Practically, every fundamental equation in physics can be found with the support of a variational principle, taking appropriate Lagrangian or action in different cases.

Hamilton's variational principle asserts that Lagrange's equation of motion. δS = 0 ⇒ extremization of S ⇔ equations of motion. Here, boundary conditions are provided externally. This can be generalized from Newtonian mechanics to classical field theory as in Maxwell's electrodynamics or Einstein's general relativity.

Newtonian Mechanics

Here, the action functional is given by

Here, the integration is over a specific path of the generalized coordinates q(t). For a variation δq(t) of this path, δq(t1) = δq(t2) = 0:

Extremization of the action function ⇒ δS = 0, i.e.,

which is Euler–Lagrangian equation for a one-dimensional mechanical system.

Field Theory

Here, we are interested in the dynamics of a field q(xα) in curved spacetime.

Let us consider an arbitrary region W of the spacetime manifold, bounded by a closed hypersurface 𝜕W. The Lagrangian L.q, q,α) depends on a scalar function of the field and its first derivative.

Thus, action function is given by

The variation of q is arbitrary within W but vanishes on 𝜕W, [δq]𝜕W = 0.

Now,

(by Gauss divergence theorem)

Thus, using [δq]𝜕W = 0 we get

which is Euler–Lagrange equation for a single scalar field q.

Examples of Lagrangian for some fields

(a) A scalar field ψ

This can represent, e.g., the π0 meson. The Lagrangian

Euler–Lagrange equations are

This is Klein–Gordon equation in curved space.

A charged scalar field Ψ

Here Ψ = Ψ1 + iΨ2, which could represent, e.g., π+ and π- meson.

The total Lagrangian of the scalar field and electromagnetic field is

e → constant,

is complex conjugate of Ψ.

Varying Ψ, and Aa independently, one obtains the following Euler–Lagrange equations

its conjugate, and

General Relativity

In general relativity, the action functional consists of two different entities namely, SG[g] from gravitational field gab and SM[ϕ, g] from matter distribution.

Introduction

The Lie derivative is a significant concept of differential geometry, named after the discovery by Sophus Lie in the late nineteenth century. It estimates the modification of a tensor field (containing scalar function, vector field), along the flow defined by an additional vector field. Lie derivative can be defined on any differentiable manifold as this change is coordinate invariant.

A vector field X is a linear mapping from C∞ function to C∞ function on a manifold, satisfying

such that

Suppose Xμ(x) is a vector field defined over a manifold M. Trajectory of Xμ is obtained by solving

Let us consider a coordinate transformation

where ε is a parameter. This is known as one parameter set of transformation. This transformation designates a mapping of the spacetime onto itself. If the transformation takes the form

then it is called infinitesimal one parameter transformation or infinitesimal mapping. Here, ξμ(x) is a contravariant vector field defined by

[Infinitesimal mapping means that for each point P(xμ) of the spacetime there corresponds another neighboring point Q(xμ + εξμ) in the same coordinate system (see Fig. 10).]

Suppose a tensor T(x) defined in our spacetime. At the point Q, one can assess the tensor (T.x) in two different ways:

• 1. One can have the value of T at Q, i.e., T(xμ).

• 2. The T(xμ) can be obtained as transmuted tensor T using normal coordinate.

Thus, transformation for tensor at the point Q has two techniques. The difference among these two values of the tensor calculated at the point Q(xμ) hints to the concept of Lie derivative of the tensor T. One can differentiate the functions, tensor fields with respect to a vector field.

Lie Derivative of a Scalar

Let at point Q, the value of scalar ϕ(x) be ϕ(x), i.e.,

[neglecting higher power of ε]

Also, transformation of scalar function ϕ(x) remains unaffected. Thus,

One can define the Lie derivative of scalar function ϕ(x) (denoted as Lξϕ) as

As ϕ is a scalar, therefore, then partial derivative can be replaced by a covariant derivative to yield,

Motion of Test Particle

Let us consider the motion of a massive test particle or a massless particle, i.e., photon in Schwarzschild spacetime. It is known that all massive particles move along time-like geodesics, whereas photons move along null geodesics. We shall consider geodesics of test particles (either time-like or null) in Schwarzschild spacetime.

Let us take the Lagrangian in the following form as (with p is an affine parameter)

We know

where

⇒ Euler–Lagrangian equation

Thus, the corresponding Euler–Lagrange's equations are

[“1” implies differentiation with respect to p]

We know the first integral of geodesics equation is

where ε = 1 for the time-like geodesics and ε = 0 for null geodesics.

For the above Lagrangian (7.1), we get the following equations of motion

Let us consider the motion in equatorial plane then. Also equation (7.3) implies This indicates that if initially θ assumes the constant value, then throughout the geodesic it assumes the same value. This means planar motion is possible in general relativity, in other words, the geodesic is confined to a single plane as in Newtonian mechanics.

Here, ϕ and t are cyclic coordinates.

Equation (7.4) ⇒

The above equation indicates the principle of conservation of angular momentum with h as the angular momentum.

From (7.5), we obtain,

Here E is the energy per unit mass. This is the conservation principle of energy.

(Actually h and E are the conservation principles corresponding to the cyclic nature of the coordinate ϕ and t.).

Using (7.6) and (7.7), we get from Eq. (7.2) as

Equations (7.6) and (7.8) yield

Changing the r coordinate by a new one as, we get,

Now, we differentiate both sides of the above equation with respect to ϕ to yield

This is the relativistic equation of the motion of particles (massive or massless) in the gravitational field of Schwarzschild spacetime. In the theory of gravitation, the Newtonian equation of motion is

Thus in relativistic treatment, the last term 3mU2 is extra. However, this relativistic correction is very small.

At the beginning of the twentieth century, Einstein spent many years developing a new theory in physics. The newly developed theory is known as the theory of relativity. This is basically a combination of two theories: the first one is known as the special theory of relativity and latter one is dubbed as the general theory of relativity. The special theory of relativity is based on two postulates, namely the principle of relativity or equivalence, that is, the laws of physics are the same in all inertial systems, which means no preferred inertial system exists, while the second postulate is the principle of the constancy of the speed of light. The general theory of relativity asserts that there is no difference between the local effects of a gravitational field and that of acceleration of an inertial system. In other words, spacetime is warped or distorted by the matter and energy in it as an effect of gravity. According to the general theory of relativity, massive objects cause the outer space to twist due to gravity like a heavy ball bending a thin rubber sheet that is holding the ball. Heavier balls bend spacetime far more than lighter ones. Like the special theory of relativity, the general theory of relativity attracted scientists a lot, immediately after its discovery by Einstein. As a result, it has been included as a compulsory subject in graduate and postgraduate courses of physics and applied mathematics all over the globe. Einstein proposed the field equations for the general theory of relativity by applying his own intuition. Later, many other methods were developed to construct Einstein's field equations.

This book on the general theory of relativity is an outcome of a series of lectures delivered by me, over several years, to postgraduate students of mathematics at Jadavpur University. I should mention that it is not a fundamental book. This book has been written, from a mathematical point of view, after consulting several books existing in the literature. I have provided the list of the reference books. During my lectures, many students asked questions that helped me know their needs as well as the shortcomings in their understanding. Therefore, it is a well-planned textbook that has been organized in a logical order and every topic has been dealt with in a simple and lucid manner.

Minkowski Spacetime

Exact solutions of Einstein equations mean that the spacetime metric, which satisfies the Einstein field equations with stress-energy tensor Tab

Now, we will study the causal structures of some exact solutions of Einstein field equations.

The most simple empty spacetime in general theory of relativity is Minkowski spacetime. This is actually the spacetime in special theory of relativity. Using the natural coordinates (x1, x2, x3, x4) on R4 = M (M = manifold), one can express the metric in the form

with the range of coordinates as -∞ < x1, x2, x3, x4 < ∞. In this spacetime, all the components of Riemann tensor, therefore, it is a flat spacetime. The vector offers a time orientation of this spacetime.

For the choice of spherical polar coordinates (t, r, θ, ϕ) where

the metric assumes the following form,

In these new coordinate system the ranges are

Here all the Christoffel symbols will not all vanish. However, due to flat spacetime, all the Riemann curvature components will vanish.

To know the structure of infinity in Minkowski spacetime is our next target. For this, we use the interesting representation of this spacetime proposed by Roger Penrose.

A two-dimensional diagram of the spacetime where the causal relations and infinity structure are depicted through the use of conformal transformations is known as the Penrose diagram. They are also called the conformal diagram.

Actually conformal is a methodological word related to the rescaling of size. The light cones could not be changed under conformal transformation. The metric on a Penrose diagram and the actual metric in spacetime are conformally equivalent. The conformal factor is selected in such a way that the whole infinite spacetime is converted into a finite size in the Penrose diagram.

Result: If any two metrics G and g are chosen from the same manifold M such that they are connected by a positive definite conformal factor Ω2(x) as

then the null geodesics with respect to the metric G are the same to the null geodesics with respect to the metric g and vice-versa.

Stellar Structure and Evolution of Stars

Everyone, whether he or she is literate or illiterate, has a query what the stars are!

Our universe consists of billions of galaxies of various shapes: elliptical, spiral, irregular, etc., and the universe contains 1011 galaxies and each galaxy consists of 1011 stars. We live in a spiral galaxy visible in the night sky, known as Milky Way. It is convex shaped with diameter, 105 l.y. and thickness about 5000 l.y.

The apparent brightness and color serve as the characteristics for identifying a star. All stars do not have the same brightness. In second-century BC, the Greeks had grouped naked eye visible stars into six classes, called the magnitudes [brightest, the first magnitude; faintest stars being the sixth magnitude].

All measurements of the brightness and other characteristics of a star are relative, i.e., one has to compare either two stars with each other or a star with an artificial standard source of light. For easy access, people use the sun as standard one, mass: M⊙ = 1.98892 × 1030 kg, diameter: 1,391,000 km, radius: R⊙ = 695, 500 km, the surface gravity of the sun: 27.94 g, volume of the sun: 1.412×1018 km3, the density of the sun: 1.622×105 kg/m3, luminosity of sun L⊙ = 3.846×1026 W.

Before 1840, people used the eye; in 1840-1940, people used the photographic method. From 1940 onwards, people use the photoelectric method (Fig. 104) to measure the brightness (light) of a star. For the latter case, starlight is allowed to fall on the cathode of a photocell and the resulting photocurrent is amplified and recorded on a pen recorder. The deflection is then directly proportional to the intensity.

Two of the basic observable parameters in stars are:

• (i) The luminosity L (or absolute magnitude M)

• (ii) The effective temperature TE (or spectrum)

In the years 1911 and 1913, two scientists Hertzsprung and Russell individually observed that the luminosities and surface temperatures of stars are interrelated. This interrelation is generally demonstrated in a two-dimensional diagram, known as Hertzsprung–Russell or H-R diagram in which the vertical axis represents the luminosity and the horizontal axis represents the surface temperature (Fig. 105). The stars can be identified by a point with coordinate (TE, L) on this diagram.

A hypersurface Σ is a three-dimensional sub-manifold of a four-dimensional spacetime manifold M. The hypersurface may be time-like, space-like or null.

Let us consider that xμ (μ = 1; 2; 3; 4) are the coordinate of the four-dimensional spacetime, and ya(a = 1, 2, 3) are the inherent coordinates of the hypersurface.

The parametric equation of the hypersurface is

The three basis vectors provide the tangent vectors to Σ as

These basis vectors provide the induced metric or first fundamental form of the hypersurface by the following scalar product

[Actually, the elementary distance “ds” between two neighboring points in Σ (which are, therefore, also in M) is given by

where is the induced metric tensor in the hypersurface Σ.]

A unit normal nα can be defined if the hypersurface is not null.

Let us consider the surface Σ, defined by the Eq. (A1), which can be written in the form

Hence, the unit normal vector nμ is defined as

Here, and

Note that unit normal is defined when Σ is non-null as for null surface gμ𝜈 f,μf,𝜈 = 0.

The extrinsic curvature or second fundamental form of the hypersurface Σ is defined by

Extrinsic curvature is symmetric tensor, i.e., kab = kba.

Another form

Here, Ln stands for Lie Derivative.

trace of the extrinsic curvature.

Result

• (i) If k > 0, then the hypersurface is convex

• (ii) If k < 0, then the hypersurface is concave

hab is purely the inherent property of a hypersurface geometry, whereas kab is concerned with extrinsic aspects.

Remember

where hab is the inverse of the induced metric.

It is possible that any arbitrary tensor Tαβ can be projected down to the hypersurface with nonzero tangential components. The quantity that effects the projection is

One can make a 3 + 1 split of the spacetime M in a slightly different manner as follows (see Appendix C for more details):

Let us consider an arbitrary scalar field t(xα) such that t = constant describes a family of nonintersecting space-like hypersurfaces Σt.

Introduction

From a physical point of view, an outcome cannot occur before its cause. In special relativity, by causality we mean that an outcome cannot occur from a cause that does not lie in the back (past) light cone of that event. Likewise, a cause cannot affect outside its front (future) light cone. These limitations are expected as causal impacts cannot propagate quicker than the speed of light and/or backward in time. Usually, in relativity, the spacetime is characterized by a Lorentzian manifold. Actually, a Lorentzian manifold is a specific case of a pseudo-Riemannian manifold, where the signature of the metric is +2 or –2. Generally, the causal relationships among the points in the manifold are understood by defining which events in spacetime can affect which other events. Causal structures in Minkowski spacetime assume, naturally, a simple form as the spacetime is flat. However, the causal relationships between points of the curved spacetime will be more complicated due to the presence of curvature. The description of the causal structure in curved spacetime is expressed in terms of smooth curves joining pairs of points. The causal relationship is defined by the conditions on the tangent vectors of the curves. In general relativity, the metric of the spacetime up to a conformal factor can be recuperated from the causal structure. In this chapter, we explore different properties of such causal relationships instituting a number of results that will be used for advanced topics in general relativity, particularly to prove the existence of singularities.

Causality

Causal structure in the special theory of relativity, i.e., in Minkowski spacetime or flat spacetime is characterized by the fact that no massive particle can travel faster than light. In general relativity, also, locally there is no difference between the causality relation with that in Minkowski spacetime. However, globally the causality relation is significantly different due to various spacetime topologies.

Let (M, gab) be a spacetime that is connected, Hausdorff, paracompact, and admits a Lorentzian metric gab of signature (+, -, -, -). Let p be an arbitrary event in M, then the tangent space Vp is isomorphic to Minkowski spacetime. As a consequence, one can note that the light cone of p is a subspace of Vp.