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logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. GR course 15 Gravitation Bin Chen School of Physics Peking University April 23, 2010 Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Jj 1 Principle of Equivalence 2 Physics in Curved spacetime 3 Properties of Einstein Eq. Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Gravitation in GR How the curvature of spacetime acts on matter to manifest itself as gravity? 1 Geodesic equation; 2 From the action of point particle in a curved spacetime; How energy and momentum influence the spacetime to create curvature? 1 Einstein’s equations: Rµν − 12gµνR = 8piGTµν ; (1) 2 It tells us how the matter curve the spacetime; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Weak Equivalence Principle (WEP) The earliest form dates from Galileo and Newton It states that the “inertial mass” and “gravitational mass” of any object are equal. mi = mg (2) The inertial mass clearly has a universal character, related to the resistance you feel when you try to push on the object; it is the same constant no matter what kind of force is being exerted: f = mia . (3) The gravitational mass is a quantity specific to the gravitational force. If you like, it is the “gravitational charge” of the body. fg = −mg∇Φ . (4) Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Implications of WEP An immediate consequence is that the behavior of freely-falling test particles is universal, independent of their mass (or any other qualities they may have) a = −∇Φ . (5) The WEP implies that there is no way to disentangle the effects of a gravitational field from those of being in a uniformly accelerating frame; This follows from the universality of gravitation; Compare to EM, it would be possible to distinguish between uniform acceleration and an electromagnetic field, by observing the behavior of particles with different charges. But with gravity it is impossible, since the “charge” is necessarily proportional to the (inertial) mass. Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Einstein Equivalence Principle (EEP) In SR, the concept of mass lost some of its uniqueness, E = mc2; Not only must gravity couple to rest mass universally, but to all forms of energy and momentum; Einstein: how to generalize WEP? There should be no way whatsoever for the physicist in the box to distinguish between uniform acceleration and an external gravitational field, no matter what experiments she did (not only by dropping test particles). EEP: “In small enough regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field.” Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. The relation between WEP and EEP In fact, it is hard to imagine theories which respect the WEP but violate the EEP. Consider a hydrogen atom, a bound state of a proton and an electron: mHi < m P i +m e i , (6) due to a negative binding energy; WEP ⇒ mHg < mPg +meg; However, the gravitational field couples to electromagnetism (which holds the atom together) in exactly the right way to make the gravitational mass come out right. Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Strong Equivalence Principle (SEP) Sometimes, a distinction is drawn between “gravitational laws of physics” and “non-gravitational laws of physics,”; EEP is defined to apply only to “non-gravitational laws of physics,”; “Strong Equivalence Principle” (SEP) to include all of the laws of physics, gravitational and otherwise. For our purposes, the EEP (or simply “the principle of equivalence”) includes all of the laws of physics. Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Inertial Frame In GR, no global inertial frame; The best we can do is to define locally inertial frames(LIF), those which follow the motion of freely falling particles in small enough regions of spacetime; LIF corresponds to Riemann normal coordinates, which could be constructed at any one point on a manifold — coordinates in which the metric takes its canonical form and the Christoffel symbols vanish; Einstein: Gravity ≡ Curvature; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Free particle EEP: in LIF, physics looks like those of SR In RNC, the laws of physics take the same form as they would in flat space. In flat space, a free particle obeys d2xµ dλ2 = 0 . (7) In curved spacetime, a free particle obeys its geodesic equation: d2xµ dλ2 + Γµρσ dxρ dλ dxσ dλ = 0 , (8) which reduces to (7) in RNC; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Principle of Covariance Take a law of physics, valid in the inertial coordinates in flat spacetime; Write it in a coord.