Do You Need to Understand General Relativity to Understand Gravitation?

Transcription

1 Do You Need to Understand General Relativity to Understand? Institute of Mathematical Sciences, Chennai IIAP-Bangalore 13 June 2006

2 Newton s Three Laws Figure: Newton s Laws.

3 Newton The fundamental law of gravitation was first given by Newton through his famous equation F = GM r 3 r This simple law accounted for a myriad of phenomena including subtle effects like Precession of the Equinoxes. However, with the advent of Special Relativity it was clear that Newton s laws were incomplete. This was dramatically and radically altered by Einstein s General Theory of Relativity(GR).

4 Einstein Figure: Einstein and gravity.

5 Einstein The crux of GR is the Principle of Equivalence. The Kinematical effects of an accelerated frame and the Dynamical effects of a gravitational field are locally indistinguishable. A very powerful dynamical principle! To find out how gravity affects a physical system, simply view it in an appropriately accelerated frame.

6 Some Consequences As is well known this simple dynamical principle has far reaching consequences. The most famous are: Bending of light by gravitating bodies. Shift of Perihelion. al Red Shift. The other effects are: al time delay Precession of Gyroscopes Dragging of inertial frames al radiation.

7 Some Consequences As is well known this simple dynamical principle has far reaching consequences. The most famous are: Bending of light by gravitating bodies. Shift of Perihelion. al Red Shift. The other effects are: al time delay Precession of Gyroscopes Dragging of inertial frames al radiation.

8 Some Consequences As is well known this simple dynamical principle has far reaching consequences. The most famous are: Bending of light by gravitating bodies. Shift of Perihelion. al Red Shift. The other effects are: al time delay Precession of Gyroscopes Dragging of inertial frames al radiation.

9 Some Consequences As is well known this simple dynamical principle has far reaching consequences. The most famous are: Bending of light by gravitating bodies. Shift of Perihelion. al Red Shift. The other effects are: al time delay Precession of Gyroscopes Dragging of inertial frames al radiation.

10 Some Consequences As is well known this simple dynamical principle has far reaching consequences. The most famous are: Bending of light by gravitating bodies. Shift of Perihelion. al Red Shift. The other effects are: al time delay Precession of Gyroscopes Dragging of inertial frames al radiation.

11 Some Consequences As is well known this simple dynamical principle has far reaching consequences. The most famous are: Bending of light by gravitating bodies. Shift of Perihelion. al Red Shift. The other effects are: al time delay Precession of Gyroscopes Dragging of inertial frames al radiation.

12 Some Consequences As is well known this simple dynamical principle has far reaching consequences. The most famous are: Bending of light by gravitating bodies. Shift of Perihelion. al Red Shift. The other effects are: al time delay Precession of Gyroscopes Dragging of inertial frames al radiation.

13 Some Consequences As is well known this simple dynamical principle has far reaching consequences. The most famous are: Bending of light by gravitating bodies. Shift of Perihelion. al Red Shift. The other effects are: al time delay Precession of Gyroscopes Dragging of inertial frames al radiation.

14 Bending of light Figure: Equivalence principle bends light rays.

15 Bending of light Figure: The famous confirmation.

16 al red shift Figure: al red and blue shifts.

17 Perihelion Motion Figure: The perihelion motion.

18 al Radiation Figure: Evidence for gravitational radiation.

19 Curved Space-Time The most radical paradigm shift is that space-time is not a static scaffolding for dynamics to take place but is itself dynamical. Matter and space-time geometry are one and the same thing.

20 Curved Space-Time Figure: Curved space-time.

21 Mathematical Formulation The basic object is the metric tensor g µν (x). Christoffel connection: Γ λ µν. Curvature Tensor: Rαβδ λ. Einstein s equations: R µν 1 2 g µνr = κt µν Technical problems: i) equations are non-linear ii) they possess a local invariance akin to gauge invariance of electrodynamics given by the freedom to transform space-time coordinates (almost)arbitrarily. Conceptual problems: finding suitable observables that reflect the invariant character of the theory. These two make GR highly unintuitive as well as technically difficult.

22 An alternative view? Can we describe gravitation as we would describe other (special) relativistic phenomena? Show-piece of relativistic system is electrodynamics. Can we describe gravitation similarly without tampering with the structure of space-time?

23 Julian Schwinger Figure: Julian Schwinger.

24 Steven Weinberg Figure: Steven Weinberg.

25 Basic Electrodynamics Coulomb s law F = q 1q 2 r 3 r Identical in form to Newton s law! A better form E int = 1 ρ( x) x x ρ( x ) Relativistic generalisation E int = J µ (x)d ret (x x )J µ (x )

26 Electrodynamics There are two very important properties to notice: Conservation µ J µ = 0 Universality J µ = j µ 1 + jµ 2...

27 Conserved Energy-Momentum Tensor We have another conserved quantity in physics, namely, the Energy Momentum Tensor: T µν : Point particle: Perfect fluid: Electromagnetic field: µ T µν = 0 t µν (x, t) = pµ p ν E δ3 ( x R(t)) T µν = (p + ρ)u µ u ν + η µν T µν = F µα F ν α + η µν F αβ F αβ

28 A Relativistic al Theory Now consider E int (x 0 ) = κ 8π [T µν (x)t µν (x ) 1 2 T (x)t 1 (x )] x x Consider the gravitational effects of a body of mass M in its rest frame T 00 = Mδ 3 (0) Applying the formula E int = κ 8π M dx 1 x [2t00 + t](x, x 0 ) where t µν is the energy momentum tensor of the test particle. We will now apply this to several cases.

