Spin Motion in General Relativity

Nicolas Maldonado Baracaldo

2020

Universidad de los Andes Facultad de Ciencias Departamento de Física

Spin Motion in General Relativity Movimiento de Spin en Relatividad General Nicolas Maldonado Baracaldo universidad de los andes facultad de ciencias departamento de física bogotá, d.c., colombia

A monograph submitted in partial fulﬁllment of the requirements for the degree of Bachelor of Science (Physics), under the supervision of Professor Marek Nowakowski, Ph.D.

August 2020

To Ramón.

He never got to see what I became.

I hope I make him proud.

Acknowledgements

I’d like to thank my supervisor, Marek Nowakowski, who always asks the questions that lead me to the necessary answers. I have drawn so much from him, as well as several authors he led me to, that few, if any, of the ideas presented herein are originally my own. Except for the mistaken ones, those are entirely on me.

I’d also like to thank Alejandra and Andrea who proofread my work through and through and whose opinion I sought on several stylistic choices. They enriched my writing despite all the parts they couldn’t understand and by now must be sick of it. Now for something of an unacknowledgement. After a whole semester helping you design housing for cancer patients, I wasn’t formally acknowledged. After another semester helping you design a tarot deck to teach sexual education, I wasn’t formally acknowledged. So now after having helped me a whole semester delve into the motion of spin in gravitational ﬁelds, you don’t get a formal ac- knowledgement. You know who you are.

I’d be remiss if I didn’t also thank Stack Exchange whose users’ ex- tensive knowledge of physics, mathematics, and LATEXgot me through a number of roadblocks along the way. As well as coffee, hundreds of cups of which I must have drunk while writing this, and Spotify, for providing just the right soundtrack.

Finally I must thank my family: my father who has always championed my education and my pursuit of science; my mother who has loved me and whose support of me never falters; my sister who grew up with me then and grows up with me still; Jack Daniel who reminds me of the importance of playfulness and sound sleep; my extended family who in one way or another have made me into who I am. My gratitude towards them all I cannot faithfully put into words.

Abstract

In direct analogy to the quantum-mechanical study of a particle’s spin when said particle is placed in an external magnetic field, we herein present a brief exploration of the general-relativistic behavior of a particle’s spin when said particle is placed in an external gravitational field. The concept of geodetic effects is explained as motivation through a literature review before explicit calculations are presented for a few of the most commonly encountered metrics using two separate formalisms, finally returning to the geodetic effects for some closing remarks.

Resumen

En analogía directa con el estudio del spin de una partícula en mecánica cuántica cuando dicha partícula se coloca en un campo magnético externo, aquí presentamos una breve exploración del com- portamiento general-relativista del spin de una partícula cuando dicha partícula se coloca en un campo gravitacional externo. El con- cepto de efectos geodéticos se explica como motivación a través de una revisión bibliográfica antes de presentar cálculos explícitos para algunas de las métricas más comúnmente encontradas usando dos formalismos distintos, finalmente volviendo a los efectos geodéticos para algunas observaciones finales.

Contents

Introduction 13

PARTISPINMOTIONINEXTERNALFIELDS

Some Formalisms for Spin 17

Spin in External Electromagnetic Fields 19

Spin in Gravity 21

Thomas, de Sitter, and Lense-Thirring Precessions 23

PARTIITHEFRIEDMANN-LEMAÎTRE-ROBERTSON- WALKER(FLRW)METRIC

The Metric 27

Point Particles in the FLRW Metric 31

Extended Bodies in the FLRW Metric 39

PARTIIITHESCHWARZSCHILDMETRIC

The Metric 51

Point Particles in the Schwarzschild Metric 55

Extended Bodies in the Schwarzschild Metric 59

PARTIVFINALREMARKS

On the Results from the Different Formalisms 65

De Sitter and Lense-Thirring Precessions, Revisited 67

Conclusion 69

Bibliography 71

Appendix A: Operator of Proper-Time-Derivative 75

Introduction

Particles in quantum physics are not only characterized by a mass and an electric charge, they also posses an intrinsic angular momentum s, called spin, which is an invariant property of the particle [ ] and thus completely independent of its state of motion1 p. 224 . The 1 F. Scheck, Quantum Physics, 2nd ed. physical manifestation of spin is usually taken to be the particle’s (Springer-Verlag, Berlin, 2007) magnetic moment, which is proportional to its spin. It is then natural to consider the behavior of a particle’s spin when said particle is introduced in an external magnetic ﬁeld, leading to the well-known Larmor precession.