-inv. (tensorial) form; Assert that the resulting law remains true in curved spactime; Comma goes to semi-colon Rule: ηµν → gµν , ∂µ → ∇µ; (9) (As ∂µφ = φ,µ and ∇µφ = φ;µ) Sometimes, this principle is also called minimal-coupling principle; Minimal coupling: the interactions of matter fields to curvature are minimal, they do not involve direct coupling to the Riemann tensor or its contractions; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Examples of application d2xµ dλ2 = dx ν dλ ∂ν dxµ dλ → dx ν dλ ∇ν dx µ dλ , i.e. d dλ → Ddλ ; Therefore, d 2xµ dλ2 = 0→ Ddλ ddλxµ, which is just the geodesic equation; Energy-momentum conservation: ∂µT µν = 0→ ∇µTµν = 0. (10) Maxwell eq.: ∂µF νµ = 4piJν → ∇µF νµ = 4piJν , ∂[µFνλ] = 0 → ∇[µFνλ] = 0, (11) which changes nothing. Actually, it is more transparent to write them in terms of differential forms d(?F ) = 4pi(?J), dF = 0; (12) which are always the tensorial equations;Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Caveat There is ambiguity in applying the rule; For example: Y µ∂µ∂νX ν = 0→ { Y µ∇µ∇νXν Y µ∇ν∇µXν (13) which have difference −RµνY µXν . The problem of ordering covariant derivatives is similar to the problem of operator-ordering ambiguities in quantum mechanics. How to choose? Experiments; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. EEP: revisited Is the principle of equivalence a fundamental principle of nature? Modern point of view: we do not expect the EEP to be rigorously true. Consider the following relation: ∇µ[(1 + αR)F νµ] = 4piJν , (14) where R is the Ricci scalar and α is some coupling constant. EP: require α = 0; However, this relation make perfect sense, is consistent with charge conservation and other desirable features of electromagnetism, which reduces to the usual equation in flat space; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. EEP: revisited why is it reasonable to set α = 0? Answer: because of scale. From dimension analysis, we know that α must have dimensions of (length)2. But since the coupling represented by α is of gravitational origin, the only reasonable expectation for the relevant length scale is α ∼ l2P , (15) where lP is the Planck length lP = ( G~ c3 )1/2 = 1.6× 10−33 cm , (16) In other words, the possible physical effect due to the modification only appears at Planck scale; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Einstein equation Einstein equation: Gµν = Rµν − 12gµνR = 8piGTµν (17) Equivalently, it could be rewritten as Rµν = 8piG(Tµν − 1 D − 2gµνT ) (18) where D is the dimension of the spacetime; Vacuum Einstein equation: Rµν = 0; (19) In D < 4, Rµν = 0⇒ Rµνσρ = 0; It is only in four dimensions or more that gravitational fields can exist in empty space; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Independent equations Gµν has ten components, suggesting ten equations; However, Bianchi identity requires that ∇µGµν = 0, giving four constraints; Therefore, there are six independent equations; If gµν is a solution of Einstein equation in xµ, it should still be a solutio in x′; This means that there are four unphysical degrees of freedom in gµν (represented by the four functions x µ′(xµ)); We should expect that Einstein’s equations only constrain the six coordinate-independent degrees of freedom. Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Difficult to solve As diff. eqs., Einstein eqs are extremely complicated; The Ricci scalar and tensor are contractions of the Riemann tensor, which involves derivatives and products of Γ’s, which in turn involve the inverse metric and derivatives of the metric. The Tµν will generally involve the metric as well. Nonlinear eqs: no superposition, no Green’s function; Looking for the exact solution of Einstein equation is an important subject in GR; In History, the first exact solution, describing a static sph. symm. star or BH, was discovered in 1916 by Schwarzschild; Then until 1962, the solution describing the rotating star or BH, was discovered by Kerr; Actually, at least two books on the exact solutions of Einstein eq.; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Nonlinearity In Newtonian gravity, we have ∇2Φ = 4piGρ; (20) It is linear, which means that the Newtonian potential due to two point masses is simply the sum of the potentials for each mass; In GR, the gravitational field couples to itself: a consequence of the equivalence principle — if gravitation did not couple to itself, a “gravitational atom” (two particles bound by their mutual gravitational attraction) would have a different inertial mass (due to the negative binding energy) than gravitational mass. The nonlinearity of Einstein eq. is a reflection of the back-reaction of gravity on itself; Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Feynman diagrams for EM The electromagnetic interaction between two electrons can be thought of as due to exchange of a virtual photon: e e- - photon Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Feynman diagrams for gravity: around the flat spacetime e e- - graviton gravitons Bin Chen GR course 15 logo JJJjjj Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq. Feynman diagrams for gravity: around the flat spacetime Gravitation: exchange of spin 2 graviton; Nonlinearity: also shared by non-Abelian gauge theory; However, different from gauge theory, GR is unrenormalizable; Nevertheless, as an effective field theory, GR makes perfect sense; Bin Chen GR course 15 Ìá¸Ù Principle of Equivalence Physics in Curved spacetime Properties of Einstein Eq.