Relativistic Hydrodynamics 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 R 4 . Examples: 1- The circle S 1 = {(x,y)∈ IR 2 | x 2 +y 2 = 1} is a manifold of dimension one. 2- The circle S n = {(x 1 , x 2 , x 3 , …x n ,)∈ IR n+1 | x1 2 + x2 2 + x3 2 + …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 (p 1 ,p 2 ,p 3 ) 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(x μ ). Example: Φ(x 1, x 2 , x 3 ) = x 3 . 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 e μ , whose number equal to the dimension of the manifold U, i.e., v( ) () () x v xe x , where () v x are the contravariant components (actually numbers) associated with each basis vector e μ . Similarly, the vector v(x) can be written as a linear combination of the dual basis vectors e (the reciprocal vectors of e μ ) : v( ) () () x v xe x , () v x here is the covariant components and . e e
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Relativistic Hydrodynamics A brief introduction to general relativity · 2016-04-14 · Relativistic Hydrodynamics S. 2 This implies that v,vQQ P G but also vePP v. Now an infinitesimal
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Relativistic Hydrodynamics
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
,
( )v x here is the covariant components and .e e
Relativistic Hydrodynamics
S.2
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:
ij in nj
n
c = a b
Relativistic Hydrodynamics
S.3
[ ]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.
Kummulativegesetzt???
Relativistic Hydrodynamics
S.4
2. Curvature in a manifold
Let us consider the following 3 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.
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.
Pc
P2
Relativistic Hydrodynamics
S.5
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
The curvature of 2D surfaces can be measured by taking a point O, draw a
normal vector to the surface at O, where the two lines intersect perpendicularly,
then find the radii of the cycles corresponding to these two curves.
C1
C2
In this case the Gaussian curvature measure can be written as:
1 2
1K =
R 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
n
o
It is nor clear here, how
you calculate the
curvature? Why do you
need n and R1 and R2???
Relativistic Hydrodynamics
S.6
Example:
Consider the metric:
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 dΩ ,
which is said to describe an isotropic and homogeneous space.
Let K designate the different curvature possibilities, then K=const.:
rrrr2 2
rr
g / r 1= 2rK g =
g 1-Kr
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
S.7
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:
λ αλ1μν αμ,ν αν,μ μν,α2
Γ = g g +g -g
Relativistic Hydrodynamics
S.8
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: