Gravito-Electromagnetism (GEM) Weak (Linearizable) Slowly Changing Gravitation

Transcription

1 Gravito-Eletromagnetism GEM Weak Linearizable Slowly Changing Gravitation Andrew Forrester June, 200 Contents Introdution 2 Notation 2 3 The Metri, Perturbation, and Potentials 3 3. Dimensional Analysis Notable Metris The Connetion and Tensors of Interest 5 4. Dimensional Analysis The Linearized Einstein Field Equation 8 5. Dimensional Analysis Gravito-Eletromagnetism 0 6. Differenes Between EM and GEM Equations Issues to Resolve Gauge Freedom in Potentials The Geodesi Equation and Gravito-Lorentz Fore 3 7. Dimensional Analysis A Simple Geometry: The Lense-Thirring Effet 6 9 Further Topis and Resoures 8 Introdution Instead of imagining spae-time as being warped by mass and energy, one an speak of a lassial spin-2 graviton field in flat, Minkowski spae-time that generates gravitation. Although we don t know yet how to quantize this field, we an think of it in a way similar to how we think of eletromagnetism being mediated by photons. And just as a /r 2 Coulomb fore generates magnetism when the finite speed of the mediating photon is taken into aount, a /r 2 Newtonian gravitational fore generates gravito-magnetism when the finite speed of the mediating graviton is taken into aount. Magnetism is fundamentally an eletri-fore effet 2, and gravity must have some analogous magneti fore, meaning a gravitational fore proportional and perpendiular to the veloity of a test mass. Einstein showed that gravity should be non-linear, so we know that the graviton should self-interat. General relativity also implies that the graviton should be spin 2. Taking that self-interation and spin-2 into aount should bring us all the way to the equivalent of general relativity. But it may be that in most of the See Chris Clark s paper Magnetism is not fundamental at < 2 Even the quantum spin magneti moment is seen to be a natural eletri effet in quantum field theory; see What is Spin? by Hans C. Ohanian, available on my website at <

2 universe barring blak holes, supernovae, et etera, all you really need to know about gravitation is the eletromagneti-analogue part gravito-eletromagnetism or GEM to have an aurate desription. In this paper, instead of taking the bottom-up approah just disussed in whih one onstruts GEM by starting from the Newtonian gravity and taking the finite speed of the graviton into aount, we take the top-down approah and start with the equations of general relativity, apply some simplifying assumptions, and obtain the GEM equations. After doing some preparation in setions 2 5, we show that under two assumptions, low mass-energy density or weak gravitational fields so we may linearize the Einstein equations 2 slow hanges 3 in matter-energy so we may neglet seond order time derivatives Einstein s field equations of general relativity, beome G µν = 8πG N 4 T µν, G ε g ρ ef ε g P 4 H 2 t G µ g J ef + µ g M + µ g t P, 2 very muh like the eletromagneti field equations in matter, E = ε 0 ρ f ε 0 P 3 B 2 t E = µ 0 J f + µ 0 M + µ 0 t P. 4 We also disuss the meaning of the differenes between these equations. In the preparation work, we linearize all the tensors and the Christoffel onnetion symbols, whih takes are of assumption above, by taking the metri to be the Minkowski metri plus a perturbation. In setion 6 we apply the seond assumption. Furthermore, in setion 7 we show that the geodesi equation for partile trajetories in spae-time, beomes the gravito-lorentz fore, analogous to the eletromagneti Lorentz fore: dp µ dλ + Γµ ρσp ρ p σ = 0, dp = F gl = E G + v H], dp = F L = Q E + v B ]. And finally, in setion 8, we do an example problem dealing with the Lense-Thirring effet, whih relates to inertial frame dragging in general relativity. 2 Notation We use 4 2 = µ µ = = 2 t 2 + x 2 + y 2 + z 2 = To be more aurate, we should use a word like aeleration rather than hange, sine slow hange implies small first derivative with respet to time. 4 Sometimes 2 is written as, in analogy to, and sometimes it s written as 2, in analogy to 2, usually in physis settings. I prefer the 2 notation beause it reveals its squared nature while the box itself reveals its four-variable nature. 2

3 K e 4πε 0 and K m µ 0 4π We use h to denote some form of ontration of the tensor h, for example, taking the pseudo-trae of h µν : h = h µ µ = g µρ γ α γ α h ρµ ; or the following ontration: hβ ɛ = h β αɛ = g αδ h γδ β ɛ. We will thus put bars over the Riemann tensor symbol R to denote the Rii tensor R and Rii salar R and over the Einstein tensor symbol G to denote the Einstein salar Ḡ, showing that they are ontrations. We will use G N for Newton s gravitational onstant exept when aompanied by a mass M: Ḡ = G µ µ G G N GM G N M Also, we will use G and G i to denote the gravito-eletri field. With these notations, the only ambiguity is between the Einstein tensor G when writing it without super- or subsripts and the magnitude of the gravito-eletri field G = G whih I think does not turn up in this doument, so long as they are not multiplied by a mass M. 3 The Metri, Perturbation, and Potentials We may desribe the metri g as the Minkowski metri η = diag,,, plus a perturbation h, where h µν for some appropriately hosen oordinate system: g µν = η µν + h µν The inverse is sine g µν η µν h µν g µν g νρ η µν + h µν η νρ h νρ = δµ ρ + h µν η νρ η µν h νρ + h µν h νρ = δµ ρ + Oh 2 We will deompose h into omponents that later be seen as analogous to the eletri salar potential and magneti vetor potential under the appropriate irumstanes. This potentials deomposition of h is the following: h µν = = 2Φ w w 2 w 3 w 2s Ψ 2s 2 2s 3 w 2 2s 2 2s 22 Ψ 2s 23 w 3 2s 3 2s 32 2s 33 Ψ 2Φ w j 2Φ w i = 2s ij Ψδ ij w j where h and s are symmetri sine g and η are symmetri, and µν w i h ij µν µν Ψ 6 hi i s ij 2 h ij 3 hk k δ ij = 2 h ij + Ψδ ij 3

4 so that s is traeless: s = s + s 22 + s 33 = δ ij s ij = 2 δij h ij 3 δij h k k δ ij = 2 h j j h k k = 0. Note that i h i = δ ij h ji = 6Ψ h = h µ µ η µν h νµ and that h annot be alled the trae of h, although it might be alled the pseudo-trae of h. We have h = h µ µ = g µν h νµ η µν h νµ equality to first order in h = h 00 + h + h 22 + h 33 = 2Φ + 2s Ψ + 2s 22 Ψ + 2s 33 Ψ = 2Φ + s 3Ψ = 2Φ 3Ψ Also, for referene, we have h 00 = 2Φ, h 0i = w i, and h ij = 2s ij Ψδ ij. 3. Dimensional Analysis Note that g = g µν dx µ dx ν. While the metri tensor g µν is a dimensionless type-0,2 tensor field, the metri g is an area-valued 2-form field Notable Metris The general, linearized metri is Given that Ψ = Φ and s = 0, The Shwarzshild metri is g = g = + 2Φ w i dx i + 2Ψδ ij + 2s ij ] dx i dx j g = + 2Φ Φdx 2 + dy 2 + dz 2 2GM r GM r dr 2 + r 2 dω 2 5 Are both g and g µν alled the metri tensor? If so, what would be a good way to verbally distinguish them? 4

5 4 The Connetion and Tensors of Interest In terms of the perturbation h, and to first order in h, we have that the Christoffel onnetion Γ, Riemann tensor R, Rii tensor R, Rii salar R, and Einstein tensor G are Γ ρ µν = 2 gρλ µ g νλ + ν g λµ λ g µν 2 ηρλ µ h νλ + ν h λµ λ h µν R ρ µσν = σ Γ ρ νµ ν Γ ρ σµ + Γ ρ σλ Γλ νµ Γ ρ νλ Γλ σµ σ Γ ρ νµ ν Γ ρ σµ σ 2 ηρλ ν h µλ + µ h λν λ h νµ ν 2 ηρλ σ h µλ + µ h λσ λ h σµ = 2 ηρλ σ µ h λν σ λ h νµ ν µ h λσ + ν λ h σµ σ µ h ρ ν σ ρ h νµ ν µ h ρ σ + ν ρ h σµ = 2 R µν = R ρ µρν ρ µ h ρ ν ρ ρ h νµ ν µ h ρ ρ + ν ρ h ρµ 2 = 2 = 2 ρ µ h ρ ν 2 h νµ ν µ h + ν ρ h ρ µ ρ µ h ρ ν + ρ ν h ρ µ µ ν h 2 h µν R = R ν ν 2 ρ ν h ρ ν + ρ ν h ρν ν ν h 2 hν ν ρ ν h ρ ν + ρ ν h ρ ν 2 h 2 h = 2 = ρ ν h ρ ν 2 h = µ ν h µν 2 h rearranging and renaming indies rearranging and using symmetry of h µν G µν = R µν 2 g µν R R µν 2 η µν R = 2 ρ µ h ρ ν + ρ ν h ρ µ µ ν h 2 h µν 2 η µν µ ν h µν 2 h ρ µ h ρ ν + ρ ν h ρ µ µ ν h 2 h µν η µν ρ λ h ρλ + η µν 2 h = 2 We ould also derive this linearized Einstein tensor by varying the following Lagrangian with respet to the perturbation h µν : ] L = 2 µ h µν ν h µ h ρσ ρ h µ σ + 2 ηµν µ h ρσ ν h ρσ 2 ηµν µ h ν h 5

