riemannian manifold is defined through the metric that
measures the distance between two arbitrary points in this manifold. In general
relativity, the interval between two such points is given by the expression 2 ,ds g dx dx
where gμν is a set of coordinate functions that determines the
geometry of the manifold. The intervals in Riemannian manifolds are positive
definite and invariant under coordinate transformation.
On manifolds, one may define scalar and vector fields.
Φ on a manifold U is said to be a scalar field, if Φ associates to each point of U
a unique real-value Φ(xµ).
Example: Φ(x1, x2, x3) = x3.
Vector fields on manifolds are the assignment
of a single vector to each point P of U.
Each vector is a linear combination of the set of
independent basis vectors eμ , whose number equal to the dimension of the manifold U, i.e.,
v( ) ( ) ( )x v x e x
, where ( )v x are the contravariant components
(actually numbers) associated with each basis vector eμ. Similarly, the vector v(x) can be written as a linear combination of the dual
basis vectors e𝝁 (the reciprocal vectors of eμ) : v( ) ( ) ( )x v x e x
,
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.2
( )v x here is the covariant components and .e e
This implies that v ,e v e e v v
but also
v .v e Now an infinitesimal interval between two arbitrary points on U, can be expressed as the inner product the two vectors:
2 ds ds ( ) ( ) , .ds dx e dx e e e dx dx g dx dx where g e e
Equivalently, 2 , .ds g dx dx where g e e
Consequently, the inner product of two vectors should be the same, whether we use the contravariant or covariant versions of the vectors, i.e.,
( ) ( ) ( ) .V W v e w e e e v w g v w g v w
It can be easily
verified that the contavariant and covariant basis vectors can be
transformed from one to other using the metric gμν, i.e.,
, .e g e e g e
But as
.e e g g
Viewing as a matrix, then , is nothing but the inverse
matrix of and vice versa.
g g
g
Helpful notes:
1. A two dimensional table can be viewed as a matrix
2. Generalization of matrices are tensors
3. A 2D matrix ⟶ tensor of rank 2
4. A 3D matrix ⟶ tensor of rank 3
5. Tensors are transformable objects that could have geometrical meanings
Tensor products:
Let A, B, and C be matrices of arbitrary dimensions, so that AB = C.
An element of C reads:
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.3
ij in nj
n
c = a b
[ ]Cij
=
A B C
But in tensor world, the sum means a contraction. As in matrix algebra, we
introduce the inner product of two second-rank tensors as follows:
β β
αβ υ αβ αυ
β
A B = A B =T
We note that in matrix algebra, the products:
AB BA Example:
1 0 1 1 1 1=
1 2 1 1 3 3
1 1 1 0 2 2=
1 1 1 2 2 2
In tensor analysis the product of two tensors, such as:
β β
α βσ βσ αA B and B A
are sets of instructions that carry out the same process and, if carried out
correctly, will yield the same results.
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.4
2. Curvature in a manifold
Let us consider the trajectories of two bees along the following three different
geometrical shapes:
One way to determine the curvature of a surface is to take two
points, Pc and P2 on the surface, that are located at a distance d from each
other. Then to draw a cycle around Pc that goes through P2.
1. In flat space, the circumference of the resulting cycle is:
measured
FC = 2 d.
The distance d in this case is identical to the radius of the
cycle. The curvature is said to be zero, i.e., K=0.
Pc
P2
C, d
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.5
2. If the shape is parabolic (dome-shaped surface), we obtain
measured
PC < 2 d. This corresponds to positive curvature, i.e., K> 0.
3. If the shape is hyperbolic (;saddle-shaped surface), then
measured
HC > 2 d. In this case the curvature is said to be negative.
The degree of curvature may be approximated by osculating a
cycle whose circumference has maximum intersection points
with the sought-surface.
A strongly curved path would be approximated
by a smaller cycle, whereas cycles with large
radii would correspond to less curved paths.
