Introduction to tensor calculus - Angelfire

23 Pages·2007·0.14 MB·English

Introduction to tensor calculus
Dr. J. Alexandre
King’s College
These lecture notes give an introduction to the formalism used in Special and General
Relativity, at an undergraduate level.
Contents
1 Tensor calculus in Special Relativity 2
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Proper-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 4-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Scalar product in Minkowsky space-time . . . . . . . . . . . . . . . . . . . 6
1.6 Tensors in Minkowsky space-time . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Maxwell’s equations in tensor notations . . . . . . . . . . . . . . . . . . . . 12
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Tensor calculus in General Relativity 15
2.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Properties of the Cristo(cid:11)el symbol . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1
1 Tensor calculus in Special Relativity
1.1 Motivations
An inertial frame is a frame where an observer at rest does not feel any force and where a
free motion leads to a constant velocity (direction and intensity). If one wishes to go from
an inertial frame to another, i.e. to describe the physics with another point of view, the
changes of coordinates in Newtonian mechanics are given by the Galil(cid:19)ee transformations:
t0 = t
r~0 = ~r+~vt; (1)
where ~v is the velocity of one frame with respect to the other. The transformations (1)
imply that the acceleration ~a of a point particle is invariant in this change of frame and
~
thus a force f felt by the particle is also invariant if you impose the fundamental law of
~
newtonian mechanics f = m~a to be valid in any inertial frame.
Since the time is an invariant quantity in Galil(cid:19)ee transformations, the composition of
velocities is obviously given by
dr~0 d~r
= +~v: (2)
dt0 dt
It happens that the transformation (2) is not respected by the experiments (historically:
Michelson’s experiment on di(cid:11)raction patterns which implies that the speed of light is
independent of the inertial frame considered). The issue Einstein found was "simply" to
abandon the transformations (1) and to assume that t0 6= t.
The independence of the speed of light (that we note c) with respect to the inertial
frame leads us to the new postulate: the free motion of a point particle over the distance
j~rj during the time t is such that the quantity (cid:0)c2t2 +j~rj2 is independent of the inertial
observer.
Therefore if the free particle moves over the distance j~rj measured in a frame O (and
j~r0j measured in a frame O0) during the time t measured in O (and t0 measured in O0), we
will have
(cid:0)c2t2 +j~rj2 = (cid:0)c2t02 +jr~0j2: (3)
Eq.(3) must be considered as the starting point from which everything follows. It has the
status of a postulate.
1.2 Proper-time
Before going to the Lorentz transformations, let us de(cid:12)ne the proper time (cid:28) as the time
measured in a frame by a clock at rest. In another frame, this clock is labelled by (t;~r)
and from Eq.(3) for an in(cid:12)nitesimal motion, we can write
2
1
d(cid:28)2 = dt2 (cid:0) d~r2; (4)
c2
such that
dt 1
=
d(cid:28) 2
1(cid:0) 1 d~r
c2 dt
r
(cid:16) (cid:17)
Since the proper-time is measured in the rest frame, d~r =~v is the relative velocity between
dt
the two frames. We denote then (cid:13) = 1= 1(cid:0)v2=c2 and have
q
dt
= (cid:13): (5)
d(cid:28)
We will see in the next section the new law of velocities composition from which we can
justify that we always have v (cid:20) c, such that (cid:13) is always de(cid:12)ned.
For a general motion and not only uniform, we can consider the tangent inertial frame
at any moment and Eq.(5) will be valid all along the trajectory, with (cid:13) depending on the
time t. Thus the proper time measured on a (cid:12)nite motion between the times t and t will
1 2
be
t2 dt
(cid:1)(cid:28) = < (cid:1)t;
(cid:13)(t)
Zt1
such that the proper-time is always smaller than the time measured in any other frame.