6 In terms of the potentials deomposition of h, we have Γ ρ µν 2 ηρλ µ h νλ + ν h λµ λ h µν Γ η00 0 h h 00 0 h 00 = 0 Φ Γ i 00 2 ηii 0 h 0i + 0 h i0 i h 00 = i Φ + 0 h 0i = i Φ + 0 w i Γ 0 j0 2 η00 j h h 0j 0 h j0 = j Φ Γ i j0 2 ηii j h 0i + 0 h ij i h j0 = 2 jw i + 0 h ij i w j = j w i] + 2 0h ij Γ 0 jk 2 η00 j h k0 + k h 0j 0 h jk = 2 jw k + k w j 0 h jk = j w k + 2 0h jk Γ i jk 2 ηii j h ki + k h ij i h jk = j h ki 2 ih jk R τµσν = g τρ R ρ µσν η τρ + h τρ σ Γ ρ νµ ν Γ ρ σµ = η τρ σ Γ ρ µν ν Γ ρ µσ R 0j0l η 00 0 Γ 0 jl l Γ 0 j0 = 0 j w l + 2 0h jl l j Φ = j l Φ + 0 j w l h jl R 0jkl η 00 k Γ 0 jl l Γ 0 jk = k j w l + 2 0h jl l j w k + 2 0h jk = k j w l 2 k 0 h jl l j w k + 2 l 0 h jk = 2 k j w l + l w j 2 l j w k + k w j + 2 l 0 h jk k 0 h jl = 2 k j w l l j w k lh kj k h lj = j 2 kw l l w k + 0 l h k]j = j k w l] + 0 l h k]j R ijkl η ii k Γ i jl l Γ i jk = k j h li 2 ih jl l j h ki 2 ih jk = k j h li 2 k i h jl l j h ki + 2 l i h jk = k 2 jh li + l h ji l 2 jh ki + k h ji + 2 l i h jk k i h jl = j 2 kh li l h ki + i 2 lh kj k h lj = j k h l]i + i l h k]j 6

7 R µν = R ρ µρν R 00 = R ρ 0ρ0 = R R 00 + R R R R 00 + R R 3030 lowering indies with η, thereby keeping only terms first order in h = R R 00 + R R 0303 by antisymmetry in last pair and first pair Φ + 0 w h Φ w h Φ w h 33 = 2 Φ + 0 w w w h + h 22 + h 33 = 2 Φ + 0 w 0 2 s Ψ + s 22 Ψ + s 33 Ψ = 2 Φ + 0 w 0 2 s 3Ψ = 2 Φ + 0 w Ψ R 0j = R ρ 0ρj = R 0 00j + R 0j + R 2 02j + R 3 03j R 000j + R 0j + R 202j + R 303j = R 0j R 022j R 033j w j] + 0 j h ] 2 2 w j] + 0 j h 2]2 3 3 w j] + 0 j h 3]3 = w j] w j] w j] 0 j h ] + j h 2]2 + j h 3]3 = 2 w j w j w j + 2 j w + 2 j w j w j h + j h 22 + j h h j + 2 h j2 + 3 h j3 = 2 2 w j + 2 j w 2 0 j h k k k hj k = 2 2 w j + 2 j w j Ψ 2 k sj k δj k Ψ = 2 2 w j + 2 j w j Ψ + k sj k j Ψ = 2 2 w j + 2 j w + 0 k s k j R ij = R ρ iρj = R 0 i0j + R ij + R 2 i2j + R 3 i3j = = = j Ψ R 0i0j + R ij + R 2i2j + R 3i3j i j Φ + 0 i w j h ij + i h j] + j h ]i + i 2 h j]2 + 2 j h 2]i i j Φ 0 i w j h ij + i h j] + 2 h j]2 + 3 h j]3 + i 3 h j]3 + 3 j h 3]i + j h ]i + 2 j h 2]i + 3 j h 3]i i j Φ 0 i w j h ij + i 2 h j + 2 h j2 + 3 h j3 i 2 jh + j h 22 + j h j h i + 2 j h 2i + 3 j h 3i 2 h ji h ji h ji i j Φ 0 i w j h ij + 2 i k hj k 2 i j δ kl h kl + 2 j k hi k 2 2 h ij = i j Φ 0 i w j 2 2 h ij + k i h k j 2 i j δ kl h kl = i j Φ 0 i w j 2 2 2s ij Ψδ ij + k 2 i s k j i δj k Ψ 2 i j 6Ψ = i j Φ 0 i w j 2 s ij + 2 Ψδ ij + 2 k i s j k k i δ k j Ψ + k j δ k i Ψ + 3 i j Ψ = i j Φ 0 i w j 2 s ij + 2 Ψδ ij + 2 k i s j k 2 i j Ψ + 3 i j Ψ = i j Φ Ψ 0 i w j 2 s ij + 2 k i s j k + 2 Ψδ ij 7

