Introduction

The principal target of tensor calculus is to investigate the relations that remain the same when we change from one coordinate system to any other. The laws of physics are independent of the frame of references in which physicists describe physical phenomena by means of laws. Therefore, it is useful to exploit tensor calculus as the mathematical tool in which such laws can be formulated.

Transformation of Coordinates

Let there be two reference systems, S with coordinates (x1, x2,…xn) and with coordinates. (Fig. 1). The new system S depends on the old system S as

Here ϕi are single-valued continuous differentiable functions of (x1, x2,…xn) and further the Jacobian

Differentiation of Eq. (1.1) yields

Now and onward, we use the Einstein summation convention, i.e., omit the summation symbol Σ and write the above equations as

The repeated index r or m is known as dummy index. The index i is not dummy and is known as free index.

The transformation matrices are inverse to each other

The symbol is Kronecker delta, is defined as

Obviously vectors in system are linked with (S) system

Covariant and Contravariant Vector and Tensor

Usually one can describe the tensors by means of their properties of transformation under coordinate transformation. There are two possible ways of transformations from one coordinate system .xi) to the other coordinate system .

Let us consider a set of n functions Ai of the coordinates xi. The functions Ai are said to be the components of covariant vector if these components transform according to the following rule

Also, one can find by multiplying and using and

Gradient of a scalar is a covariant vector.

Here, Ai is known as the covariant tensor of first order or of the type (0, 1).

The functions Ai are said to be the components of the contravariant vector if these components transform according to the following rule

Also, one can find by multiplying both sides with and using

Here, Ai is known as the contravariant tensor of first order or of the type (1, 0).

Geodesics Equation

According to general theory of relativity, gravitation is not a force but a property of spacetime geometry. A test particle and light move in response to the geometry of the spacetime. Actually, curved spacetimes of general relativity are explored by reviewing the nature of the motion of freely falling particles and light through them. Freely falling particles are those particles that are free from any effects except curvature of spacetime. This chapter provides the derivation of the equations of motion of the test particles and light rays in a general curved spacetime.

The path or the differential equation of the curve having an external length, i.e., path of extremum distance between two points is called the geodesic equation.

Therefore, for a geodesic must be extremum, where the limits of integration are taken to be two fixed points A and B.

Thus,

For Riemannian space

Here,

is known as Lagrangian.

This implies Euler–Lagrange equation as

Here, p = affine parameter, describing the trajectory.

[In general, dτ(proper time) is proportional to dp, such that for a material particle, we can normalize p so that p = τ. Nevertheless, for a photon, the proportionality constant vanishes (as ds = 0, for photon)]

We can describe the geodesic line between two fixed points A and B. Let us consider the shortest path, i.e., the curve C is the geodesic line. The other two curves C′ and C′′, e.g., are different curves, other than geodesic C, with a variation δxμ (see Fig. 5). The geodesic line is described by xμ = xμ(s) = xμ(p), where s and p are parameters along the curve. The metric is defined as ds2 = gμ𝜈 dxμdx𝜈. The tangent vector to the curve xα = xα(s) at P is the unit vector. A and B are two fixed points.

Derivation of Euler–Lagrange Equation

We prove Euler–Lagrange equation from Eq. (2.1)

We can write the second term of (2.4) as the difference of two terms

The first expression yields zero after integration as the variations vanish at the end points of the curve. Thus, Eq. (2.4) can be written in the form

Introduction

Cosmology is the study of the universe as a whole. That means cosmology deals with the structure and evolution of the universe on the largest scales of space and time. Gravity plays the central role for the structure of the universe on these scales and regulates its evolution. Thus, general relativity is the main part of cosmological study and cosmology is one of the most important applications of general relativity. The universe consists of stars, gases, collections of matter (with zero rest mass or massive particles) with known and unknown characters and vacuum energy.

Note 11.1

Zero rest mass particles (photons, gravitons, etc.) are referred to as radiation and nonzero rest mass particles (protons, neutrons, electrons, etc.) are called matter. It is argued that protons and neutrons are built through more fundamental particles dubbed as quarks. The general terms for particles made up of these quarks are baryons. The neutrinos are extremely weak interacting particles, which are created, e.g., in radioactive decay. Neutrinos with very small masses behave approximately like radiation in some circumstances and matter in others.

The visible matter in the universe is mostly contained in galaxies—gravitationally bound collections of stars, gas, and dust. A typical galaxy has about 1011 stars and a total mass of 1012 times of solar mass, i.e., = 1012M⊙. In the observable universe, very roughly 1011 galaxies are seen. Assuming the visible matter in the galaxies is smoothed and uniformly distributed over the largest scales, then at the present day the density is approximately estimated as

which is equivalent to one proton per cubic meter.

