MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 7 pdf

The signifi- cance of the Riemann curvature tensor is, that all of its components vanish only in flat space, that is we cannot find a coordinate system where R i j k l = 0 10.232 unles

Similarly, the covariant derivative of a contravariant vector is defined as

(AB)., = A,;B + AB,i (10.210) and

(uA + bB);i = u A ; ~ + bB,i (10.211) where A and B are tensors of arbitrary rank and a and b are scalars

10.9.8 Some Covariant Derivatives

In the following we also show equivalent ways of writing these operations commonly encountered in the literature

1 Using definition Equation (10.123) we can write the covariant derivative

of a scalar function @ as an ordinary derivative:

This is also the covariant component of the gradient

(9.);

(10.212)

(10.213)

2 Using the symmetry of Christoffel symbols, the curl of a vector field 3

can be defined as the second-rank tensor

(bx 3) , = aiv, - a,vj = vi;j - vj;i (10.214)

2 3

(10.215)

Note that because we have used the symmetry of the Christoffel symbols, the curl operation can only be performed on the covariant components

of a vector

3 The covariant derivative of the metric tensor is zero:

a k g i j = &;k = 0, (10.216) with Equation (10.209) and the definition of Christoffel symbols the proof is straightforward

4 A frequently used property of the Christoffel symbol of the second kind

is

In the derivation we use the result

from the theory of matrices, where g = det g i j

5 We can now define covariant divergence as

If vz is a tensor density of weight +1, divergence becomes

V.v+==U!Z (= d i d ) , (10.222) which is again a scalar density of weight f l

6 Using Equation (10.213) we write the contravariant component of the gradient of a scalar function as

We can now define the Laplacian as a scalar field:

(10.223) (10.224)

10.9.9 Riemann Curvature Tensor

Let us take the covariant derivative of ui twice The difference

Actually, there is one more symmetry that we will not discuss The signifi- cance of the Riemann curvature tensor is, that all of its components vanish

only in flat space, that is we cannot find a coordinate system where

R i j k l = 0 (10.232)

unless the space is truly flat

which is obtained from R i j k l by contracting its indices as

An important scalar in Riemann spaces is the Riemann curvature scalar,

.

R == y'lgikR ajkl = j L R i 221 = R a , 3 32 (10.233)

Note that &jkl = 0 implies R = 0, but not vice versa

Example 10.1 Laphcian as a scalar field: We consider the line element

ds2 = dr2 + r 2 d 2 + r2 sin2 Bd$, (10.234) where

x1 = r, x 2 = 8, x3 = 4 (10.235)

and

911 = 1, g22 = r2, 933 = r 2 sin28 (10.236)

Contravariant components 9'3 are:

(10.237) Using Equation (10.225) and g = r4 sin2 0, we can write the Laplacian

as

(10.238)

After simplifying, the Laplacian is obtained as

(10.239) Here we have obtained a well-known formula in a rather straightfor- ward manner, demonstrating the advantages of the tensor formalism Note that even though the components of the metric tensor depend on position [Eq (10.236)], the curvature tensor is zero,

R i j k i = 0; (10.240) thus the space of the line element [Eq (10.234)] is flat However, for the metric

it can be shown that not all the components of R i j k l vanish In fact, this line element gives the distance between two infinitesimally close points

on the surface of a hypersphere (S-3) with constant radius &

10.9.10 Geodesics

Geodesics are defined as the shortest paths between two points in a given geometry In flat space they are naturally the straight lines We can gener- alize the concept of straight lines as curves whose tangents remain constant along the curve However, the constancy is now with respect to the covariant derivative If we parametrize an arbitrary curve in terms of arclength s as

its tangent vector will be given as

10.9.11 lnvariance and Covariance

We have seen that scalars preserve their value under general coordinate trans formations Certain other properties like the magnitude of a vector and the trace of a second-rank tensor also do not change under general coordinate transformations Such properties are called invariants They are very im-

portant in the study of the coordinate-independent properties of a system

An important property of tensors is that tensor equations preserve their form under coordinate transformations For example, the tensor equation

transforms into

(10.247)

This is called covariance Under coordinate transformations individual com-

ponents of tensors change; however, the form of the tensor equation remains the same One of the early uses for tensors in physics was in searching and expressing the coordinate independent properties of crystals However, the covariance of tensor equations reaches its full potential only wit$h the intro-

duction of the spacetime concept and the special and the general theories of relativity