8 R = R µ µ = R R i i R 00 + R i i to first order in h 2 Φ + 0 w Ψ + i i Φ Ψ 0 i w i 2 si i + 2 k i s k i + 2 Ψδi i = 2 Φ 0 w Ψ 2 Φ Ψ 0 w + 2 k i si k Ψ = 2 2Φ Ψ 2 0 w Ψ + 2 k i s ik Ψ = 2 2 Φ 2Ψ 2 0 w + 2 k i s ik Ψ = 2 2 Φ 2Ψ 0 w + i j s ij Ψ and G µν = R µν 2 g µν R R µν 2 η µν R G 00 R 00 2 η 00 R 2 Φ + 0 w Ψ Φ 2Ψ 0 w + i j s ij Ψ = 2 2 Ψ + i j s ij G 0j R 0j 2 η 0j R 2 2 w j + 2 j w + 0 k s k j j Ψ G ij R ij 2 η ij R i j Φ Ψ 0 i w j 2 s ij + 2 k i s k j + 2 Ψδ ij 2 δ ij2 2 Φ 2Ψ 0 w + k l s kl Ψ = i j Φ Ψ 0 i w j 2 s ij + 2 k i s j k + δ ij Ψ δ ij 2 Φ 2Ψ + δ ij 0 w δ ij k l s kl + 3δ ij 0 2 Ψ = i j Φ Ψ 0 i w j 2 s ij + 2 k i s j k δ ij 0 2 Ψ δ ij 2 Φ Ψ + δ ij 0 w δ ij k l s kl + 3δ ij 0 2 Ψ = δ ij 2 i j Φ Ψ + δ ij 0 w 0 i w j δ ij k l s kl 2 s ij + 2 k i s j k + 2δ ij 0 2 Ψ 4. Dimensional Analysis Note that the Christoffel onnetion 6 symbol Γ i jk has dimensions of inverse length and that the Riemann ρ tensor R µσν, Rii tensor R µν, Rii salar R, and Einstein tensor Gµν all have dimensions of inverse area. 5 The Linearized Einstein Field Equation The Einstein field equations is are or G µν = 8πG N 4 T µν G µν = κt µν Γ i jk? 6 Is there suh a thing as the abstrat Christoffel onnetion Γ that is distint from the Christoffel onnetion symbol 8