Apart from galaxies, the universe comprises of radiation consisting of zero rest mass particles—photons and also some Newtonian and gravitational waves. The velocity of this radiation is similar to the velocity of light. The radiation is not gravitationally bound, clustered; that moves up the galaxies. Detected electromagnetic radiation that has been left-over from the hot big bang, with the greatest energy density, is the cosmic background radiation. To remarkable experimental precision, this is very similar to the black body spectrum with the temperature of 2.725 ± .001 K.

Spherically Symmetric Line Element

Spherically symmetric means an invariance under any arbitrary rotation of axes at a particular point, called the center of symmetry. Using θ and ϕ (polar coordinates) and choosing the center of symmetry at origin, we have the general form of the line element with spherical symmetry.

For the surfaces r = constant and t = constant, the line elements reduces to form two spheres on which a typical point is labeled by coordinate θ and ϕ and line element takes the form

This spherical symmetric line element is invariant when θ and ϕ are varied. The center of symmetry is the point O, which is given by r = 0.

Now we introduce new coordinates by the transformations:

where the function K will be chosen later.

From the above transformation equations, we have

Then the line element becomes

Now we choose K such that coefficient of dr′dt′ is zero.

Thus, we have

Hence we get general line element on

where σ, ω, and μ are functions of r′ and t′.

Now we take another transformation,

where q is so chosen that the coefficient of dR dT in ds2 is zero.

Thus, finally, we get the line element

where 𝜈 and μ are functions of R and T.

Note 6.1

Here R coordinate has specific significance:

The area of the surface of the sphere, R = constant is given by A = 4πR2.

Three volume of the sphere with radius R

e.g., for eμ = (1 - ar2)2, one can get

For the four-dimensional tube that is bounded by the sphere with radius R and two planes, t = constant, separated by a time T, the four volume is

e.g., for eμ = e𝜈 = (1 - ar2)2, one can get

Schwarzschild Solution or Exterior Solution

The exact solution of the Einstein field equation in empty space was obtained by Schwarzschild in 1916, which describes the geometry of spacetime outside a spherically symmetric distribution of matter.

Consider the metric of the empty spacetime outside of a spherically symmetric distribution of matter of mass M as

To describe the Hamiltonian formulation of a field theory in curved spacetime it is necessary to foliate W, a region in the spacetime, with a family of space-like hypersurfaces, Σ, in every “instant of time.” This is the purpose of the breakup of spacetime into space and time, i.e., 3+1 decomposition.

To express this decomposition one needs a scalar field t(xα) such that the surface of constant time (t = constant) represents a family of nonintersecting space-like hypersurfaces Σt.

Let nα α 𝜕αt, the unit normal to the hypersurfaces, be a future-directed time-like vector field. Now one can introduce new coordinates, ya in each hypersurface Σt. Let us assume a congruence of curves γ intersecting the hypersurfaces Σt (see Fig. 111).

[Let O be an open region in spacetime. A congruence in O is a family of curves such that through each point in O there passes one and only one curve from this family (the curves do not intersect)]

Let tα be the tangent to the congruence satisfying

Let a specific congruence curve γp pass through a point P on Σt and meet a point p′ on, and then a point p′′ on, etc. Now one can fix coordinates of p′ and P′′ by imposing

for a given ya(P) on Σt. Therefore, we can fix ya for every member of the congruence. Thus, in this way we can describe a new coordinate system (t, ya) in W.

Here spacetime is foliated in terms of space-like three-dimensional hypersurfaces Σt and the spacetime manifold W is diffeomorphic to R × Σt, where the manifold Σ t represents space and t ∈ R represents time. One can note that the particular slicing of spacetime into instants of time is fully arbitrary.

It is obvious that spacetime coordinate xμ must be some function of ya and t, i.e., xα = xα(t, ya).

Here,

Now, one can define the tangent vector on Σt as

Also Lie derivative of the tangent vector is zero, i.e.,

Newtonian Gravity

Newton's theory of gravitation can be treated as a three-dimensional field theory. The gravitational field is characterized by a scalar field ϕ(x; y; z). This satisfies

where G = 6.67×10-8cm3 gm-1 sec-2 is the gravitational constant and ρ is the mass density of matter in space that produces the gravitational field. The above equation is known as the Poisson equation.

