aa r X i v : . [ phy s i c s . pop - ph ] O c t General Relativity for Pedestrians - First 6 lectures
Patrick Das Gupta
Department of Physics and Astrophysics, University of Delhi, Delhi - 110 007 (India) ∗ The 2017 Nobel Prize in physics awarded to Rainer Weiss, Barry C. Barish and Kip S. Thornehas generated unprecedented interest in gravitational waves (GWs). These notes are based on mylectures on various occasions - in the University of Delhi as well as in different GW schools heldin India following the exciting direct detection of GWs. I discuss GW flux and luminosity whilepointing out a curious aspect associated with the latter - physical dimensions of c /G as well as thirdtime derivative of mass quadrupole moment are that of luminosity. Formation of primordial blackholes in the early universe and progenitors of fast radio bursts could have generated GW luminositycomparable to the Planck luminosity, c /G . I also address the issue of black hole thermodynamicsin connection with the GW150914 event, demonstrating that this event is consistent with Hawking’sblack hole area theorem. In the last section, as an illustrative exercise, I estimate the GW amplitudeexpected from the fast moving plasma bullets that have been shot out from the vicinity of the carbonstar V Hydrae, as reported recently by Sahai et al. (2016). I. INTRODUCTION
Gravity is universal. Everything creates gravity as well as gets affected by gravity since everything has mass (or,equivalently, energy). However, unless the mass of an object is very large, the gravity it generates is very weak.According to Newton’s laws of gravity, acceleration of a test particle due to the gravitational field of a massiveobject, is proportional to latter’s mass, is directed towards the massive body, is inversely proportional to the squareof the distance between the two objects, and furthermore, is independent of the test particle’s mass.Newton, with a flash of brilliance, had realized that Moon’s orbiting of the Earth is nothing but its continuous fallunder Earth’s gravity. He estimated from the Moon’s orbital period of about 28 days that its acceleration directedtowards us due to Earth’s gravity is smaller than the acceleration of an apple falling on Newton’s head by a factor ofsquare of the ratio of Earth’s radius to the Moon-Earth distance. With a leap of generalization, Newton deduced theinverse square law for gravity.But Newton’s theory is inconsistent with special theory of relativity (STR). If the Sun were to disappear at thisinstant then Newton’s theory predicts that at this very instant the Earth will fly off tangentially (in the absence of acentripetal force). But according to STR, no information can travel faster than the speed of light, so the disappearanceof Sun cannot instantaneously affect Earth’s trajectory.Einstein corrected the situation by proposing in 1915, about hundred years back, a consistent theory of gravitythrough his theory of general relativity. General theory of relativity (GTR) is a relativistic theory of gravitation. GTRis based on the observation that the trajectory of a test particle in any arbitrary gravitational field is independentof its inertial mass m (as the acceleration does not dependent on m ), and therefore, it must be the geometry ofspace-time that determines test particle trajectories.Note that for no other force, acceleration of a test particle is independent of its inertial mass (e.g. in the presenceof electromagnetic fields, acceleration of a test charge is proportional to the ratio of its charge to mass). II. GRAVITY, INERTIAL FRAMES AND EQUIVALENCE PRINCIPLE
In an inertial frame, according to Newtonian laws of gravity, the magnitude of gravitational force between twoobjects 1 and 2, separated by a distance d , is given by F = − GM M d , where M and M are the gravitational masses.Gravitational mass M plays the role of gravitational charge. This is analogous to the Coulombic case of electric forcebetween electric charges. But the magnitude of acceleration of object 1 due to this gravitational force is a = Fm where m is the inertial mass (which appears in F = ma or momentum p = mv , etc.).But from experiments we know that a is independent of m (e.g. Galileo’s, and later the torsion balance experi-ments). In other words, the gravitational acceleration of a test particle is independent of its inertial mass. This iscalled the weak equivalence principle. Because of this, the gravitational mass M divided by the inertial mass m hasto be a constant for all objects. Hence, we can choose units for masses so that this constant has the value 1.But what is an inertial frame? One operational way of defining an inertial frame is that it is a frame of reference inwhich if there is no real force acting on an object, then the object either remains at rest or it moves with a uniformvelocity. Such a definition rules out an accelerating frame to be an inertial frame.But the key condition is that there should be no real force acting on the object. One can always shield it fromelectromagnetic forces, and weak and nuclear forces are anyway short-ranged. But what about gravity? Anythingthat has energy has mass too, and therefore will be a source of Newtonian gravity. So, how does one create a framethat has no gravity in it?Einstein, with his brilliant insight, offered an ingenious solution to this predicament. Basically, he employed theGalilean-Newtonian weak equivalence principle which states that in a given region, acceleration of a body due toexternal gravity is independent of its inertial mass. Imagine that a small bundle of test particles are freely falling inan arbitrary gravitational field. Since their accelerations due to gravity are nearly identical as their relative separationsare small, if one were to sit on one such particle and observe the rest, one would find that the other test particles arefreely floating as though gravity has simply disappeared!This is Einstein’s principle of equivalence according to which no matter how strong or how time varying the gravityis, one can always choose a small enough frame of reference, for a sufficiently small time interval such that gravityvanishes in this frame. So, one has obtained a truly inertial frame, albeit of a limited size one! In other words,from Einstein’s argument, no matter where, one can always construct a local inertial frame, the size of the framedepending on the scale on which the gravity varies. However, we have a queer situation here: according to a freelyfalling observer A, there is no gravitational force in her/his neighbourhood, while on the other hand, according to anoutside observer B who is at rest on the surface of the earth, there is gravity acting on the falling observer. In theNewtonian paradigm, the existence of a genuine force cannot depend on the choice of frames of reference.To highlight the above point further, let us look at Einstein’s equivalence principle from another angle. Considera frame of reference that is far removed from sources of gravitation so that there is no external gravity felt by anobserver C anywhere in this frame. But if this frame C is accelerating with respect to an inertial frame (i.e. C isa non-inertial frame), then the observer C will experience a pseudo-gravitational force. No measurement in C candistinguish between real gravity and the pseudo-gravity if weak equivalence principle is correct. Is gravity then a ‘realforce’ ? We will see shortly how this perplexing issue is resolved in Einstein’s GTR. III. GRAVITY AND SPECIAL THEORY OF RELATIVITY