1.3 Lorentz transformations
We will now look for the new transformations of coordinates, in a change of inertial frame,
which are consistent with the assumption (3). For this, we will consider a motion along the
x-axis, leaving the coordinates y and z unchanged and we will not suppose that the time is
an invariant quantity. Furthermore, we will look for linear transformations since we impose
them to be valid uniformly in space and time. We will thus look for transformations of the
form
t0 = Dx+Et
x0 = Ax+Bt
y0 = y
z0 = z: (6)
Plugging the transformations (6) in the assumption (3), we obtain
3
c2D2 (cid:0)A2 = (cid:0)1
c2E2 (cid:0)B2 = c2
c2ED(cid:0)AB = 0
Then if we consider the origin of the spatial coordinates of O, we obtain x0 = Bt and
t0 = Et, such that
B x0
= = v;
E t0
by de(cid:12)nition of v. We now have the 4 equations to determine A;B;D;E and (cid:12)nd easily
the Lorentz transformation
v
ct0 = (cid:13) ct+ x
c
(cid:18) (cid:19)
x0 = (cid:13)(x+vt); (7)
where (cid:13) = 1= 1(cid:0)v2=c2.
As a (cid:12)rst cqonsequence, let us come back to the composition of velocities. From Eqs.(7),
we obtain
dx0 dx+vdt dx +v
= = dt :
dt0 dt+ v dx 1+ v dx
c2 c2 dt
If we suppose that dx << c, we are led to dx0 ’ dx +v, which is the non-relativistic limit.
dt dt0 dt
Then we see that if dx = c, we also have dx0 = c, which was expected from the invariance
dt dt0
of the speed of light and which indicates that the latter is the maximum speed that one
can obtain.
This leads us to the following classi(cid:12)cation. Two events (t ;~r ) and (t ;~r ) related in
1 1 2 2
a way such that c2(t (cid:0)t )2 > (~r (cid:0)~r )2 are said to be separated by a time-like interval
2 2 2 1
(they can interact via an information that has a speed smaller than c). If they are such
that c2(t (cid:0)t )2 < (~r (cid:0)~r )2, they are said to be separated by a space-like interval (they
2 2 2 1
cannot interact). Finally, if they are such that c2(t (cid:0)t )2 = (~r (cid:0)~r )2, they are said to be
2 2 2 1
separated by a light-like interval (they can interact only via a ray of light).
Anotherconsequence ofthelorentz transformationsistheso-called’lengthcontraction’:
if you measure an in(cid:12)nitesimal length dx0 in a rest frame with the corresponding length dx
measured in another frame at one given time t, you obtain from (7)
dx0
= (cid:13);
dx
4
such that the length measured in the rest frame (the proper-length) is always larger than
the one measured in any other inertial frame. This implies a ’contraction’ of an object
when it is measured by a moving observer.
Finally, a general Lorentz transformation includes also the coordinates y and z, but
always in a way such that the proper-time d(cid:28) remains an invariant quantity. We note that
the invariance of d(cid:28) is also satis(cid:12)ed by rotations in space, leaving the time unchanged, and
not only in a change of inertial frame.
1.4 4-vectors
We now come to the main purpose of this lesson. We wish to take into account these new
e(cid:11)ects in a formalism showing naturally the conservation of the in(cid:12)nitesimal proper-time
(4).
In Galil(cid:19)ee transformations, the time is an invariant quantity and we consider the trans-
formationsofvectors inspace. InLorentztransformations, thetimedepends ontheinertial
observer and thus should be part of a new vector in space-time, having 4 coordinates. We
will write such a 4-component vector x = (ct;~r) such that its components are x0 = ct and
xk = rk where k = 1;2;3. Its general components will be noted x(cid:22) where (cid:22) = 0;1;2;3.
The greek letters will denote space-time indices and the latin letters the space indices.
A 4-vector is de(cid:12)ned as a set of 4 quantities which transform under a Lorentz transfor-
mation in a change of inertial frame.
By de(cid:12)nition, x = (ct;~r) is a 4-vector, called the 4-position (we will note with an arrow
the vectors in the 3-dimensional space). Then we can use the proper-time which is an
invariant quantity and de(cid:12)ne the 4-velocity u with components
dx(cid:22) dx(cid:22) dt dx(cid:22)
u(cid:22) = = = (cid:13) ;
d(cid:28) dt d(cid:28) dt
such that u0 = (cid:13)c and ~u = (cid:13)~v, where ~v is the velocity in the 3-dimensional space. u is a
4-vector since it is constructed from another 4-vector and an invariant quantity. It must be
stressed here that the quantities dx(cid:22)=dt do not form a 4-vector: a Lorentz transformation
on these quantities does not lead to the equivalent quantities dx0(cid:22)=dt0 in the new frame.