9 where κ 8πG N / 4. Sine the stress tensor is defined as T µν, the right-hand side ontains a fator quadrati in the metri: T µν = g µα g νβ T αβ. Now, writing this nonlinear field equation out in terms of the metri, we get G µν = R µν 2 g µν R = R µν 2 g µν R µ µ = R ρ µρν 2 g µνr ρ µρµ { } { } = ρ Γ ρ νµ ν Γ ρ ρµ + Γ ρ ρλ Γλ νµ Γ ρ νλ Γλ ρµ 2 g µν ρ Γ ρ µµ µ Γ ρ ρµ + Γ ρ ρλ Γλ µµ Γ ρ µλ Γλ ρµ { ρ g ρτ ν g µτ + µ g τν τ g νµ ] ν g ρτ ρ g µτ + µ g τρ τ g ρµ ] = gρτ ρ g λτ + λ g τρ τ g ρλ g λτ ν g µτ + µ g τν τ g νµ } 2 gρτ ν g λτ + λ g τν τ g νλ g λτ ρ g µτ + µ g τρ τ g ρµ 4 g µν { ρ g ρτ µ g µτ + µ g τµ τ g µµ ] µ g ρτ ρ g µτ + µ g τρ τ g ρµ ] = κg µα g νβ T αβ + 2 gρτ ρ g λτ + λ g τρ τ g ρλ g λτ µ g µτ + µ g τµ τ g µµ } 2 gρτ µ g λτ + λ g τµ τ g µλ g λτ ρ g µτ + µ g τρ τ g ρµ If we take our weak-field approximation g µν = η µν + h µν, where hµν for some appropriately hosen oordinate system, and linearize the Einstein equation, we get { } G µν 2 η ρτ ρ ν h µτ + µ h τν τ h νµ η ρτ ν ρ h µτ + µ h τρ τ h ρµ ρ µ h ρ ν + ρ ν h ρ µ µ ν h 2 h µν η µν ρ λ h ρλ + η µν 2 h = 2 κ η µα η νβ + h µα η νβ + η µα h νβ + h µα h νβ T αβ = κη µα η νβ T αβ where the seond line is obtained from the work done in the previous setion. Again using our previous work, we an write the linearized Einstein equations in terms of the potentials: G Ψ + i j s ij κt 00 G 0j 2 2 w j + 2 j w + 0 k sj k κt 0j j Ψ G ij δ ij 2 i j Φ Ψ + δ ij 0 w 0 i w j δ ij k l s kl 2 s ij + 2 k i s j k + 2δ ij 0 2 Ψ κt ij These equations are reëxpressed in partiular gauges at the end of the next setion, after the parallels with eletromagnetism are drawn. 5. Dimensional Analysis Consider G µν = 8πG N 4 g µα g νβ T αβ = κg µα g νβ T αβ 9

10 Note that sine G µν has dimensions of inverse area, G N has dimensions of fore-area per squared mass, κ has dimensions of foretime 4 /areamass 2, and g µν is dimensionless, T µν must have dimensions of energy per volume energy density, whih is the same as fore per area pressure: κ]g µα ]g νβ ]T αβ foretime 4 ] = areamass 2 6 Gravito-Eletromagnetism fore fore = area fore 2 fore = area area = G µν] If we impose two ertain onditions on the matter and energy in spaetime, we will see that the Einstein and Maxwell equations are almost exatly the same. Given that we have low mass-energy density or weak gravitational fields so we may linearize the Einstein equations 2 slow hanges 7 in matter-energy so we may neglet seond order time derivatives we will find the similarity after some mathematial manipulation. The linear Einstein equations are G Ψ + i j s ij κt 00 G 0j 2 2 w j + 2 j w + 0 k sj k j Ψ κt 0j G ij δ ij 2 i j Φ Ψ + δ ij 0 w 0 i w j δ ij k l s kl 2 s ij + 2 k i s k j + 2δ ij 2 0 Ψ κt ij If we define S k l s lk, so that i j s ij = i S i = S, and k s j k = S j, and k i s j k = i S j, then we may rewrite the above equations as G Ψ + S κt 00 G 0j 2 2 w j + 2 j w + 0 S j j Ψ κt 0j G ij δ ij 2 i j Φ Ψ + δ ij 0 w 0 i w j δ ij S 2 s ij + 2 i S j + 2δ ij 2 0 Ψ κt ij Taking the trae of the third equation and defining p 3 T i i, we see we have G i i Φ Ψ w 0 w 3 S 2 s + 2 i S i = Φ Ψ w 3 S + 2 S = 2 2 Φ Ψ w S κt i i = κ3 p 7 To be more aurate, we should use a word like aeleration rather than hange, sine slow hange implies small first derivative with respet to time. 0

11 so that 2 Ψ = 2 Φ + 0 w 2 S κ 3 2 p, and therefore Ψ = Φ + 0w 2 S ε g P ε g C, where ε g is a onstant to be defined later, C is an arbitrary vetor field, and ε g Pr κ 3 4π R 3 2 p r R R 3 dv so that ε g P = κ 3 2 p with R = R r r. Then, rewriting the equations for G 00 and G 0j, 2 2 Φ + 0 w 2 S ε g P 2 2 w j + 2 j w + 0 S j j Φ + 0 w j 2 S j ε g P j ε g C j + S κt 00 κt 0j 2 2 w j + 2 j w + 0 S j j Φ w j 0 S j κt 0j + 2 ε g 0 C j + 2 ε g 0 P j or, rearranging and dividing by 2, 2 Φ t w 4πG N T 00 4 ε g P t j Φ + 4 j w 4 2 w j 4πG N 4 T 0j + ε g t C j + ε g tp j Now let s take a look at the eletromagneti potential equations and see how similar they look to the two gravitational potential equations immediately above. 2 V t A = ε 0 ρ ε 0 P 2 t V + A 2 A = µ 0 J + µ 0 M + µ 0 t P 2 t j V + j A 2 A j = µ 0 J j + µ 0 M j + µ 0 t P j To make the gravitational equations more like the eletromagneti ones, we an add w j on the left of the seond equation whih will ombine with the 2 w j term to yield a 2 w j term sine it is negligible by assumption, and let s define M t C whih is just as arbitrary as the arbitrary vetor field C. And now, let s finish the transformation by figuring out how the onstants should relate. Using Newton s law of gravitation and Coulomb s law, we ompare F = K e q q 2 /r 2 and F = G N m m 2 /r 2 and see G N = K ge = /4πε g. Sine ε 0 µ 0 = 2 = K m /K e, we have G N = 2 K gm = 2 µ g /4π. So the gravitation equations beome after dividing the seond one by 2 Φ t w 4 ε g T 00 ε g P 2 t j Φ + 4 j w 4 2 w j 2 µ g 5 T 0j + ε g 2 M j + ε g 2 t P j A final rearranging yeilds a Maxwellian form of the slowly hanging linear Einstein potential field equations 2 Φ t w T 00 ε g 4 ε g P 8 2 t j Φ + 4 j w 2 w j ] µ g T 0j + µ 4 g M j + µ g t P j 9 whih you an see are very similar to the Maxwell potential field equations 2 V t A = ε 0 ρ f ε 0 P 0 2 t j V + j A 2 A j = µ 0 J f j + µ 0 M j + µ 0 t P j

12 Of ourse, if we define the gravito-eletri field G and the gravito-magneti field H G Φ tw and H w as analogues to the eletri field E and the magneti field B E = V t A and B = A and we define the free non-gravitational energy 8 density ρ ef and free energy urrent density J ef by ρ ef T 00 4 and J ef j T 0j 4 then we gain the gravito-eletromagneti field equations very muh like the eletromagneti field equations G ε g ρ ef ε g P 2 4 H 2 t G µ g J ef + µ g M + µ g t P 3 E = ε 0 ρ f ε 0 P 4 B 2 t E = µ 0 J f + µ 0 M + µ 0 t P 5 The fator of /4 arises sine 4 H 2 t G 4 w 2 t Φ tw = 4 w 4 2 w + 2 t Φ + 3 t w ] 4 w 2 w ] + 2 t Φ 4 w 2 w + 2 t Φ µ g J e + µ g M + µ g t P So, the equations are very similar, but what do the differenes mean? One differene is the hanges in signs, whih is easily explained: in eletromagnetism like harges repel but in gravitation like harges positive masses attrat. Another differene is presene of the fators of. I m not yet sure how to explain this. 9 There is also the fator of 4: what Wikipedia says in the artile on Gravitoeletromagnetism 0 seems to suggest that this has to do with the spin of the graviton. Finally, there is the task of disovering in what sense ρ ef and J ef are free, interpreting what P and M mean, and determining why M is arbitrary or at least seems to be arbitrary. Apparently, P has to do with bound energy, but I m not sure how to hek that yet. And M appears to do with bound irulating mass, or angular momentum, perhaps inluding quantum mehanial inertial spin, if it exists. 6. Differenes Between EM and GEM Equations signs gravitation is only attrative, mass is only positive 2 fators of trivial? 3 fator of 4 spin-2 graviton? 8 Sine T 00 and T 0j have dimensions of energy density, this terminology of energy density is not quite right. 9 We may have to do some onstant shuffling to get a more ideal and analogous ombination. It may be that the SI definition of B or A is not ideal sine we have E = V ta rather than E = V ta. 0 Wikipedia Gravitoeletromagnetism artile URI: 2

13 6.2 Issues to Resolve the meaning of ρ ef and J ef free? 2 the meaning of P bound energy? 3 the meaning of the seemingly arbitrary M bound angular momentum, spin? 6.3 Gauge Freedom in Potentials Let s ompare the gravitational and eletromagneti gauge transformations using the gravitational gauge ξ µ t, x and the eletromagneti gauge λt, x and then look at some partiular gauge hoies. Φ Φ + 0 ξ 0 V V t λ w i w i + 0 ξ i i ξ 0 A i A i + i λ Ψ Ψ 3 iξ i s ij s ij + i ξ j 3 kξ k δ ij Transverse Gauge: generalization of the onformal Newtonian or Poisson gauge, losely related to Coulomb gauge in eletrodynamis i w i = 0 analogue of Coulomb gauge, i A i = 0 i s ij = 0 Linearized Einstein Field Equations in Transverse Gauge: G Ψ κt 00 G 0j 2 2 w j j Ψ κt 0j G ij δ ij 2 i j Φ Ψ 0 i w j + 2δ ij 0 2 Ψ 2 s ij κt ij Synhronous gauge: equivalent to Gaussian normal oordinates, as disussed in Appendix D ; gravitational analogue of the temporal gauge of eletrodynamis A 0 = 0, sine it kills off the nonspatial omponents of the perturbation Φ = 0 Lorenz / harmoni gauge: note h η µν h µν µ h µ ν 2 ν h = 0 7 The Geodesi Equation and Gravito-Lorentz Fore A test partile in free-fall where there are only gravitational fores will follow a geodesi: Is ξ µ a 4-gauge? dp µ dλ + Γµ ρσp ρ p σ = 0 3

14 We also have that p µ = dxµ dλ = m dxµ dτ p 0 = dλ = E/ p i = m dτ dx i = γmv i = Ev i / 2 where λ is an affine parameter related to the proper time by an affine transformation, and λ = τ/m and E = γm 2 if the partile is massive. 2 So the geodesi equation an be rewritten, emulating Newton s seond law, dp µ = dλ dp µ dλ = 2 E Γµ ρσp ρ p σ de = dp0 = 3 E = 3 E = E dp i = 2 E Γi ρσp ρ p σ = 2 E = 2 E = 3 E Γ0 ρσp ρ p σ Γ 0 00p 0 p 0 + 2Γ 0 j0p j p 0 + Γ 0 jk pj p k] 0 ΦE/ j ΦEv j / 2 E/ + ] 0 Φ + 2 j Φv j / j w k 2 0h jk v j v k / 2 Γ i 00p 0 p 0 + 2Γ i j0p j p 0 + Γ i jk pj p k] i Φ + 0 w i E/ j w i] + 2 0h ij ] j h ki 2 ih jk Ev j / 2 Ev k / 2 ] j w k + 2 0h jk Ev j / 2 Ev k / 2 Ev j / 2 E/ + ] = E i Φ + 0 w i + 2 j w i] + 2 0h ij v j / + j h ki 2 ih jk v j v k / 2 This expression for dp i / simply shows the different kinds of fores that an observer will be able to detet in his oordinate frame. Now, we may use gravito-eletromagneti field definitions to reëxpress these fores. and sine G Φ tw H w G i i Φ 0 w i H i ε ijk j w k H i = ε iab δ jk ab δjk ba jw k = ε iab a w b b w a v H i = ε ijk v j H k abuse = ε ijk v j H k = ε ijk v j ε klm l w m = ε klm ε kij l w m v j = δ lm ij = i w j j w i v j abuse = i w j j w i v j = v w i j w i] v j = 2 jw i i w j v j = 2 v Hi 2 What about light-like trajetories? What kind of parametrizations are possible? δ lm ji l w m v j 4

15 we an restate and reformulate the geodesi equations like so: ] de = E 0 Φ + 2 j Φv j / j w k 2 0h jk v j v k / 2 ] dp i = E i Φ 0 w i 2 j w i] v j / 0 h ij v j / j h ki 2 ih jk v j v k / 2 = E G i + ] v H i 0 h ij v j / j h ki 2 ih jk v j v k / Notie that for very slowly hanging in spae and time h ij, Equation 7 is nearly idential to the eletromagneti Lorentz fore law: dp = F gl = E G + v H] dp = F L = Q E + v B ] The only differene this time is the fator of in the magneti term of the gravito-lorentz fore. 7. Dimensional Analysis Note that we obtain the gravito-eletri fore F ge = EG where E = γm 2 as defined earlier from the gravito-lorentz fore and that this agrees dimensionally with our definition of G G Φ tw whih shows that G has dimensions of inverse length sine Φ and w are dimensionless. 3 3 If tw = 0, the gravitational potential Φ tells you how muh gravitational energy there is per non-gravitational energy in spaetime. Corret? How does the feedbak or nonlinear idea of gravitational-field-beomes-soure ome into play? 5