10.10 SPACETIME AND FOUR-TENSORS

fig 10.8 Minkowski spacetime

Because there is no limit to the energy that one could pump into a system, this formula implies that in principle one could accelerate particles to any desired velocity In classical physics this makes it possible to construct infinitely fast signals to communicate with the other parts of the universe Another property of Newton’s theory is that time is universal (or absolute), that is, identical clocks carried by moving observers, uniform or accelerated, run at the same rate Thus once two observers synchronize their clocks, they will remain synchronized for ever In Newton’s theory this allows us to study systems with moving parts in terms of a single (universal) time parameter With the discovery of the special theory of relativity it became clear that clocks carried by moving observers run at different rates; thus using a single time parameter for all observers is not possible

After Einstein’s introduction of the special theory of relativity another remarkable contribution toward the understanding of time came with the introduction of the spacetime concept by Minkowski Spacetime not only strengthened the mathematical foundations of special relativity but also paved the way to Einstein’s theory of gravitation

Minkowski spacetime is obtained by simply adding a time axis orthogonal

to the Cartesian axis, thus treating time as another coordinate (Fig 10.8) A point in spacetime corresponds to a n event However, space and time are also fundamentally different and cannot be treated symmetrically For example, it

is possible to be present at the same place at two different times; however, if

we reverse the roles of space and time, and if space and time were symmetric, then it would also mean that we could be present at two different places at the

same time So far there is no evidence for this, neither in the micro- nor in the macro-realm Thus, in relativity even though space and time are treated on equal footing as independent coordinates, they are not treated symmetrically This is evident in the Minkowski line element:

ds)’ = c2dt2 - dx)’ - dy2 - dZ2, (10.249) where the signs of the spatial and the time coordinates are different It is for this reason that Minkowski spacetime is called pseudo-Euclidean In

this line element c is the speed of light representing the maximum velocity

in nature An interesting property of the Minkowski spacetime is that two events connected by light rays, like the emission of a photon from one galaxy and its subsequent absorption in another, have zero distance between them even though they are widely separated in spacetime

10.10.2

In Minkowski spacetime there are many different ways to chotlse the orienta- tion of the coordinate axis However, a particular group of coordinate systems, which are related to each other by linear transformations of the form

In 1905 Einstein published his celebrated paper on the special theory of relativity, which is based on two postulates:

First postulate of relativity: It is impossible to detect or measure uniform translatory motion of a system in free space

Second postulate of relativity: The speed of light in free space is the maximum velocity in the universe, and it is the same for all uniformly moving observers

In special relativity two inertial observers K and E, where is moving uniformly with the velocity 21 along the common direction of the d- and

x3 2 3

fig 10.9 Lorenta transformations

?$-axes are related by the Lorentz transformation (Fig 10.9):

K is arbitrary in direction, then the Lorentz transformation is generalized as

(10.260) (10.261)

8 = y [xo- (3.34

We have written y = l/Jm and 3 = 3 / c

10.10.3

Two immediate and important consequences of the Lorentz transformation equations [Eqs (10.252-10.255)] are the time dilation and length contraction formulas, which are given as

Time Dilation and Length Contraction

respectively These formulas relate the time and the space intervals measured

by two inertial observers ?7 and K The second formula is also known as the Lorentz contraction The time dilation formula indicates that clocks carried

by moving observers run slower compared to the clocks of the observer a t rest Similarly, the Lorentz contraction indicates that meter sticks carried by

a moving observers appear shorter to the observer a t rest

10.10.4 Addition of Velocities

Another important consequence of the Lorentz transformation is the formula for the addition of velocities, which relates the velocities measured in the K and 1T frames by the formula

If the axes in K and ?? remain parallel, but the velocity 3 of frame 1T

in frame K is arbitrary in direction, then the parallel and the perpendicular components of velocity transform as

(10.266) (10.267)

In this notation U I I and 3 1refer to the parallel and perpendicular components with respect to d and y = (1 - Y ~ / c ~ ) - ~ / ~

10.10.5 Four-Tensors in Minkowski Spacetime

From the second postulate of relativity, the invariance of the speed of light means

(10.268) This can also be written as

7japdEadZp = gapdxadxB = 0, ( 10.269) where the metric of the Minkowski spacetime is