Thus, a sphere with infinite radius
would correspond to a straight line.
Such a measure is conventionally called Gaussian curvature: and has the form:
2
1K =
R
In general the metric of a more general surface contains the full information on
the geometry of spacetime.
Under certain conditions, the Gaussian curvature can be calculated from the
metric coefficients as follows:
rr
2
rr
g / rK = f(g ) =
2rg
Example:
Consider the metric:
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.6
2 2 2 2 2 2 2 2
rr
2 2 2 2 2
rr
ds = c dt -g dr - r (dθ + sin θdφ )
= c dt -g dr - r d traΩ , ( ce=?)
which is said to describe an isotropic and homogeneous space.
Riemann curvature tensor:
The metric tensor μυg of a flat space, or in curvilinear coordinates are known
a priori.
The physical world may, however, be different and a definitive procedure is
required to determine the components of the correct metric.
The Riemann curvature tensor,
λ
μνσR has been constructed to properly measure
the curvature of spacetime in multi-dimensions.
Let us consider the surface of a sphere and let A be a point on this surface,
where the vector V is located . It appears to be impossible to find two-
coordinate manifold (x1,x2) throughout the surface, such that 2 1 2 2 2ds = (dx ) + (dx ) = const.
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
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By parallel-transporting the vector on the surface of the sphere, the vector may
undergo a significant change in its direction, depending upon the path followed
by the vector, as can be seen from the figure below:
θ φ
θ φ
V t=0 = (V ,V ) = (0,V)
V t= = (V ,V ) = (V,0)
This change is a consequence of the curvature of the spherical surface, and not
because of changing of the components of the vector.
The Riemann curvature tensor, RCT, measures the total variation of the
components of the contravariant vector μV parallel-transported through a
closed circuit of infinitesimal extend.
Without going into details of differential geometry, the RCT is found to have the
following form:
μ μ μ μ ν μ ν
σρλ σλ ρσ νρ σλ νλ ρσρ λR - + - ,
x x
Where
is the so called Christoffel symbol or connection coefficient.
These symbols are computed from the metric g
describing the geometry of
the manifold as follows:
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
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λ αλ1μν αμ,ν αν,μ μν,α2
Γ = g g +g -g
The tensor μ
σρλR has 44 components, but this number can be reduced
singnificantly, if the following general conditions are considered:
1. λ λ
μ νσ μ σνR = - R , i.e., antisymmetric with respect to the final two indices.
2. λμνσ μλσνR = - R
3. λμνσ νσλμR = R
4. μμνσ νσμμR = 0 = R
5. μνστ μτνσ μστνR + R + R = 0 cyclicity condition
With these conditions, the RCT reduces to just 20 independent
components.
The contracted RCT, is called Ricci tensor and it has the form:
λ λ ρ λ ρ1 1
αβ αβλ ρβ αβ αβ,λ αβ2 2,ρ ,α,βR = R = Γ Γ - Γ - Γ ln(-g) + ln(-g)
,
where λ 1αβ,λ 2 α
= - ln (-g)x
and μνg = g = det. g .
Bianchi identities:
Assuming the Christoffel’s symbols to vanish locally, we may compute the
covariant derivative of the RCT with respect to ηx as follows:
2 22 2
μν μκλν λκ
η κ μ κ λ μ ν ν λ
g gg g1λμνκ,η 2 x x x x x x x x x
R = - - +
By permuting ν,κ and η cyclically, the following equality can be obtained:
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.9
λμνκ,η λμην,κ λμκη,νR +R + R = 0
Taking symmetry into account and multiplying the latter equation by αμ βλg g ,
we get: βλ μ αμ μ βλ
βλμ,ν ανλ;μ βνμ;λg R - g R - g R 0 . Carrying out the contraction and raising
operations, we finally obtain the equation of spacetime curvature in empty
space:
μ μ1 ν ν ;μ2
(R - g R) =0 , where μ
νR is called Ricci’s tensor and R is the Ricci
scalar.