Newton’s theory of gravitation also demands that the gravitational force be instantaneously transmitted by thesource to the test particle, since it is inversely proportional to the square of the instantaneous separation between thetwo. Instant transmission is unsatisfactory, as Einstein’s special theory of relativity demands that no physical effectcan propagate faster than c = 3 × m s − , the speed of light in vacuum or, for that matter, speed of any particlewith zero rest mass. This ensues from the relativistic expression for energy E of a free particle with rest mass m givenby, E = mc q − v c (1)From the above, it is evident that if v > c , the energy becomes pure imaginary, ruling out faster than light motions.Clearly, gravitational theory needs to incorporate relativity. But, how? The clue comes from Einstein’s version ofequivalence principle. We have seen in the previous section that weak equivalence principle guarantees that in thepresence of gravitation it is always possible to choose a limited size frame of reference for a short enough time in whichgravity disappears (e.g. a freely falling frame). This limited region constitutes a local inertial frame of reference, sothat a Cartesian coordinate system can be set up for specifying spatial coordinates here, and clocks can be arrangedto measure proper time. Such a coordinate system is referred to as the Minkowskian coordinate system. Therefore,in this local inertial frame (LIF), the laws of physics (other than the gravitational phenomena) must take the sameform as they do in special theory of relativity. The proper distance ds between two nearby events in the LIF withspace-time coordinates x µ = ( ct, x, y, z ) and x µ + dx µ = ( ct + cdt, x + dx, y + dy, z + dz ) is evaluated, as in STR, using, ds = c dt − dx − dy − dz ≡ η µν dx µ dx ν . (2)Note that in eq.(2), x i , i=1,2,3 are the Cartesian coordinates of the event, and η µν is the Minkowski metric with η = 1 = − η ii , rest of the off-diagonal components of the metric being zero. Einstein summation convention hasbeen used in eq.(2), so that repetition of Greek indices imply summation over 0,1,2 and 3. (From now on we will adopt the Einstein summation convention wherein whenever a Greek indexrepeats in an expression it means that the expression is being summed over with the index runningfrom 0 to 3.) STR not only proclaimed that time is the fourth dimension but also necessitated a departure from the Euclideannotion of distances. For events that are not causally connected ds is negative, which was unthinkable in Euclideanparadigm.In other words, by going over to a small freely falling frame and choosing a locally inertial or Minkowskian coordinatesystem, one manages not only to make gravity vanish locally but also express the non-gravitational laws of physicsexactly as one does in special relativity. But, what about the laws pertaining to gravitation itself? And, what if onerequires to express laws of physics over larger regions of space-time?Let us first deal with the simplest and hypothetical ‘gravitational’ set up - the case of uniform and static gravity,so that the acceleration vector due to gravity is same everywhere, and at all times. But this situation, according tothe equivalence principle is identical to zero gravity case, for, one has to just consider a freely falling reference frameas large as and for as long as one wants, and in this frame gravity simply vanishes. Hence, one can choose Minkowskicoordinates globally and eq.(2) describes the space-time geometry everywhere in such a frame. Thus, uniform gravityeverywhere is equivalent to zero gravity.Our next case is: gravity around a massive, spherically symmetric body of radius R and mass M . From theNewtonian point of view, the acceleration due to gravity caused by it at any external point P, is inversely proportionalto the distance between P and the centre of the massive object, and is directed towards the centre. Now, if oneconsiders a large freely falling frame (LFFF) at an initial distance d ≫ R , then does gravity totally vanish in thisframe?Clearly the answer is no. For, if one takes two test particles 1 and 2 separated by a vector ~L , that are freely fallingalong with this LFFF, then if ~L is perpendicular to the radial direction of fall, an observer in LFFF will notice 1 and2 to be accelerating towards each other with a magnitude, a ≈ G M Ld (3)because each of the particles will be accelerating radially towards the centre of the massive body.On the other hand, if ~L was along the radial direction of the free fall, the observer in LFFF would measure 1 and2 to be accelerating away from each other with a magnitude given by eq.(3), as the particle nearer to the massiveobject would be falling with a greater acceleration than the one farther away. These are nothing but instances of tidalacceleration, ubiquitous whenever gravity is non-uniform.Although gravitational acceleration vanishes in a local inertial frame (LIF), tidal acceleration does not. It is justthat in a LIF, the magnitude of ~L is small as the frame itself is of limited size, so that according to eq.(3), the valueof the tidal acceleration is negligibly small here. But when the frame is large, its different parts encounter varyingdegree of tidal stretching or tidal compression. For instance, we do not experience Sun’s gravity as Earth is freelyfalling towards the Sun.Nevertheless oceans exhibit high and low tides, since our planet is large enough for Sun’s tidal forces to be non-negligible. The above example shows that, in general, one cannot eliminate the effects of gravitation entirely. TheLIFs, however, are very useful through the use of STR, for the extraction of physical meanings of various mathematicalexpressions.Since one cannot have in general global inertial (i.e. Minkowski) coordinates in the presence of gravity, it isnecessary to develop a formalism that employs arbitrary coordinates like curvilinear coordinates in the analysis. Onecan motivate the necessity of using coordinate systems other than the Minkowskian ones, from a physical standpoint.Consider the case of a sufficiently large reference frame that is made up of 3-dimensional Cartesian grid of standardrods with clocks arranged at their intersections. Such a framework of Minkowskian coordinate system cannot bemaintained as a LFFF, when there is non-uniform gravitation present, because of the following reason.From STR, the condition that nothing can travel faster than c implies that no object can be absolutely rigid.Otherwise, one could simply transfer energy (and therefore, signals) from one spatial point to another with infinitespeed, just by tapping one end of a long ‘rigid’ rod causing the other end to move instantaneously. Now, non-uniformgravity would mean that different portions of the LFFF would fall with different accelerations, leading to stretchingand compression of the (initially) cubical grid of rods and clocks (forces other than gravity will enter the analysis), sothat it is no longer possible to maintain a global Minkowskian coordinate system in any LFFF. Curvilinear coordinates,therefore, become indispensable in relativistic gravitational physics.In STR, square of the proper (i.e. Lorentz invariant) distance between any two infinitesimally events is givenby eq.(2) when Minkowskian coordinates are chosen. Preceding arguments make it clear that when gravitation isincluded, one would need to modify eq.(2) and, instead of the Minkowski metric, one would require a general metrictensor. Similarly, the concept of tidal acceleration has to be made precise from the point of view of arbitrary framesof reference that use general coordinate systems. IV. CURVILINEAR COORDINATES, SCALAR, VECTOR AND TENSOR FIELDS