Another example of 4-vector is the 4-momentum p = mu, where m is the mass of a
particle (a constant under Lorentz transformations). Its components are p0 = (cid:13)mc and
p~ = (cid:13)m~v, which is thus the relativistic momentum (in the 3-dimensional space). A Taylor
expansion of (cid:13) for v=c << 1 gives
1 1
p0 = mc2 + mv2 +::: ;
c 2
(cid:18) (cid:19)
where we recognize the kinetic energy mv2=2 and the mass energy mc2 de(cid:12)ned as the
energy of the particle in the rest frame. We will thus call the energy of the particle the
quantity E = (cid:13)mc2 and the 4-momentum can be written
5
E
p = ;(cid:13)m~v
c
(cid:18) (cid:19)
Other examples of 4-vectors will be seen after the de(cid:12)nition of their scalar product.
1.5 Scalar product in Minkowsky space-time
The idea is to de(cid:12)ne a scalar product which is left invariant in a change of inertial frame,
just as the usual scalar product is left invariant under a rotation in space.
Let x and y be two 4-vectors. We note x = x(cid:22)e and y = y(cid:23)e , where the 4-vectors e
(cid:22) (cid:23) (cid:22)
form a basis of the 4-dimensional space-time and the summation over repeated indices is
understood. We want the scalar product x:y of x and y to satisfy
x:y = y:x
x:(y +z) = x:y +x:z
(ax):y = x:(ay) = a(x:y)
where ais anyrealnumber. Withthese properties we have thenx:y = x(cid:22)y(cid:23)e :e = x(cid:22)y(cid:23)(cid:17)
(cid:22) (cid:23) (cid:22)(cid:23)
where
(cid:17) = (cid:17) = e :e :
(cid:22)(cid:23) (cid:23)(cid:22) (cid:22) (cid:23)
By de(cid:12)nition of the scalar product, we want it to be invariant in a change of frame, such
that we have to take
(cid:0)1 0 0 0
0 1 0 0
(cid:17) = 0 1;
(cid:22)(cid:23) 0 0 1 0
B C
B 0 0 0 1 C
B C
@ A
and (cid:0)x0y0+xkyk is left invariant. (cid:17) is called the metric tensor. We will see the de(cid:12)nition
(cid:22)(cid:23)
of tensors in the next section and see that (cid:17) is indeed a tensor. For the moment, we can
(cid:22)(cid:23)
consider it as an array 4 (cid:2) 4. Note that we could have taken the opposite signs for the
components (cid:17) . This would not have changed the Physics, of course, and is just a matter
(cid:22)(cid:23)
of convention.
We de(cid:12)ne then the new quantities
x = x(cid:23)(cid:17) ;
(cid:22) (cid:22)(cid:23)
6
which are called the covariant components of x, whereas those with the indice up are called
the contravariant components. We have then for any 4-vector x: x = (cid:0)x0 and x = xk
0 k
(k = 1;2;3). With this de(cid:12)nition, we can write the scalar product in a compact way
x:y = x(cid:22)y(cid:23)(cid:17)
(cid:22)(cid:23)
= (cid:0)x0y0 +x1y1 +x2y2 +x3y3
= x0y +x1y +x2y +x3y
0 1 2 3
= x(cid:22)y = x y(cid:22)
(cid:22) (cid:22)
We note that in a scalar product we always sum a covariant indice with a contravariant
one, and not two of the same kind.
We also de(cid:12)ne (cid:17)(cid:22)(cid:23) as being the components of the inverse matrix of (cid:17) , such that
(cid:22)(cid:23)
(cid:17)(cid:22)(cid:26)(cid:17) = (cid:14)(cid:22);
(cid:26)(cid:23) (cid:23)
where (cid:14)(cid:22) is the Kronecker symbol (1 if (cid:22) = (cid:23) and 0 otherwise). We have obviously
(cid:23)
x(cid:22) = (cid:17)(cid:22)(cid:23)x and
(cid:23)
(cid:0)1 0 0 0
0 1 0 0
(cid:17)(cid:22)(cid:23) = 0 1:
0 0 1 0
B C
B 0 0 0 1 C
B C
@ A
Finally, we notethatx2 = 0does notimply x = 0, and we can have x2 < 0. The space-time
with this scalar product is called the Minkowsky space-time.