The tensor μ μ μ1 ν ν ν2
G = R - g R is called the Einstein’s tensor.
Example:
Consider the 2D space on the surface of a sphere of radius a.
The metric of this space is: 2 2 2 2 2 2ds = a d + a sin d . Therefore:
μν
μν 2 2
1 0 1 0g = , g =
0 sin θ 0 1/ sin θ
The Christoffel’s symbols read:
λ αλ1μν αμ,ν αν,μ μν,α2Γ = g g +g -g
θ θα θθ1 1φφ φα,φ αφ,φ φφ,α φθ,φ θφ,φ φφ,θ2 2
Γ = g g +g -g = g g +g -g
θθ1φφ,θ2
= g -g = -sinθ cosθ
φ φα φφ1 1θφ θα,φ αφ,θ θφ,α φφ,θ θφ,φ2 2
Γ = g g +g -g = g g -g
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.10
φ φφ1θφ φφ,θ2
Γ = g g =cotθ
The determinant of the metric tensor reads:
4 2g = a sin θ ln (g) = 2 cotθ
Thus, the θθ- component of the RCT is:
λ ρ θ θ1θθ ρθ θλ θθ,0 θθ ,θ2
R = Γ Γ - Γ - cotθΓ + cot θ
Summing over λ and ρ , we obtain then:
φ 0 2
θθ 0θ θφ ,θ 2
1R = Γ Γ - cot θ = cot θ - = -1
sin θ
and 2
φφR = -sin θ.
Therefore, the curvature invariant, R, is then:
θθ φφ 2
θθ φφR= g R + g R = -2/a
The curvature of a sphere decreases with increasing its radius.
Some useful connections:
1. The covariant derivative of a contravariant vector Vμ is composed of two
contributions: the change of the vector along a certain coordinate and its
change through parallel translation.
Mathematically, this reads:
: , ,V V V
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.11
where the semicolon refers to covariant differentiation and the comma to
ordinary partial differentiation.
On the other hand, the covariant derivative of a covariant vector V𝛍 reads:
: , .V V V
2. The covariant derivative of tensors of rank-2 reads:
: ,
If it is symmetric 1( )
T T T T
gT Tg
Whereas the covariant derivative of tensors of higher ranks:
: ,T T T T T T
The equation of the geodesic lines:
Assume we are given two events A and B on a surface S of a sphere.
The events correspond to the motion of a particle between the starting point A
and where the final point B located.
The motion of the particle is assumed to be field-free.
Therefore, we expect the particle to move along a
trajectory, that corresponds to the shortest distance
between the events A and B.
This choice of path has been experimentally observed,
and therefore, we may retain this hypothesis as a
principle for the future considerations.
Mathematically, such trajectories must fulfill the minimum variation principle:
HWQ: In the case of a scalar function, ; , . Explain why?
Q: Prove its
correctness!
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.12
ds= 0
A world-line satisfying this extremal condition is called a geodesic.
As a consequence, particles in free-fall through curved spacetime will also move
along the shortest possible length between events measured by the proper time.
In the absence of external forces, the Hamiltonian principle
states that the integral of the Lagrangian function describing the motion of a
particle must have an extremal values between two instants, i.e.,
L(q,q,t)dt = K(q,q,t) - (q,t) dt , where K,
denote the kinetic and potential energies of the system.
As this applies for any system at arbitrary time, this integral
can be expanded to yield the Euler-Lagrange equation, i.e.,
2 2
1 1
t t
t t
L L L LL(q,q,t)dt= q+ q dt - =0.
q q t q q
Taking into account that =0(force free motion) , and using
the Euler-Lagrange equation of the calculation of variation, we obtain the so
called geodesic equation:
2
1
t 2 μ ν σ2 μ1
νσ2 2
t
d x dx dxmx dt + Γ =0
ds ds ds
μ
νσThe Christoffel symbols Γ and the Euler equations:
Consider a general metric of the form:
2 μ ν
μνds = g dx dx
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.13
Using the minimal variation principle, it appears that
geodesic lines fulfill the Euler-Lagrange equation:
L L- =0,
t q q
, where
μ ν
μνL= g dx dx
This relation between the Lagrangian function and the metric tensor can be used
to compute the Christoffel’s symbols.
Gravitation & Fields
Events in inertial frames cannot be formulated by means of transformation of
coordinates in the (3+1)-spacetime.
There is no transformation that relates the coordinates of an inertial frame to that
of a space with α
βγδR 0
This is a consequence of the fact that:
α
βγδ ,R =f( ) = f( (g ))
.
In inertial frames α α
βγ βγδ = 0 R =0
Therefore the transformation of a curvilinear tensor from inertial frame into
other space with curvature would have
the form:
α ρ σ τα μ
βγδ ρστμ β γ δ
x x x xR = R .
x x x x
But if α α
βγδ βγδR =0 R =0 .
Gravitational fields, which are assumed to curve spacetime, are generically
non-symmetric. Therefore, it is almost impossible to get the curvature by
means of just coordinate transformation.
A
A
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.14
The metric tensor equations in an empty space:
In Newtonian theory the gravitational potential satisfies the one of the following
equations:
= 0 in an empty space
= -4 G in gaseous clouds
.
The metric tensor of an Euclidian space is: μν μνg =
Here, we need to solve just a single equation to determine
In a curved spacetime, however, μνg may have 10 non-zero components, which
implies that 10 equations should be solved to determine .
Taking into account that α
βγδ ,λR =f( , ) , the requirement that λ
μνσR =0
can be fulfilled by only a flat spacetime.
However, a zero-contracted RCT, i.e., λ
μν μνλR = R =0 , which has 10
independent components, would certainly allow a much bigger degree of
freedom.
In the weak field approximation, μνR = 0 should lead to the Laplace’s
equation to determine the gravitational potential.
The metric tensor equations in the presence of matter:
In presence of matter the gravitational potential in Newtonian theory satisfies
the Poisson equation:
= -4 G (spatial operator).(potential)= matter
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.15
In general relativity, the LHS is described by the Einstein tensor, μνG . In an
analogous manner, we may assume that the RHS to be described by matter-
tensor μνT . Hence the general equation to be solved is:
μν μνG = A T ,
where A is a constant to be determined, and where μνT is said to have a
covariant derivative.
In an empty space, we have: 1
μν μν μν2G =0 R - g R=0
Note that if μνR =0 then its contraction R=0. This implies that
μν μνG =0 R =0 .
If the system has a non-zero density/energy 0 0 , then we may assume the
matter tensor to have the form: μν 0
dx dxT =
ds ds
, where ds is an element of
the worldline of the particle. dx
ds
has the character of a relative velocity
dx 1 dx v=
ds c dt c
. Thus, 00 0 0
2 2
dt dtT = =
c dt dt c
In order to find A, we multiply the general equation with μνg :
μσ μσ μσ1μν μν μν2
σ σ σ1ν ν ν2
g R - g g R= A g T
R - δ R= AT
Setting σ=ν and contract: 42
R - R= AT R = -AT , where λ
λT= T .
Substituting these back into the general equation, we obtain: 1
μν μν μν2R = A(T - g T) .
If T corresponds to a stationary dust cloud, then the only non-vanishing
component of this matter tensor is: 00 0
2
ρT =
c.
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.16
Comparing this equation with the normal Poisson equation, it is easy to verify
that if GM
= -r
, then 00 2
2f = -
c
and therefore 2
8 GA= -
c
.
Consequently, in the presence of matter the metric equation
that describes the interaction of spacetime and matter is the
famous Einstein’s equation:
μν μν μν12 2
8 GR - g R= - T
c
, where
μνT
is the so called the energy-momentum tensor.