Let an event occur at some space-time point P, which is assigned a coordinate x µ (P) by an observer O, with µ = 0 , , ,
3, which are four real numbers corresponding to one time and three space coordinates. The same pointP, in general, will have a different coordinate x ′ µ (P) according to another observer O’. Note that the observers Oand O’ may not be inertial observers so that the coordinates x µ and x ′ µ are, in general, curvilinear coordinates. A space-time manifold is defined to be the set of all events. In GTR, the mathematical forms of physical laws remainthe same even when one makes an arbitrary coordinate transformation.In general, any arbitrary event belonging to a space-time manifold can be assigned coordinates x µ and x ′ µ byO and O’, respectively. Since labeling of an event with coordinates by an observer involves a mapping from thespace-time manifold to R , it follows that there is a mapping between x µ and x ′ µ , i.e. there exists a function thatrelates coordinate system employed by O to that of O’. Therefore, one may either treat x ′ µ to be a continuous anddifferentiable function of x α or vice-versa (i.e. x µ as a smooth function of x ′ α ), with µ , α = 0 , , , x α to another set x ′ α is called a general coordinate transformation . In a given coordinate system, each coordinatecomponent of an event is functionally independent of the other coordinate so that, ∂x µ ∂x ν = δ µν (4)Consider some physical variable (e.g. comoving energy density or pressure of a fluid) that can be described by observerO as a real valued function φ ( x α ) of space-time coordinates. Observer O’, however, will find the same physical variableto be represented by a different function φ ′ ( x ′ α ). The function φ is said to be a scalar field if everywhere on thespace-time manifold, φ ′ ( x ′ α ) = φ ( x α )given that x α and x ′ α are the space-time coordinates assigned by observers O and O’, respectively, to the same event.Physically, what a scalar field signifies is that, at every event P, the value of the physical variable φ ( x α ( P )) asmeasured by the observer O is identical to the value φ ′ ( x ′ α ( P )) as measured by O’, although the functional form ofthe physical variable depends on the observer.How does one define vector components when one is using general curvilinear coordinates? Intuitively, a vectorhas magnitude as well as direction, and hence it resembles an arrow. Suppose, we have two events P and P’ whichare temporally as well as spatially near each other, so that they have coordinates x µ and x µ + dx µ , respectively, forobserver O. Clearly, the directed line PP’ from P to P’ looks like an infinitesimally short arrow and thereby qualifiesto be called a vector with dx µ as the vector components.According to O’, however, the directed line PP’ has components dx ′ µ since P and P’ have coordinates x ′ µ and x ′ µ + dx ′ µ , respectively, in her/his frame. The relation between the components is given by the usual rules of partialderivatives. dx ′ µ = ∂x ′ µ ∂x ν dx ν (5)The above equation suggests that a contravariant vector field V µ ( x α ) ought to be defined as an entity that transformsunder a general coordinate transformation x γ → x ′ γ in the following manner, V µ ( x α ) → V ′ µ ( x ′ α ) = ∂x ′ µ ∂x ν V ν ( x α ) (6)The next question that arises is: What about objects like ∂φ ( x ) ∂x µ , where φ ( x α ) is a scalar field? Let us see how thisentity transforms under general coordinate transformation. When x γ → x ′ γ , we find that, ∂φ ( x ) ∂x µ → ∂φ ′ ( x ′ ) ∂x ′ µ = ∂φ ( x ) ∂x ′ µ = ∂x ν ∂x ′ µ ∂φ ( x ) ∂x ν (7)Clearly, the transformation given by eq.(7) is different from the one in eq.(6). This motivates one to introduce anotherkind of vector field called covariant vectors.A covariant vector field V µ ( x ) is defined to be an object such that under x γ → x ′ γ , V µ ( x ) → V ′ µ ( x ′ ) = ∂x ν ∂x ′ µ V ν ( x ) (8)To summarize, while the transformation property and the directional nature of infinitesimal vector dx µ leads tothe notion of contravariant vectors, similar considerations concerning the partial derivative ∂∂x µ entails the concept ofcovariant vectors. In 3+1 dimensional space-time, vector fields have 4 components corresponding to µ = 0 , , ,
3. Inpictorial terms, contravariant vectors are like arrows while the covariant vectors are like normal vectors to surfaces.We can now wrap up the above considerations to arrive at a generalization - tensor fields of arbitrary ranks. Atensor field V µ µ ...µ n − µ n ν ν ...ν m − ν m ( x ) of rank n + m is an entity such that under the coordinate transformation x α → x ′ α , V µ µ ...µ n − µ n ν ν ...ν m − ν m ( x ) → V ′ µ µ ...µ n − µ n ν ν ...ν m − ν m ( x ′ )where, V ′ µ µ ...µ n − µ n ν ν ...ν m − ν m ( x ′ ) = ∂x ′ µ ∂x α ∂x ′ µ ∂x α ... ∂x ′ µ n ∂x α n ∂x β ∂x ′ ν ∂x β ∂x ′ ν ... ∂x β m ∂x ′ ν m V α α ...α n − α n β β ...β m − β m ( x ) (9)We should note that in the above equation the arguments x α and x ′ α of V and V ′ , respectively, are the coordinatesof the same event, as emphasized in the first paragraph of this section. In other words, transformation of tensors arecompletely local because of which tensors of identical ranks can be added and subtracted.An important result that follows from eq.(9) is that if a tensor vanishes at an event in one coordinate system, it isidentically zero at that event in all coordinate systems.A fundamental entity in GTR that describes space-time geometry is the space-time dependent metric tensor g µν ( x α ),which determines the invariant proper distance ds between any two nearby events with coordinates x µ and x µ + dx µ , ds = g µν ( x ) dx µ dx ν (10)Here, x µ , µ, ν =0,1,2,3, now represents a general curvilinear coordinate, specifying the location of an event. The metric g µν ( x ) is a generalization of η µν , the Minkowski metric tensor.Is g µν ( x ) a tensor field? If ds is invariant under general coordinate transformation, then we can readily prove that g µν is a covariant tensor of rank 2. This is because, under x α → x ′ α , we have, ds = g µν ( x ) dx µ dx ν → ds = g ′ αβ ( x ′ ) dx ′ α dx ′ β = g ′ αβ ( x ′ ) (cid:18) ∂x ′ α ∂x µ dx µ (cid:19)(cid:18) ∂x ′ β ∂x ν dx ν (cid:19) = ∂x ′ α ∂x µ ∂x ′ β ∂x ν g ′ αβ ( x ′ ) dx µ dx ν (11)Comparing eqs.(10) and (11) as well as using the fact that dx µ is an arbitrary infinitesimal separation, we get theresult, g µν ( x ) = ∂x ′ α ∂x µ ∂x ′ β ∂x ν g ′ αβ ( x ′ ) (12)implying that g µν is a covariant tensor of second rank. This result readily connects with equivalence principle in thefollowing manner.If the space-time geometry was not curved, one could choose a coordinate system such that everywhere the metrictensor is just the Minkowski metric tensor. But GTR states that energy and momentum associated with matter warpthe space-time geometry, entailing that in general it is not possible to choose inertial coordinates everywhere so thatthe metric is globally Minkowskian.However, according to the principle of equivalence, by choosing an appropriate coordinate system, even in anarbitrarily curved space-time, the metric tensor can be made to take the form of η µν in a sufficiently small space-timeregion (physically, this corresponds to choosing a sufficiently small freely falling frame). This is precisely what eq.