Let us check explicitely that this scalar product is invariant under the transformations
(7). We have for two 4-vectors p = (p0;p~) and q = (q0;~q)
p0(cid:22)q0 = (cid:0)p00q00 +p01q01 +p02q02 +p03q03
(cid:22)
v v v v
= (cid:0)(cid:13)2 p0 + p1 q0 + q1 +(cid:13)2 p1 + p0 q1 + q0 +p2q2 +p3q3
c c c c
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
= (cid:0)p0q0 +p1q1 +p2q2 +p3q3
= p(cid:22)q
(cid:22)
and (cid:12)nd the expected result. In general, we de(cid:12)ne a Lorentz scalar as a quantity invariant
under a change of inertial frame.
To give another illustration of this invariance, let us compute the square of the 4-
momentum:
p2 = p(cid:22)p = (cid:0)((cid:13)mc)2 +((cid:13)m~v)2 = (cid:0)m2c2;
(cid:22)
7
which is indeed an invariant quantity. This last relation can also be written
E2 = c2jp~j2 +m2c4:
Let us now see an example taken from classical electrodynamics. Maxwell’s equations
are of course relativistic since they describe the propagation oflight and are at the origin of
the theory of relativity! Thus the scalar quantities appearing in the solutions of Maxwell’s
equations must be Lorentz scalars. We have for example the phase of a monochromatic
plane wave which can be written
(cid:18) = (cid:0)!t+~k~r = k(cid:22)x ;
(cid:22)
where k(cid:22) = (!=c;~k). Since x is a 4-vector and its scalar product with k(cid:22) gives a Lorentz
(cid:22)
scalar, we conclude that k(cid:22) is a 4-vector. We can derive from this example the relativistic
Doppler e(cid:11)ect. Suppose that a source of light of proper frequency ! (i.e. in a frame where
0
the source is at rest) and proper-time (cid:28) moves with a constant velocity ~v with respect to
an inertial observer who measures the frequency !, the time t and the position ~r of the
source. If ~n is the unit vector pointing from the observer towards the source at the time of
emission, we know from the invariance of the phase (cid:18) that during an in(cid:12)nitesimal motion
!
~
! d(cid:28) = !dt(cid:0)k:d~r = !dt+ ~n:d~r;
0
c
such that,
dt ~n:d~r dt ~n:~v
! = ! + = (cid:13)! 1+
0
d(cid:28) cdt d(cid:28)! c !
We note that, unlike the non-relativistic case, we have a Doppler e(cid:11)ect when ~n:~v = 0.
Let us now consider the situation of the derivatives with respect to space-time indices.
We have
@(x2) @
= (x(cid:26)x(cid:27)(cid:17) )
@x(cid:22) @x(cid:22) (cid:26)(cid:27)
= (cid:17) ((cid:14)(cid:26)x(cid:27) +(cid:14)(cid:27)x(cid:26))
(cid:26)(cid:27) (cid:22) (cid:22)
= (cid:17) x(cid:27) +(cid:17) x(cid:26)
(cid:27)(cid:22) (cid:22)(cid:26)
= 2x
(cid:22)
such that the derivative with respect to a contravariant component is a covariant compo-
nent. Wecansee inthesameway thatthederivative withrespect toacovariantcomponent
is a contravariant component. Thus we note
8
@ @
= @ and = @(cid:22); (8)
@x(cid:22) (cid:22) @x
(cid:22)
and the derivatives (8) are 4-vectors. If we come back to classical electrodynamics, the
conservation of the electric charge can be written
@(cid:26) @(c(cid:26)) @jk
0 = +r~ :~(cid:17) = + = @ j(cid:22) (9)
@t @x0 @xk (cid:22)
where j(cid:22) = (c(cid:26);~(cid:17)) is a 4-vector (the 4-current) since its scalar product with @ gives zero
(cid:22)
in any inertial frame (the ’continuity equation’ (9) is of course Lorentz invariant since it is
derived from Maxwell’s equations).
1.6 Tensors in Minkowsky space-time
Now that we have de(cid:12)ned 4-vectors, we are naturally led to a generalization of this classi(cid:12)-
cation of physical quantities with respect to their transformation under a change of inertial
frame, what will be done with tensors. Let us (cid:12)rst de(cid:12)ne for this the tensorial product
M = M (cid:10) M, where M is the Minkowsky space-time. As M, M is a vector space. Its
dimension is 4(cid:2)4 and its elements have the following properties
x(cid:10)(y+z) = x(cid:10)y+x(cid:10)z
(ax)(cid:10)y = x(cid:10)(ay) = a(x(cid:10)y);
where x, y and z are 4-vectors and a is a scalar. With these properties, we can write in
terms of components
x(cid:10)y = x(cid:22)y(cid:23)e (cid:10)e (10)
(cid:22) (cid:23)
where e (cid:10)e are 4(cid:2)4 basis tensors of M. The 16 components of x(cid:10)y are thus given by
(cid:22) (cid:23)
x(cid:22)y(cid:23). A general tensor of rank two is a linear combination of terms like (10) and can be
written
T = T(cid:22)(cid:23)e (cid:10)e ; (11)
(2) (cid:22) (cid:23)
where T(cid:22)(cid:23) are its contravariant components. By making r tensorial products, one can
de(cid:12)ne a tensor of rank r which has the general form
T = T(cid:22)1:::(cid:22)re (cid:10):::(cid:10)e :
(r) (cid:22)1 (cid:22)r
We de(cid:12)ne the covariant components of a tensor as we did with the 4-vectors:
T (cid:23) = T(cid:26)(cid:23)(cid:17) and T = T(cid:26)(cid:27)(cid:17) (cid:17) ;
(cid:22) (cid:22)(cid:26) (cid:22)(cid:23) (cid:22)(cid:26) (cid:23)(cid:27)
9
and similarily for higher order tensors. Finally we note that a 4-vector is a tensor of rank
1.
Let us now come to a fundamental property of tensors. This property deals with the
transformation law of a tensor under a change of inertial frame and can actually be seen
as a de(cid:12)nition of tensors in Minkowsky space-time. In a general Lorentz transformation,
the components of a 4-vector transform like
x(cid:22) (cid:0)! x0(cid:22) = (cid:3)(cid:22)x(cid:23):
(cid:23)
The speci(cid:12)c example that was derived in section 3 is a particular case which took into
account the transformations of x0 and x1 only. In this situation, the matrix (cid:3)(cid:22) was
(cid:23)
(cid:13) v(cid:13) 0 0
c
v(cid:13) (cid:13) 0 0
(cid:3)(cid:22) = 0 c 1;
(cid:23) 0 0 1 0
B C
B 0 0 0 1 C
B C
@ A
but we consider now a general Lorentz transformation, a(cid:11)ecting all the 4 components x(cid:22).
The coe(cid:14)cients (cid:3)(cid:22) being constant, we have @x0(cid:22)=@x(cid:23) = (cid:3)(cid:22) and thus
(cid:23) (cid:23)
@x0(cid:22)
x0(cid:22) = x(cid:23): (12)
@x(cid:23)
A general 4-vector can then be written in the two di(cid:11)erent frames
@x0(cid:22)
x = x(cid:23)e and x = x0(cid:22)e0 = x(cid:23)e0 ;
(cid:23) (cid:22) @x(cid:23) (cid:22)
such that the transformation of the basis 4-vectors is given by
@x0(cid:22)
e = e0 :
(cid:23) @x(cid:23) (cid:22)
Then we conclude from the de(cid:12)nition (11) that the contravariant components of a tensor
of rank 2 transform as
@x0(cid:22) @x0(cid:23)
T(cid:22)(cid:23) (cid:0)! T0(cid:22)(cid:23) = T(cid:26)(cid:27) : (13)
@x(cid:26) @x(cid:27)
Finally, the contravariant components of a tensor of rank r obviously transform as
@x0(cid:22)1 @x0(cid:22)r
T(cid:22)1:::(cid:22)r (cid:0)! T0(cid:22)1:::(cid:22)r = T(cid:26)1:::(cid:26)r ::: ;
@x(cid:26)1 @x(cid:26)r
10