29 Stationary Body Let us consider a stationary body located at R so that t 00 = mδ 3 (x R) with t = t kk t 00 giving E int = GMm R

30 Light Bending Now consider the test particle to be a light ray. In this case t = 0 and t 00 = Eδ 3 (x R) giving E int = 2GME Rc 2 displaying the famous factor of 2 (which Einstein had actually missed originally!). One can also straight away compute the deflection angle φ = 2GM( ρ ) which is the GR result. dz z 2 + ρ 2 = 4GM/ρ

31 Time delay The same interaction reduces the speed of light by the factor 1 2GM/R. Considering the superrior conjunction of a planet with the line of sight from earth passing at a distance ρ from Sun t = 4GM( ze 0 + zp 0 dz ) (z 2 + ρ 2 ) = 4GMlog(2z e ρ )(2z p ρ )

32 Red Shift A slowly moving excited atom of energy E 1 in the absence of a gravitating body has now the total energy E 1 GME 1 /R and likewise for its ground state. Hence the frequency emitted is reduced by the factor 1 GM/R.

33 Perihelion Motion The derivation of the (excess)perihelion motion is rather subtle. The derivation of GMm/R ignored the t kk contribution for the test particle. t kk (p/m) 2 t 00 = (2T /m)t 00 T now stands for the kinetic energy of the test particle. This increases the potential by a factor 1 + 2T /m. Earlier we had taken t 00 to be m while it really is m + T and this gives an additional factor 1 + T /m. The relavistic correction to kinetic energy p 2 + m 2 m T T 2 /2m gives an additional correction of T 2 /2m.

34 Perihelion Motion - A new feature A new feature appears at this order of accuracy. We had so far neglected the energy density of the gravitational field. In electrostatics the energy density of the electric field is E 2 = φ 2. We can work out the gravitational analog by replacing φ by the Newtonian potential. This gives an additional contribution V 2 /2m. Total additional contribution is V (3T /m) T 2 /2m + V 2 /2m. Using E 0 = T + V the total potential becomes E 2 0 /2m + 4E 0V /m + V 3V 2 /m A constant multiple of V only produces a small change in the scale of the orbit. Thus the effective potential is V eff = V 3V 2 /2m

35 Perihelion Motion This produces a perihelion precession angle Which is again the GR result. φ = 6πG 2 M 2 /L 2 1

36 al Spin Precession Now consider the test particle to be replaced by a test body that is spinning around some axis of rotation. The velocity of a small element is given by V = V 0 + ξx ω Going through the same kind of exercise gives an additional interaction energy of E spin = 3 2 GM (RxV 0 S 2 R 3 It is again easy to see that this correctly reproduces Lense-Thirring Effect

37 Lense-Thirring Precession Figure: Lense-Thirring Precession.

38 al Radiation With V. Soni I showed in 1980 that the Einstein Quadrupole Formula can also be reproduced in a non-gr way.

39 Inertial Frame Dragging Figure: Frame dragging. Now consider the gravitating body itself to be rotating. Instead of only T 00 its energy momentum tensor will have all components.this will induce additional couplings between light ray and the gravitating object. These couplings are such as to make the velocity of light anisotropic.

40 Inertial Frame Dragging An operational definition of an Inertial Frame is that it is that frame in which the velocity of light is isotropic. If we adopt this criterion we will see that inertial frames are actually (partially) dragged by the rotating shell. It can be further demonstrated that in this frame the Lense-Thirring precession also vanishes.

41 Spin Precession in Binary Pulsars Figure: A Binary Pulsar.

42 Spin Precession in Binary Pulsars Figure: A Pulsar.

43 Spin Precession in Binary Pulsars Due to Lense-Thirring precession the pulsar rotation axis must precess. This will affect the pulse profile and polarisation sweep in an observable manner. But the formulae for Lense-Thirring effect available in text books is inapplicable as the gyroscope can no longer be treated as a test particle being as massive as the companion. With C.F. Cho I derived the correct formula in 1975 using the methods described here. GR people derived the same much later. Using this formula I predicted in collaboration with V. Radhakrishnan the observable effects in binary pulsars. 30 years later in 2005 (World Year of Physics) Hotan, Bailes and Ord have reported observing these effects in PSR J

44 Can we reconcile the two views? We see that ALL the high precision tests of GR can indeed be reproduced, and some even independently predicted, by the more intuitive picture which puts gravitation on par with other interactions of physics. It is however true that as the strength of the gravitational fields increases this method becomes more unwieldy. Nevertheless methods of Quantum Field Theory can be used to show that at large distances the Spin-2 theories are exactly equivalent to GR(Duff). In the spin-2 approach the principle of equivalence follows as a self-consistency requirement and need not be postulated as in GR.

45 Can we reconcile the two views? In the spin-2 approach we started with special relativistic space-time which can be thought of as a fiducial space-time which is flat. But even there we saw that the physical properties of light itself gets modified. The physical space-time based on the modified properties of light is indeed curved. For weak gravitational fields it is still sensible to treat the fiducial ST as real and gravity as an interaction. For strong gravitational fields it is conceptually better to only deal with the physical ST.