- ga(3 = gap = [ i ; [I i]-

We use the notation where the Greek indices take the values 0,1,2,3 and the Latin indices run through 1,2,3 Note that even though the Minkowski space time is flat, because of the reversal of sign for the spatial components it is not Euclidean; thus the covariant and the contravariant indices differ in space time Contravariant metric components can be obtained using (Gantmacher)

(10.274) -a

x =a;xP

For the Lorentz transformations [Eqs (10.252-10.255) and (10.260-10.261)],

a$ are given respectively as

~2 = [gaoa~ag] d x ~ d z 6 (10.280)

If we restrict ourselves to transformations that preserve the length of a vector

we obtain the relation

[&p";'L?j] = gra- (10.281) This is the analog of the orthogonality relation [Eq (10.38)) The position vector in Minkowski spacetime is called a four-vector, and its components

transform as Zn = agxo, where its magnitude is a four-scalar

An arbitrary four-vector

A =A" = (Ao, A', A2,A3), (10.282)

is defined as a vector that transforms like the posit,ion vector x" as

Fig 10.10 Worldlines and four-velocity

Similarly we can define four-acceleration as

du"

d r

d2xa d.r2 '

-

From the line element [Eq (10.278)], it is seen that the proper time

&=-= ds ( I - - $&,

C

is the time that the clocks carried by moving observers measure

10.10.7 Four-Momentum and Conservation Laws

Using four-velocity, we can define a four-momentum as

magnitude of the four-momentum as

pap, = muZLaua = const (10.295)

To evaluate the constant value of pupu we use the line element and the defi- nition of the proper time to find uUuu as

(10.306)

( 10.307) (10.308)

The second term inside the square brackets is the classical expression for the kinetic energy of a particle; however, the first term is new to Newton's mechanics It indicates that free particles, even when they are at rest, have energy due to their rest mass This is the Einstein's famous formula

This is the mass of a particle moving with velocity w It says that as the speed

of a particle increases its mass (inertia) also increases, thus making it harder

to accelerate As the speed of a particle approaches the speed of light, its inertia approaches infinity, thus making it impossible to accelerate beyond c

This immediately suggests a wave four-vector as

We have written y = l/d- and 3 = $/c

For light waves

~ / 2 , that is, when light is emitted perpendicular to the direction of motion

10.10.10 Derivative Operators in Spacetime

Let us now consider the derivative operator

a

thus - transforms like a covariant four-vector In general we write the four-gradient operator f3XB as

a a"= 8%" - (&> -")

or

8" = ( & , d )

(10.327)

(10.328)

Four-divergence of a four-vector is a four-scalar:

The wave (d'Alembert) operator in space time is written as

of A; hence we can write

Fig 10.11 Orientation of the axis with respect to the K frame

We rearrange this as

A%, = AB (&a$) (10.337) Since a and p are dummy indices, we can replace p with a to write

A%, = A" (&a:') , (10.338) which gives us the transformation law of the basis vectors as

Fig 10.12 Orientation of the K axis with respect to the 5? frame

and its inverse as

(10.343) (10.344)

The second set gives the orientation of the R axis in terms of the K axis Since ,L? < 1, relative orientation of the axis with respect to the K axis can

be shown as in Figure 10.11

Similarly, using the first set, we can obtain the relative orientation of the

K axis with respect to the IT axis as shown in Figure 10.12

10.10.12

Before the spacetime formulation of special relativity, it was known that Maxwell's equations are covariant (form-invariant) under Lorentz transfor- mations However, their covariance can be most conveniently expressed in terms of four-tensors

First let us start with the conservation of charge, which can be expressed

Maxwell's Equations in Minkowski Spacetime

as

(10.345)

where p is the charge density and f is the current density in space Defining

(10.358) where is the spatial momentum and T is the velocity of the charged particle We can write this in covariant form by introducing four-momentum

(10.359) (10.360) where mo is the rest mass, urn is the four-velocity, and po = E/c Using the derivative in terms of invariant proper time we can write Equation (10.358)

as

(10.361)

10.10.13 Transformation of Electromagnetic Fields

Because Fa@ is a second-rank four-tensor, it transforms as

(10.363) Given the values of Fys in an inertial frame K , we can find it in another inertial frame 17 as

(10.364)

a@ a B y6

F = aya6F

If IT corresponds t o an inertial frame moving with respect to I( with velocity

21 along the common Z1- and 21-axes, the new components of 3 and + B are

(10.365) (10.366) (10.367)