(12)entails. One can choose a new set of coordinates ξ α such that in a small region, η µν = ∂x ′ α ∂ξ µ ∂x ′ β ∂ξ ν g ′ αβ ( x ′ ) (13)For the dynamics of bodies moving in pure gravity, the notion of gravitational mass becomes superfluous in GTRsince particle trajectories are geodesics of space-time geometry determined from the line-element, ds = g µν dx µ dx ν Hence, it is not surprising that the world lines of freely falling test particles are independent of their inertial masses.One can also define a contravariant metric tensor g µν ( x ) by demanding that, g µα ( x ) g αν ( x ) = δ µν (14)If one considers g µν to be a 4 × g µν can be viewedas the inverse of the corresponding matrix.Both g µν ( x ) and g µν ( x ) are symmetric tensor fields, i.e. g µν = g νµ and g µν = g νµ everywhere in the space-timemanifold.Employing g µν ( x ), one can raise indices of a covariant tensor, just like lowering the indices of a contravariant tensorfield can be achieved by using g µν . Therefore, V µ ( x ) = g µν ( x ) V ν ( x ) (15)is a contravariant vector field corresponding to the covariant vector field V µ ( x ). While, W µ ( x ) = g µν ( x ) W ν ( x ) (16)is a covariant vector field corresponding to the contravariant vector field W µ ( x ). Hence, raising and lowering of indicescan be done freely for any tensor field of any rank by making suitable use of the metric tensors.So far we have done some amount of tensor algebra. Let us now take up some tensor calculus. Suppose A µ ( x ) is acontravariant vector field. Is ∂A µ ( x ) ∂x ν a second rank tensor? Under a general coordinate transformation x γ → x ′ γ , ∂A µ ( x ) ∂x ν → ∂A ′ µ ( x ′ ) ∂x ′ ν = ∂x α ∂x ′ ν ∂∂x α (cid:18) ∂x ′ µ ∂x β A β ( x ) (cid:19) = ∂x α ∂x ′ ν ∂x ′ µ ∂x β ∂A β ( x ) ∂x α + ∂x α ∂x ′ ν ∂ x ′ µ ∂x α ∂x β A β ( x ) (17)It is obvious from eq.(17) that because of the second term in its right hand side, ∂A µ ( x ) ∂x ν does not transform like atensor. What is the remedy?This is where the concept of covariant derivative comes in. We introduce a new mathematical object Γ µαβ ( x λ )referred to as Christoffel symbol (and also as affine connection and Levi Civita connection), and define the covariantderivative of A µ ( x ) as follows, A µ ; ν = A µ , ν + Γ µνα A α (18)where, A µ , ν ≡ ∂A µ ( x ) ∂x ν (19) (From now on partial derivative w.r.t. x α will be denoted by , α ) Using eq.(17), it can be easily shown that A µ ; ν , defined by eq.(18), transforms as a tensor of rank 1+1 providedthe Christoffel symbol Γ µαβ ( x λ ) transforms under a general coordinate transformation as,Γ µαβ ( x λ ) → Γ ′ µαβ ( x ′ λ ) = ∂x ′ µ ∂x ν ∂x σ ∂x ′ α ∂x γ ∂x ′ β Γ νσγ ( x λ ) + ∂x ′ µ ∂x ν ∂ x ν ∂x ′ α ∂x ′ β (20)Clearly, because of the second term in the right hand side of eq.(20), Christoffel symbol is not a tensor. One canuse precisely this feature to choose local Minkowski coordinates ξ µ to make Γ µαβ vanish at a space-time point. Thisdovetails nicely with equivalence principle, since we know that in a freely falling frame, the metric is η µν in a smallneighborhood so that it has vanishing derivatives at a point.Now, since for any arbitrary scalar field φ ( x ), ∂φ ( x ) ∂x µ is already a covariant vector field (see eq.(7)), the covariantderivative of any scalar field is its usual partial derivative, φ ; ν = φ, ν (21)Now, if V µ ( x ) and U µ ( x ) are any two contravariant and covariant vector fields, respectively, V µ ( x ) U µ ( x ) is a scalarfield (see Prob.1(a)). Hence, its covariant derivative according to eqs. (18) and (21) is given by,( V µ ( x ) U µ ( x )); ν = V µ ; νU µ + V µ U µ ; ν = ( V µ ( x ) U µ ( x )) , ν = V µ , νU µ + V µ U µ , ν = ( V µ , ν + Γ µνα V α ) U µ + V µ U µ ; ν (22)From eq.(22) it ensues, U µ ; ν = U µ , ν − Γ ανµ U α (23)as V µ ( x ) is an arbitrary contravariant vector field. This procedure can be deployed to obtain covariant derivatives ofany tensor field of arbitrary rank.Therefore, as particular examples, T µν ; α = T µν , α − Γ βαµ T βν − Γ βαν T µβ (24)and, A µν ; α = A µν , α + Γ µαβ A βν + Γ ναβ A µβ (25)In 3-dimensional Euclidean geometry, the line-element dl = dx + dy + dz has the same form whether youshift the Cartesian coordinate system by any constant vector or rotate the coordinate system about any axis byany constant angle. The line-element given by eq.(2) is similarly invariant under Lorentz transformations as well asconstant space-time translations. According to the equivalence principle, whatever is the gravity around, in a locallyinertial frame (i.e. freely falling frame), the line-element is given by eq.(2) and non-gravitational laws of physics takethe same form as in special relativity. But, what is the connection between this feature of gravitation and geometry?Consider a generally curved two-dimensional surface (e.g. the surface of, say, a pear). No matter how greatlythe surface is curved, one can always choose a tiny enough patch on it, such that it is sufficiently flat for Euclideangeometry to hold good over it. As one increases the size of the patch, the curvature of the pear’s surface becomesapparent. This is so similar to the main characteristic of gravity that we discussed in the preceding paragraph. Thesmall patch on the pear over which the line-element is Euclidean ( dl = dx + dy ) is analogous to the local inertialframe in the case of 4-dimensional space-time where the line-element is described by eq.(2). V. CHRISTOFFEL SYMBOL, CURVATURE TENSOR AND THE EINSTEIN EQUATIONS
Now, from eqs.(15) and (16), in order that, V µ ; α = g µν ; αV ν + g µν V ν ; α = g µν V ν ; α and, W µ ; α = g µν ; αW ν + g µν W ν ; α = g µν W ν ; α we require, g µν ; α = 0 = g µν ; α (26)Making use of eq.(26), we have, g µα ; ν + g αν ; µ − g µν ; α = 0 (27)From the eqs.(24) and (27), it can be easily proved that,Γ µαβ = 12 g µλ ( g αλ , β + g βλ , α − g αβ , λ ) (28)displaying an important fact that Christoffel symbol is related to metric tensor and its derivatives.Because of eq.(28), one can show that the trajectory (i.e. the worldline) x µ ( λ ), where λ is an affine parametercharacterizing the worldline, that extremizes the proper length (invariant under general coordinate transformations), S ≡ Z ds = Z p g µν dx µ dx ν = Z r g µν dx µ dλ dx ν dλ dλ (29)satisfies the geodesic equation, d x α dλ + Γ αµν dx µ dλ dx ν dλ = 0 . (30)In weak fields (small departure from Minkowski space-time), one can choose quasi-Minkowskian coordinates so that, g µν = η µν + h µν (31)with, | h µν | ≪ µ, ν =0,1,...,3.For static and weak gravitational fields where test particles move with speeds much less than c , one must haveeq.(30) reduce to Newtonian gravitational dynamics, where a particle with spatial coordinate x i , i = 1 , , d x i dt = − ∂φ N ( ~r ) ∂x i (33)where φ N ( ~r ) is the Newtonian gravitational potential at ~r .Eq.(30) indeed leads to eq.(33) in the weak and static field approximation provided, g = 1 + h ≈ φ N ( ~r ) c (34) g i ≈ , g ij ≈ − δ ij (35)so that, Γ i ≈ c ∂φ N ∂x i (36)We know from STR that the time elapsed in a clock (comoving with an observer O’) cruising with uniform velocitywith respect to an inertial observer O is given by, τ = 1 c Z p η µν dx µ dx ν = p − v /c t , (37)where x µ , v and t are the space-time coordinates of O’, speed of O’ and time as measured by the inertial observerO, respectively. This is called the proper time that elapses in the frame of O’, and is invariant under Lorentztransformations. The time dilation result ensues from eq.(37).What can we say about the proper time elapsed for a test particle as it moves along an arbitrary worldline in acurved space-time? Let the worldline in a space-time manifold whose geometry is described by metric g µν ( x α ) bedescribed by x µ ( λ ) from λ to λ , λ being an affine parameter characterizing the worldline.Since one understands good clocks and good measuring rods in the framework of STR, the way to measure propertime τ elapsed in an arbitrarily accelerating clock in arbitrary gravity is clearly by adding the infinitesmal timeintervals elapsed in local inertial frames that lie along the trajectory of the clock at different instants of time and thatco-move with the clock at those instants of time (for comoving clocks v = 0 so that the proper time is just the timeelapsed in these clocks as seen from eq.(37)). But by virtue of eqs.(11) and (13), this sum is just, τ = 1 c Z r g µν dx µ dλ dx ν dλ dλ (38)Another way of stating the above argument is that the infinitesimal proper time interval between two time-likeseparated close by events is ds/c , and since time is additive, the total proper time elapsed is simply the integral givenby eq.(38).We can apply the above result to determine the proper time elapsed in a clock at rest in a weak and static gravity.Using eqs.(34) and (35) in eq.(38) for a clock at rest ( dx i = 0) at a point A, one obtains the proper time elapsed tobe given by, τ A ≈ (1 + φ N ( A ) c ) t , (39) t being the proper time elapsed in the frame of a static inertial observer at infinity where h = 0. From this oneconcludes that not only time runs slow in attractive gravitational fields but also radiation emitted from regions withstronger and attractive gravitational potentials get redshifted as they move out to weaker gravity regions.To summarize, in order to connect LIFs at different space-time points, and to express physical laws in terms ofarbitrary coordinates in reference frames of size as large as one wishes, one needs the language of tensor calculus sothat one acquires an affine connection Γ µαβ derivable from the metric tensor g µν and its derivatives. Although, thisaffine connection (or, Christoffel symbol) vanishes at a point in a LIF, its derivative does not.This brings us to the Riemann curvature tensor R µναβ which represents how curved is the space-time geometry,and is constructed out of Christoffel symbol and its derivatives in the following way, R µναβ = Γ µνβ , α − Γ µνα , β + Γ µσα Γ σνβ − Γ µσβ Γ σνα (40)From eq.(40), it is obvious that, R µναβ = − R µνβα One can obtain a symmetric second rank tensor called the Ricci tensor from the Riemann tensor, R νβ = R µνµβ = R βν (41)The Ricci scalar is simply, R = g µν R µν (42)From eq.(40), one can easily prove that if, R µναβ = g µλ R λναβ then, R µναβ = − R µνβα = − R νµαβ = R αβµν (43)To summarize, Christoffel symbol is like gravitational field since it involves derivatives of the metric. However,because it is not a tensor (see eq.(20)), it can also represent pseudo-gravity. For instance, consider a hypotheticalcase in which there is no gravity anywhere, so that one may choose global Minkowskian coordinates in order that themetric tensor is simply η µν everywhere in this coordinate system. In such a situation, for an observer O’ acceleratingwith respect to an inertial observer O, one can easily show that the Christoffel symbol for O’ is non-zero and, in fact,corresponds to a fictitious force in the accelerating frame that mimics gravity.The true gravitation, represented by the Riemann curvature tensor, is the tidal gravity. Although in a local inertialframe, gravity disappears, tidal gravitational force does not. Since the Christoffel symbol is not a tensor, in a LIF itis zero at a point, while the Riemann tensor in general is nonzero. This reminds us of the acceleration due to gravityvanishing while the tidal gravitational force being nonzero, in a LIF.0For instance, earth is freely falling towards the sun because of latter’s pull. But we do not feel sun’s gravity sincethe freely falling earth constitutes a local inertial frame. However, as sun’s gravity is non-uniform, portions of earthcloser to the sun feel a greater tug than those located farther. This differential pull is the source of tidal force whichcauses the commonly observed ocean tides. In GTR, the tidal acceleration is due to the fourth rank Riemann tensorthat is constructed out of the metric and its first as well as second derivatives. Therefore, the ocean tides owe theirexistence to the non-zero Riemann tensor describing the curvature of space-time geometry around the sun (as well asthe moon).Therefore, true gravity represented by the tidal gravitational field is related to the Riemann curvature tensor R µναβ ,a fourth rank tensor constructed out of the connection Γ µαβ and its derivatives. In mathematics, R µναβ determineswhether the geometry is flat or curved. This, in a sense, completes the identification of gravity with geometry. Whilein the gauge theory framework, Γ µαβ is analogous to gauge potential with R µναβ as the corresponding gauge covariantfield strength.In later lectures, we will see that the dynamics of space-time geometry is determined by the Einstein equations, R µν − Rg µν = 8 πGc T µν (44)where the Ricci tensor and Ricci scalar are R µν ≡ R αµαν and R = g µν R µν , respectively. T µν is the matter energy-momentum tensor whose various components represent the flux of energy and momentum carried by matter inappropriate directions. When the gravity is weak and static, eq.(33) reduces to Newton’s gravity, ∇ φ = 4 πGρ (45)for a non-relativistic source with mass density ρ and negligible pressure. The Newtonian gravitational potential φ isidentified with the geometrical entity ( g − c / VI. GRAVITATIONAL RADIATION
GTR as a theory of gravitation gained immediate acceptance among the physics community as soon as its predic-tion of bending of light was actually seen during the solar eclipse of 1919. Of course, GTR had already correctlyexplained the anomalous precession of the perihelion of Mercury. Since GTR is based on special relativity, gravita-tional perturbations too propagate as gravitational waves (or, undulations in space-time geometry) with finite speedc. Later, indirect evidence for gravitational waves predicted by Einstein was corroborated with the discovery of slowlyinspiralling Hulse-Taylor binary pulsar (PSR 1913 + 16).Very far away from a source whose energy and momentum distributions are changing asymmetrically, if the ensuingperturbation in the space-time geometry, represented by h µν , is sufficiently weak, one can choose a quasi-Minkowskiancoordinate system and express the metric tensor as, g µν ≈ η µν + h µν ( ~r, t ) , (46)with the perturbation or the gravitational wave amplitude satisfying, | h µν ( ~r, t ) | ≪ h µν ( ~r, t ) is determined by the second time derivative of the mass quadrupolemoment of the source that is undergoing changes in its matter distribution. Physical effects of GWs on test particlesare best understood in the transverse, traceless (TT) gauge. In this gauge, only the space-space components of theGW amplitude are non-zero along with the conditions of vanishing trace and components orthogonal to the directionof propagation. Hence, if the GW is propagating in the z-direction, it can have only h = − h and h = h as thenon-zero components (so that they are manifestly traceless and orthogonal to the z-direction).1Do GWs transport energy? Now, only entities that have energy can possibly be perceived or measured, sinceexchange of energy between an object and the sensors is crucial for its detection. Even in quantum theory, twosubsystems can influence each other only via an interaction Hamiltonian. So, in order for a measurement device todetermine an eigenvalue corresponding to a quantum observable of a system, there has to be an exchange of energybetween the system and the apparatus through a suitable interaction Hamiltonian.Returning to GWs, Feynman and Hermann Bondi had used the following thought experiment to demonstrate thatGWs transport energy [1]: Consider two loose metal rings around a rod that is held in a horizontal position. If aGW passes by, the rings will move and oscillate with respect to the rod (elasticity of the rod that gives the latterrigidity will prevent appreciable change in its length because of the incident GW). Hence, the rings and the rod willget heated up because of friction. This energy certainly has to be at the expense of the energy carried by the GW. Infact, the observed slow decrease in the orbital period of the two neutron stars in the Hulse-Taylor binary pulsar PSR1913+16 demonstrates unequivocally that the loss of binary system energy is due to the radiated GWs that carryenergy away, agreeing extremely well with the prediction ensuing from GTR [2].In the TT-gauge, for a source with non-relativistic internal motion and mass density ρ ( t, ~r ), the GW amplitude asseen by an observer at time t and position ~r ( r ≫ source size) from the source is given by [3], h ij ( t, ~r ) = 2 Gc r d ( I ij ( t − r/c )) dt (48 a )where the reduced mass quadrupole moment is defined as, I ij ( t ) ≡ I ij ( t ) − δ ij I kk ( t ) (48 b )with the mass quadrupole moment being, I ij ( t ) ≡ Z ρ ( t, ~r ) x i x j d r . (48 c )In eq.(48a), causality is ensured because of the retarded time t − r/c appearing in the RHS.The quantity d I ij ( t − r/c ) dt is, in some sense, a measure of asymmetric motion of matter in the GW source, representingapproximately the non-symmetric portion of the source’s kinetic energy. Hence, from eq.(48a), one can write down aformula to make back-of-the-envelope estimate of the GW magnitude, h ≈ GE nonsym c r . (49)The energy-momentum pseudo-tensor corresponding to GWs is given by [3], T µν = c πG (cid:28) h jk,µ h jk,ν (cid:29) (50)where h jk is the GW amplitude in the TT-gauge and that, < ... > represents averaging over many wavelengths(Raising and lowering of indices of GW amplitude are done using Minkowski metric tensor.).The difficulty of constructing a proper energy-momentum tensor for the gravitational degrees of freedom is relatedto the fact that g µν → η µν and Γ µαβ → g µν and Γ µαβ will be simply zero at a point P in a local inertial frame. Butthis point P is arbitrary since one can choose a local inertial frame anywhere in the entire space-time. Hence, it impliesthat such a tensor is identically zero every where. If one includes first derivatives of Γ µαβ , then symmetric tensors like R µν or G µν can certainly be constructed. But R µν and G µν vanish where there is no matter by virtue of Einsteinequations (eq.(44)), belying their representing energy and momentum flux of GWs propagating through vacuum.Therefore, it is common practice to employ the Landau-Lifshitz energy-momentum pseudo-tensor, that contains onlyfirst derivatives of the metric tensor, to study energy and momentum associated with gravitational degrees of freedom[4].Since T is the energy density, the GW energy flux is given by, F GW = cT = c πG (cid:28) h jk, h jk, (cid:29) = c πG (cid:28) ˙ h jk ˙ h jk (cid:29) (51)where ˙ h jk ≡ ∂h jk ∂t . Making use of eq.(48a) in eq.(51) one obtains, F GW = 14 πr (cid:18) G c (cid:28) ··· I jk ··· I jk (cid:29)(cid:19) (52)2When the background space-time is nearly flat (so that 4 πr ∼ = surface area of a sphere of radius r ) and the emissionof GWs is isotropic, it is evident from eq.(52) that the GW luminosity is given by, L GW ∼ = 4 πr F GW = G c (cid:28) ··· I jk ··· I jk (cid:29) . (53)Now, if mass distribution in a GW source changes over a typical time scale of ∼ T GW = 2 π/ω = 1 /f , then fromeq.(53) one can pen down a back-of-the-envelope expression to estimate L GW , L GW ∼ Gc ω E nonsym ≈ × (cid:18) E nonsym erg s − (cid:19) (cid:18) f (cid:19) erg s − . (54) A. GW Luminosity, c /G , Planck Scales and Hawking Radiation It is very interesting to observe that the physical dimensions of both c G = 3 . × erg s − and ··· I jk are identically ML T = [Luminosity]. Does this indicate something deep about gravitation?As an amusing exercise related to the above coincidence, one may consider the gravitational collapse of a compactastrophysical object like a supra-massive neutron star of mass M and initial radius R = α R s , where α is a numberjust in excess of unity and R s ≡ GM/c is the Schwarzschild radius. If the collapse to a black hole is non-spherical,with asymmetric kinetic energy E nonsym = α M c (where α is a number less than but of order unity), and takesplace in dynamical time scale p R /GM ∼ T GW then, according to eq.(54), L GW ∼ α α c G = 9 × α α erg s − (55 a )and ··· I jk ∼ E nonsym /T GW ∼ α √ α c G , is also roughly of the same order as L GW . In this case, curiously enough,neither L GW of eq.(55a) nor E nonsym /T GW depend on the mass of the compact, collapsing object. Hence, formationof primordial black holes, either due to collision of bubble walls or rapid collapse of false vacuum pockets in the earlyuniverse [5,6], could generate GW luminosity comparable to c /G . Similarly, compact supra-massive neutron stars(likely progenitors of fast radio bursts) undergoing near free fall, as they collapse very rapidly to form black holes,might also lead to such high GW luminosity [7].A further connection to c /G emerges, if one considers Planck energy, E P l ≡ m P l c ≡ p c ¯ h/G and the Plancktime, t P l ≡ p ¯ hG/c so that one may define Planck luminosity, L P l [8], L P l ≡ E P l t P l = c G . (55 b )Eq.(55b) indicates three interesting features - (i) close to the time of big bang when quantum gravity effects weredominant, quantum fluctuations could have generated GWs with luminosity ∼ c /G , (ii) Planck luminosity does nothave the quantum imprint ¯ h and, hence, (iii) for an astrophysical source to generate such a large luminosity, if itsnon-symmetric kinetic energy E nonsym scales as ∼ α E P l , the time scale over which its mass quadrupole momentchanges substantially must also scale as ∼ α t P l , α being a very large positive number.Above considerations suggest that c /G may represent the maximum possible GW luminosity (see also [8]). If itembodies a fundamental upper limit for L GW then it leads to a lower limit for the time scale T GW (or equivalently,an upper limit on the characteristic frequency f ) over which matter in a GW source gets redistributed. This followsfrom, L GW ≈ E nonsym T GW = α M c T GW ≤ c G ⇒ T GW ≥ α GMc = 1 . × − α (cid:18) M M ⊙ (cid:19) s ⇒ f ≤ c α GM ∼ = 67 α − (cid:18) M M ⊙ (cid:19) − kHz . (55 c )3Although GW luminosity may approach c /G in situations involving collapse to form primordial black holes orimplosion of supra-massive neutron stars, the total energy radiated in the form of GWs will be limited by E nonsym = α M c . One may also compare Planck luminosity with the luminosity associated with the evaporation of black holesthat was predicted by Stephen Hawking in 1974 [5,6]. Existence of Hawking temperature, T H = c ¯ h/ πGM k B , for ablack hole of mass M implies that the event horizon acts like a black body, emitting radiation with a flux, F H = σT H ,so that the corresponding luminosity is given by, L H = F H × πR s = 115360 π (cid:18) m P l M (cid:19) c G . (55 d )From eq.(55d) it is evident that even for a Planck scale primordial black hole, the luminosity corresponding to Hawkingevaporation is four orders of magnitude below the Planck luminosity.Since most of the GW sources that are going to be detected are likely to be extra-galactic (e.g. even the first directlydetected GWs from GW150914 originated at a cosmological redshift of z ≈ . r . If this object radiates GWs witha luminosity density L GW ( t, ν ) at cosmic time t then energy radiated in the time interval ( t, t + ∆ t ) and frequencyband ( ν, ν + ∆ ν ) is L GW ( t )∆ t ∆ ν . Luminosity L GW ( t ) is related to the luminosity density L GW ( t, ν ) as, L GW ( t ) = Z ∞ L GW ( t, ν ) dν (56)Hence, the number of gravitons emitted in the time interval ( t, t + ∆ t ) and frequency range ( ν, ν + ∆ ν )is given by,∆ N ( t, ν ) = L GW ( t, ν )∆ t ∆ νhν (57)Assuming that GWs are radiated isotropically from the source, the radiated energy and the number of gravitonswill be spread over an area 4 πa ( t ) r by the time it reaches an observer ( r = 0) at present times over the time interval( t , t + ∆ t ). Due to the expansion of the universe, both frequency ν and ∆ t will be cosmologically redshifted to ν = ν a ( t ) a ( t ) = ν z and ∆ t = ∆ t a ( t ) a ( t ) = (1+ z )∆ t , respectively. (Here we have used the Robertson-Walker coordinateswith a ( t ) being the expansion scale factor and redshift z = a ( t ) a ( t ) − r = 0 per unit area in the time interval ( t , t + ∆ t ) and frequency band( ν , ν + ∆ ν ) then is given by, ∆ N = ∆ N ( t, ν )4 πa ( t ) r = L GW ( t, ν )∆ t ∆ νhν πa ( t ) r (58)Therefore, GW energy received in the frequency range ( ν , ν + ∆ ν ) per unit area per unit time is given by, F ( t , ν )∆ ν = hν ∆ N ∆ t = hν L GW ( t, ν )∆ t ∆ νhν πa ( t ) r ∆ t = L GW ( t, ν )∆ ν πa ( t ) r (1 + z ) (59)Thus, from eqs.(56) and (59), the GW flux is simply, F GW ( t ) = Z ∞ F ( t , ν ) dν = L GW ( t )4 πa ( t ) r (1 + z ) = L GW ( t )4 πD L ( z ) , (60)the required generalization of eq.(52), where D L ( z ) is the standard luminosity distance of Friedmann-Robertson-Walker cosmology. B. Measuring GW amplitude
Physical effects of GWs on test particles are best understood in the transverse, traceless gauge. Since the metrictensor governs the proper distance (or equivalently, proper time) by virtue of eq.(29), the proper distance between twotest particles will undulate when a GW is incident on them. By measuring the relative separation between the test4particles as a function of time, one can gain information about h µν . Consider now, for simplicity, the case of a nearmonochromatic GW for which h = − h = h and h = h = 0. Suppose one has two test particles scattered onthe xy-plane. In order to monitor the change in the proper distance between the test particles because of the metricperturbation, one has to have the initial separation L between these particles to be much less than the radius ofcurvature of space-time geometry associated with the incident GW, so that one can use either eq.(29) or the geodesicdeviation equation to determine the change in the separation.For a GW, the radius of curvature of space-time geometry is of the order of its wavelength. Hence, the abovecondition then is, L ≪ c ( ω/ π ) − (61)where ω is the angular frequency of the GW. If the two test particles lie on the x-axis then according to eq.(29), theproper distance l ( t ) is given by, l ( t ) = Z p − g µν dx µ dx ν = Z L p − h dx ∼ = (cid:18) − h (cid:19) L , (62)where one has used the condition given by eq.(47). The negative sign inside the square-root sign in eq.(62) occursbecause the two test particles at the same instant of time are space-like separated.Therefore, the strain corresponding to the change in length is simply related to the GW amplitude in the followingmanner, ∆ LL ≡ l ( t ) − LL = − h . (63)Eq.(62) tells us that as the GW passes by, the proper distance between the test particles changes with time. Therefore,if light is emitted from one test particle at time t towards the other, the arrival time t of light at the other testparticle is given by, t ≈ t + l ( t ) /c = t + (cid:18) − h (cid:19) L/c . (64)It is essentially this feature, represented by eq.(64), that is used in a laser interferometer to detect GW, since changesin the arrival times from the two arms of the interferometer correspond directly to the changes in the phase differencethat lead to GW amplitude measurements from the fringe shifts in the interference pattern (see [10] for a pedagogicalintroduction to this subject).In a time interval T = 2 π/ω (where ω is the angular frequency of the near monochromatic GW), the variation inthe proper distance ∆ L between two test particles intercepting the GW can change by an amount of the order of hL/ h and L are the magnitude of GW amplitude and initial separation between the particles (provided L is much much less than the wavelength of the GW, which is ∼ πc/ω ). An interesting question to ask is: Can thechange ∆ L happen so rapidly that ∆ L/T > c ? Now, ∆
L/T = hL/ T = hLω/ π , which is much less than hc becauseof eq.(61). Since, h is much much less than unity (eq.(47)), ∆ L/T is much much less than c . Therefore, the rate ofchange of separation between the test particles can never exceed the speed of light. VII. THE FIRST DIRECT MEASUREMENT OF GW AMPLITUDE
The two LIGOs, laser interferometric GW detectors of USA, independently achieved direct detection of GWs from ablack hole binary merger event on September 14, 2015, with a time delay of about 6.9 ms [11]. This GW source is at aredshift z ≈ .
09, corresponding to a luminosity distance of about 410 Mpc. The GW150914 binary system consistedof two coalescing black holes of mass 29 ± M ⊙ and 36 +5 − M ⊙ that eventually collided with each other to settle downinto a bigger rotating black hole of mass M f = 62 ± M ⊙ . The spin angular momentum of the final black hole isestimated to be 0 . +0 . − . GM f c . The mass deficit of about 3 M ⊙ was carried away by GWs. Significantly, this eventnot only corroborates GW results that ensue from GTR but is also consistent with the prediction of quasi-normalmode emission of GWs from a perturbed black hole [12].Therefore, it is pertinent to ask whether GW150914 is consistent with black hole thermodynamics (BHT) too. Atheorem proved by Hawking several decades back implies, in this case, that area of the event horizon of the finalblack hole must be larger than the sum of the event horizon areas of the binary components. The radius of the eventhorizon of a black hole of mass M and spin angular momentum L is given by (see, for example,[13]), R EH = Gc (cid:18) M + p M − ( L c/G M ) (cid:19) (65 a )5while the area of the event horizon is given by (see, for instance,[14]), A = 8 π (cid:18) Gc (cid:19) M (cid:18) M + p M − ( L c/G M ) (cid:19) (65 b )There is considerable uncertainty about the spin angular momenta of the two initial black holes of GW150914. Inour analysis, we may take them to be Schwarzschild black holes (i.e. L = 0) so that we may start with the maximumevent horizon area. Then, from eq.(65b) with L = 0, the initial total area of the event horizons is given by, A i = 16 π ( G/c ) [ M + M ] (66)where M = 29 M ⊙ and M = 36 M ⊙ . The black hole formed after the merger has a mass M f = 62 M ⊙ and spinangular momentum L f = 0 . GM f c . Hence, the final area of the event horizon according to eq.(65b) is given by, A f = 8 π ( G/c ) M f (cid:18) M f + q M f − ( L f c/G M f ) (cid:19) (67)Therefore, from eqs.(66) and (67), the ratio of final area to the initial is, A f A i = M f (cid:18) q − ( L f c/G M f ) (cid:19) (cid:18) M + M (cid:19) = 1 .
57 (68)where one has used, L f c/G M f = 0 . . Thus, even after overestimating the initial area by assuming that the initial black holes were Schwarzschild blackholes, eq.(68) demonstrates that the parameters deduced from the event GW150914 are consistent with Hawking’sarea theorem, which states that in any classical physical process, the event horizon area of the final black hole mustbe larger than the sum of the event horizon areas of the initial black holes involved in the process (see, for example,[14]).One may go one step ahead by including the estimated errors in the parameters, and check if Hawking’s black holearea theorem is violated in the worst case scenario by considering M = 33 M ⊙ , M = 41 M ⊙ , M f = 58 M ⊙ and L f = 0 . GM f c (consistent with the errors quoted in [11]). It turns out then, (cid:18) A f A i (cid:19) min = 1 . , (69)which is still in agreement with the area theorem (I have made a simplifying assumption in the above analysis thatthe errors in the parameters are mutually independent).On the other hand, if one considers the case where M = 25 M ⊙ , M = 32 M ⊙ , M f = 66 M ⊙ and L f = 0 . GM f c (again, being consistent with the errors reported in [11]), one finds that, (cid:18) A f A i (cid:19) max = 2 . . (70)The right hand side of eq.(70) can, of course, be larger if the initial black holes are Kerr black holes. One may combineeqs.(69) and (70) to express the ratio as, A f A i = 1 . +0 . − . , that suggests GW150914 upholding the classical BHT.LIGOs, subsequently, have detected GWs from three more black hole binary mergers, with the latest been observedby VIRGO, an European GW detector, too. In a straight forward exercise, analogous to the preceding analysis, onecan easily verify that the parameters deduced from the three later GW detections are in agreement with Hawking’sblack hole area theorem.6 VIII. GWS FROM STELLAR OUTFLOWS
In a very recent paper, it has been shown that a carbon-rich red giant star, V Hydrae, is linked with ejection offast moving ( V =200 km/s to 250 km/s) plasma blobs as heavy as planet Mars [15]. The carbon star is at a distanceof about r = 500 pc from us [16]. The paper argues that ejection of these knotty outflows originate from an accretiondisc around an unseen companion of V Hydrae that is moving in a highly eccentric orbit around the common centre ofmass with an orbital period of 8.5 yrs. The plasma ‘cannonballs’ are shot out when the companion of mass M c < M p passes close to the stellar envelope of the primary (of mass M p =1 to 2 M ⊙ ) at periastron [15].It may be an instructive exercise to calculate the GW amplitude expected from such a source using the back-of-the-envelope estimate discussed in section VI. For the V Hydrae case, the mass m b of the outflowing blobs are foundto be ≈ (0 . − . × gm [15]. Therefore, E nonsym = 12 m b V ≈ × (cid:18) m b gm (cid:19)(cid:18) V
250 km/s (cid:19) erg . (71)Estimates of GW amplitude from sources with relativistic jets have been made in the past (e.g. [17] and [18]). In thecase of V Hydrae, the plasma ‘bullets’ shot out have non-relativistic speeds, and hence, one uses eq.(49) to find GWamplitude to be, h ≈ × − (cid:18) m b gm (cid:19)(cid:18) V
250 km/s (cid:19) (cid:18) r
500 pc (cid:19) − (72)Since the blobs are ejected from the innermost radius r a of the accretion disc, typical time scale characterizing theacceleration of these blobs is given by, ∆ t ∼ πr a V < ∼ . × (cid:18) r a R ⊙ (cid:19) s , (73)assuming that r a is of the order of the size of the companion star (which is likely to be a main sequence star of solaror sub-solar mass [16]). Eq.(73) implies that the GW amplitude from these plasma knots, given by eq.(72), wouldhave a characteristic frequency ν c ∼ / ∆ t > ∼ . References
1. D. Kennefick, ‘Controversies in the History of the Radiation Reaction problem in General Relativity’ (2007)(http://dafix.uark.edu/ ∼ Practice Problems V µ ( x ) and W µ ( x ) are two vector fields, then show that:(a) V µ W µ is a scalar field.(b) V µ W ν is a tensor field of rank 1+1(c) g να V µ W α is a tensor field of rank 2.2 (a) A tensor A µν is found to be anti-symmetric in a particular coordinate system { x µ } . Prove that A µν = − A νµ in all coordinate systems.(b) It is given that a zero-rest mass particle is moving along the world line x µ ( λ ) in 3+1-dimensions, where λ is anaffine parameter. Show that the tangent vector u µ = dx µ dλ satisfies the equation, g µν u µ u ν = 0 . (c) (i) If A µν ( x ) is a second rank tensor field of (1+1) type, then show that, A µν ; α = A µν , α + Γ µαβ A βν − Γ βαν A µβ (ii) Show that δ µν ; α = 03 (a) Consider the 2-dimensional manifold constituted by the surface of a sphere of radius r = 1.(i) Choosing a convenient coordinate system, obtain g µν , g µν and Γ µαβ .(ii) A vector v µ starting out with components v = A and v = B from the point P: ( θ, φ ) = ( π/ ,
0) is paralleltransported from P to ( π/ , π/
2) first, then to ( α, π/ α, ≤ α <π/
2. Find the components of v µ when it returns to P. Does your result change when α → V µ ( x γ ), prove that, V µ ; α ; β − V µ ; β ; α = − R µναβ V ν (c) If ψ ( x λ ) is a scalar field then prove that, g µν ψ ; µ ; ν = 1 √− g ∂∂x ν (cid:18) √− g g µν ∂ψ∂x µ (cid:19) (d) For weak and static gravity, show that clocks tick at a slower rate in regions of stronger gravity.4 (a) The Einstein-Hilbert action is given by, A G = − c πG Z R √− g d x g µν → g µν + δg µν , the variation in A G is given by, δA G = − c πG Z ( R µν − / g µν R ) δg µν √− g d x provided δg µν ( x α ) vanishes at infinity.(b) If δg µν is an infinitesimal variation of the metric tensor g µν and T µν is the energy-momentum tensor, provethat T µν δg µν = − T µν δg µν (c) If ¯ G µν ≡ R µν − / g µν R + Λ g µν then prove that ¯ G µν ; ν = 0. Assume that Λ is a constant.5 (a) Show that a light ray emitted radially outward from r < R s = 2 G M/c can never cross the event horizon ofa Schwarzschild blackhole of mass M .(b) For a Schwarzschild blackhole of mass M , show that a test particle of non-zero rest mass cannot have constant r trajectories when it is inside the event horizon (i.e. r < R s ≡ G M/c )).5 (a) For a particular space-time, ξ µ ( x α ) is given to be a Killing vector field. Consider a test particle falling freelyin this space-time along a geodesic x µ ( λ ), where λ is an affine parameter. Show that u µ ξ µ is a constant of motion,given that u µ ≡ dx µ dλ .(b) Suppose the metric tensor g µν is independent of the particular coordinate x σ for a fixed value of σ so that ∂g µν ∂x σ = 0. Then, show that ξ µ = δ µσ satisfies the Killing equation, ξ µ ; ν + ξ ν ; µ = 0(b) Given a spherically symmetric white dwarf of mass 2 × gm and radius 6000 km, find the maximum energythat can be extracted by lowering a test particle of mass 10 gm very very slowly towards the white dwarf, by firstobtaining an expression for the conserved energy of the test particle at rest in this space-time.6 (a) Show that a test particle of rest mass m falling freely due to gravity of a spherically symmetric body of mass M ≫ m moves along a geodesic r ( φ ) that satisfies the differential equation,(i) d udφ + u = R s l + 3 R s u l ≡ L z mc , u ( φ ) ≡ /r ( φ ), R s ≡ G M/c and L z is the angular momentum of the test particle.(ii) Under what approximation is, r ( φ ) ∼ = a (1 − e )1 + e cos φ a solution of the differential equation in (i)? [ a and e are positive constants](b) In the case of static and weak gravity, g µν ≈ η µν + h µν with | h µν | ≪ ∂h µν /∂t = 0. If h = 2 φ N /c show that Newton’s gravity, ∇ φ N = 4 πGρ follows from the Einstein equation, R µν = 8 πGc [ T µν − / g µν T ]where φ N and ρ are the Newtonian gravitational potential and mass density, respectively.(c) Consider a blackhole of mass M = 2 × gm and a test body of mass m = 10 − M orbiting around theblackhole in the θ = π/ r min = 10 cm when it has the maximum speed v max = 10 − c . The test body follows, to a good approximation, the geodesic r ( φ ) ∼ = 2 l /R s e ψ ( φ )where ψ ( φ ) ≡ cos φ + 3 R s l φ sin φ , l ≡ L z mc , u ( φ ) ≡ /r ( φ ), R s ≡ G M/c , e is the eccentricity and L z is the angular momentum of the test body.The orbital period P is given to be 10 seconds. Estimate the rate of precession of the point of closest approach.Does the result depend on m ? ∗∗