Proof of Poisson Equation

Let us consider a mass M occupying a volume V, which is enclosed by a surface S. The gravitational flux passing through the elementary surface dS is given by g:ndS, where g is gravitational vector field (also known as gravitational acceleration) and n is the unit outward normal vector to S. Now, the total gravitational flux through S is

We know is the elementary solid angle dΩ subtended at M by the elementary surface dS, where er is the radial unit vector. Thus,

Applying Gauss divergence theorem to the left-hand side, we get

Thus, we get

Using g = -∇ϕ (gravity is a conservative force, therefore, it can be written as the gradient of a scalar potential ϕ, known as the gravitational potential), we finally obtain

The gravitational field (F) is proportional to the negative of ∇ϕ, i.e.,

This is the force acting on a particle of mass m.

For a single mass M that produces the potential ϕ, then the solution of Poisson equation is given by

The force acting on another particle with mass m will be

The ratio of gravitational force and electrical force between two electrons is given by

Here, ke is Coulomb's constant with me and e are mass and charge of the electron, respectively. This indicates that the gravitational force is very weak.

Exercise 4.1

Find the gravitational potential inside and outside of a sphere of uniform mass density having a radius R and a total mass M. Normalize the potential so that it vanishes at infinity.

Hints:

The mass density in the sphere is (see Fig. 8)

For spherically symmetric distribution of matter, the gravitational potential ϕ is a function of radius r only.

Introduction

To describe how matter and energy change the geometry of spacetime, Einstein proposed a new theory of gravitation, known as the general theory of relativity. This theory is characterized by a set of dynamical equations, dubbed as Einstein field equations, which demonstrate the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy. Albert Einstein published this new theory of gravitation in 1915 as a tensor equation. The motion of a body in this gravitational field is explained almost perfectly by the geodesic equations.

Einstein used three principles to develop his general theory of relativity.

• 1. Mach's Principle: Ernest Mach gave some ideas without any proof and arguments that are known as Mach's principle. There are several forms of Mach's principle. Some forms are executed in general relativity and others are not.

  • (i) Geometry is determined by the matter distribution.

  • (ii) If matter does not exist then the geometry will not occur.

  • These statements are not absolutely true as there are vacuum solutions of the Einstein field equations (see later).

  • (iii) The local inertial frame is entirely persistent by the dynamical fields in the universe.

  • Actually, this form of Mach's principle is the key idea of Einstein.

• 2. Equivalence Principle: The laws of physics are the same in all inertial systems. No preferred inertial system exists.

• (The above statements, i.e., containing the words “Laws of Physics” is referred to strong equivalence principle and when “Law of Physics” is replaced by “Law of motion of freely falling particles” then the principle is referred to weak principle of equivalence.)

  This principle reveals that one can eliminate gravity locally by falling freely in a gravitational field and regain special relativity.

• 3. Principle of covariance: The law of gravitation should be independent of the coordinate system. This means the field equations of gravitation should be invariant in form with respect to arbitrary coordinate transformations. In other words, the field equations should have a tensorial form.

Three Types of Mass

• 1. Inertial mass: This mass is a measure of the body's resistance to change its state, i.e., either in a rest position or in motion. The mass appears in Newton's second law F = mIf, i.e., here mI is known as inertial mass.

Null Tetrad

The tetrad formalism is a transformed coordinate approach to general relativity (GR) that replaces the coordinate basis by the local basis for the tangent bundle, which is less restricted. It is constituted by a set of four linearly independent vector fields known as a tetrad or vierbein. Usually, specific tetrad basis, which is a set of four independent vector fields, is used to represent a tensor in tetrad formalism. This basis spans the four-dimensional (4D) vector tangent space at each point in spacetime. Therefore, at any point P on the curve, we can present an orthogonal frame of three unit space-like vectors

which are all orthogonal to vα (which is unit tangent vector), a time-like vector and we define

These four vectors (i = 0, 1, 2, 3) are known to form a frame or tetrad at P.

Treating as a 4×4 matrix at P, we can define its inverse (called the dual basis or dual tetrad), which follows

Actually, are four linearly independent vector fields.

We introduce a new matrix gij defined by

which is known as frame metric.

Here, are linearly independent and the global metric, is nonsingular. As a result the matrix gij is nonsingular and hence invertible, i.e., gij exists, which is contravariant frame metric. Note that

Exactly the same way as metric tensor, we can raise and lower the tensor indices with the frame metric gij.

Eq. (10.2) yields the inverse relation as

Note that tetrad vectors determine the linear differential forms

and from this the metric takes the form

For a simple physical interpretation of the frame, we can think as the four velocity of an observer whose world line is C and three space-like vectors (i = 1, 2, 3) are rectangular coordinate vectors (such as usual cartesian basis) at P. For different tetrad, one gets different frame metric.