General formulation of the energy-momentum tensor μνT :
To describe the matter distribution at each point on a manifold or equivalently at
each event in spacetime, we need a tensor with special properties.
Due to the equivalence of matter and energy in special relativity, we may
intuitively assign for the matter-energy distribution at each event in
spacetime the following function:
Energies in tensor form, i.e. in T
2
0
The simplist form is the kinetic energy v
( ) ( ) ( ) ( )
Energy-momentum tensor
x x V x V x
v v
T
T
To see Tμν in a matrix form, consider the metric tensor of a Lorentz frame:
2 22 2 2 2 2 2 2 2 2 2
2
d V c dtds = c dt -d = c dt (1-( ) )=c dt (1-( ) )= ,
dt c
i.e.,
2 22
2
c dtds =
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.17
Now, using
μ ν μ νμν 2 2 μ ν μ ν0
0 0 2 2
dx dx dx dx dt ρT = = ( ) = x x = x x ,
ds ds dt dt ds c c
x y z
22x x x y x zμν
222y y x y y z
22z z x z y z
1 V V V1 x y z
V V V V V Vx x xy zxT = =
V V V V V Vy yx y zyc
V V V V V Vz zx yz z
The appearance of 2γ as coefficient in the stress energy tensor marks two
important effects of special relativity:
1. The volume of a moving matter shrinks by a factor γ as measured in
the observer frame.
2. The mass of a particle moving with a velocity V increases by a factor
γ .
Using the 4-velocity formulation:
00 0 0 2 2
0
0 0 0 2
0
2
0
( , ), then the components of T :
T = energy density of matter
T T energy flux in the j-direction
T material flux of the i-component of t
j j j j
ij i j i j
v c v read
v v c
v v cv
v v v v
he
momentum in the j-direction.
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.18
The conservative form of the hydro-equations in the non-relativistic regime
can be obtained from general relativity through the requirement that μνT
must have a vanishing covariant derivative, i.e., μν
;νT =0 .
Recalling that σ
μν =0 in Lorentz frames, we conclude that:
μνμν μν
;ν ,ν ν
TT = T = =0.
x
These expressions can then be expanded to yield the following equalities:
00 01 02 032 0ν 2
;ν 0 1 2 3
T T T Tc T =c + + + =0
x x x x
In this frame:
200 0 2 2
T = =c c
,
y01 02 01x z 2 2 2
VV VT = , T = ,T =
c c c
Putting the terms together, we obtain:
y2 0ν x z ;ν
VV Vc T = + + + = + V=0.
t x y z t
This equation is identical to the equation of mass conservation in normal fluid
dynamics.
The momentum equation is obtained by expanding the expression 2 1ν
;νc T =0
as follows: 10 11 12 13
2 1ν 2
;ν 0 1 2 3
1 1 1 1 2 1 3 xx0 1 2 3
T T T Tc T =c + + + =0
x x x x
V V V V V V V V + + + =0 + ( V V)=0,
x x x x t
Q: Why doesn’t the modified differential operator 1( )g
g
appear?
Relativistic Hydrodynamics BAS/FS17/LectNote-7&8
S.19
where x y zV=(V ,V ,V ) is the 3-velocity vector of the flow.
However, the momentum equations in this form are said to describe the time-
evolution of non-interacting matter.
If the matter is interacting, but still non-dissipative, then the stress-energy tensor
must have the form: μ ν
μν μν
0 2 2
p dx dx pT = ( + ) - g
c ds ds c , where p is the pressure as measured in the
proper frame of the moving fluid and μνg corresponds to the contravariant
metric tensor.
If the flow is interacting and dissipative, then the stress energy tensor must be
modified to include the “highly complicated” viscous interaction tensor, Σμν and
heat conduction tensor Πμν:
μ νμν μν μν μν μ ν ν μ
0 2 2
p dx dx pT = ( + ) - g + af -2 + q u +q u .
c ds ds c