16 8 A Simple Geometry: The Lense-Thirring Effet Exerise 7.2 in Carroll ] pg 320: Consider a thin spherial shell of matter, with mass M and radius R, slowly rotating with an angular veloity Ω. a Show G = 0 inside the shell and alulate H = HM, R, Ω. b H 0 inertial frame dragging Lense-Thirring effet Calulate the rotation relative to the bakground Minkowski inertial frame of a freely-falling observer sitting at the enter of the shell. Solution to Part a Slowly rotating RΩ We assume low mass-energy density GEM eqns valid We an use dust stress tensor eqn.0: T µν = ρu µ u ν From the shell, we have ρ = ρ 0 δr R = M2 4πR 2 δr R u µ = γ v, v µ where v = RΩ Sθ so γ v, RΩ Sθ Sφ, Cφ, 0 µ µ =, RΩ Sθ Sφ, Cφ, 0 T µν = M4 δr R 4πR2 = M4 δr R 4πR2 RΩ Sθ Sφ RΩ Sθ Cφ 0 RΩ Sθ Sφ RΩ 2 S 2 θ S 2 φ RΩ 2 S 2 θ Cφ Sφ 0 RΩ Sθ Cφ RΩ 2 S 2 θ Cφ Sφ RΩ 2 S 2 θ C 2 φ RΩ Sθ Sφ RΩ Sθ Cφ 0 RΩ Sθ Sφ RΩ Sθ Cφ Consider the linearized Einstein field equations in the transverse gauge, Equations : G 00 = 2 2 Ψ = κt 00 M = 8πG N δr R 4πR2 G 0j = 2 2 w j j Ψ = κt 0j G ij = δ ij 2 i j Φ Ψ 0 i w j + 2δ ij 0 2 Ψ 2 s ij = κt ij = 0 6

17 where κ 8πG N / 4. The stress tensor T µν is independent of time, so everything else should be likewise and time-differentiated terms drop out. After using separation of variables and the appropriate boundary onditions, we get the solutions for the potentials: { GM Φ = Ψ = R, for r < R GM r, for r R Inside the shell, Outside the shell, w = w 2 = { 4GMΩ 3R 4GMΩ 3R { w 3 = 0 s ij = 0 4GMΩ 3R 4GMΩ 3R G = Φ 0 w = y, for r < R R 2 y, r 3 for r R x, for r < R R 2 x, r 3 for r R { 0, for r < R GM r, r 3 for r R H = w = z w 2, z w, x w 2 y w = 4GMΩ 3R 4GMΩ ẑ 3R = 8GMΩ 3R ẑ H = w = z w 2, z w, x w 2 y w = 4GMΩ R 2 4GMΩ R 2 zx, R r5 R r 5 zy, 4GMΩ 3R = 4GMΩ R 2 R r 3 zx r 2, zy r 2, x2 + y 2 r 2 4GMΩ R 2 3R r 3 + 4GMΩ R 2 R r 5 x2 + 4GMΩ R R 2 r 5 y2 Solution to Part b Inside and at the enter of the shell, we have an observer of mass m with energy E = γ v m 2 and momentum p = mv. ] de = E 0 Φ + 2 jφv j / j w k 2 0h jk v j v k / 2 0 dp i = E G i + v i H 0 h ij v j / ] j h ki 2 ih jk v j v k / 2 7

18 So the energy E of the partile is essentially onstant and we have dp = E p m H = E m p H Sine H is a onstant vetor in the z-diretion, we see that p is preessing: let p be the projetion of p on the x-y plane perpendiular to H so dp = E m p H = F entripetal = mr rot ω 2 = E dp E 2 2 m H = p H H m EH 2 = p m d 2 p Thus, the angular frequeny of the irular motion of the observer is and sine we have ω = E m H = E 8GMΩ m 3R m2 + p 2 /2m m 8GMΩ 3R = + v 2 / 2 8GMΩ 3R v = mr rot ω r rot = v m ω = v m 3R m E 8GMΩ v 3R = m + v 2 / 2 8GMΩ 8 9 In obtaining r rot and ω we have found the rotation relative to the bakground Minkowski inertial frame of a freely-falling observer sitting at the enter of the shell. Note that the rotation is dependent upon the initial veloity of the observer and there is no rotation if the observer is initially stationary. 9 Further Topis and Resoures Papers Brill and Cohen: Rotating Masses and Their Effet on Inertial Frames, Phys. Rev. 43, Issue 4 Marh 966] Cohen: Gravitational Collapse of Rotating Bodies, Phys. Rev. 73, Issue 5 September 968] Referenes ] Sean M. Carroll: Spaetime and Geometry: An Introdution to General Relativity, Addison-Wesley ] Charles W. Misner, Kip S. Thorne, John Arhibald Wheeler: Gravitation, W. H. Freeman and Company 973 8