Spin is itself, however, a vector, and as such it is subject to relativistic corrections. Indeed the placement of a particle in an external magnetic field, in the special-relativistic case, requires the use of the Bargmann-Michel-Telegdi (bmt) equation, which results not only in Larmor precession but also a small correction known as Thomas precession. But what of general-relativistic corrections? We should expect there to be some effect on the spin when a particle is placed in an external gravitational field. This is in fact known as the geodetic effect and can be further subdivided depending on the source of the gravitational field. When simply considering the effect due to a central mass we arrive at what is known as de Sitter precession; meanwhile, if the central mass happens to itself be rotating, there is an additional frame dragging effect which results in so-called Lense- [ − ] Thirring precession2 p. 252 254 2 W. Rindler, Relativity: Special, General, and Cosmological, 2nd ed. (Oxford University Press, Oxford, 2006) What follows is a brief exploration of spin motion in the general- relativistic case. De Sitter and Lense Thirring precessions, collec- tively the geodetic effects, are further explained as motivation after a literature review, and some explicit calculations are then carried out for the most commonly encountered metrics, the Friedmann- Lemaître-Robertson-Walker (flrw) metric used in cosmology and the Schwarzschild metric for a static central mass, in both cases us- 14

ing a formalism for point particles, called the geodesic formalism, and a formalism suited to extended bodies, through the Mathisson- Papapetrou-Dixon (mpd) equations. We ﬁnally return to the geodetic effects and their experimental conﬁrmation for some closing remarks. Part I

Spin Motion in External Fields

Some Formalisms for Spin

While defining spin can be conceptually simple, as has been done [ ] in the introduction following Scheck3 p. 224 , the way it is treated 3 F. Scheck, Quantum Physics, 2nd ed. mathematically can take several forms. (Springer-Verlag, Berlin, 2007)

The most general for quantum mechanical applications is to consider a spin operator Sˆ, which, following the Heisenberg picture of quantum mechanics, will have a time-evolution. Indeed, in the Heisenberg picture of quantum mechanics, it is the operators that change with time according to the relation 4 4 B. Zwiebach, Quantum Dynamics, Cambridge, MA, 2013 ∂ i Aˆ(t) = [Hˆ , Aˆ],(1) ∂t h¯ where Aˆ(t) is any time-dependent operator, Hˆ is the Heisenberg Hamiltonian (this is not necessarily the same as the Schrödinger Hamiltonian), and [Hˆ , Aˆ] is their commutator. In particular for the spin operator, Sˆ, we have

∂ i Sˆ (t) = [Hˆ , Sˆ ].(2) ∂t h¯

Following this it is also possible to turn the spin operator into a spin function by simply taking the operator’s expectation value, we then have

S(t) = hψ|Sˆ (t)|ψi.(3)

For relativistic treatments of spin one must consider it as a tensor. This may well be a (1,0) tensor—the spin vector—Sα, analogous to the spin function S(t), albeit dependent now on proper time τ. It may also be a (2,0) tensor—the spin bivector—Sαβ, which arises naturally from another tensor describing the rotational motion of particles in spacetime through Noether’s theorem. It can be shown for arbitrary translations that 18 spin motion in general relativity

µν ∂νT = 0, (4)

with Tµν the energy-momentum tensor. Tµν is then called the Noether current for translations and it leads to a conserved quantity upon spatial integration, the four-momentum

Z Pµ = d3xTµ0 (5) µ ∂tP = 0.

[ ] It may also be shown 5 p. 19 for arbitrary Lorentz transformations 5 J. D. Bjorken and S. D. Drell, Relativis- without translations that tic Quantum Fields, 1st ed. (McGraw- Hill, Inc., New York, 1965)

µνλ ∂µM = 0 ∂L νλ (6) Mµνλ = ν µλ − λ µν + x T x T µ ∑ φs ∂ (∂φr/∂x ) rs

with Mµνλ the corresponding Noether current for rotations—itself the sum of two terms, one again for energy-momentum, and one for inﬁnitesimal rotations. This then leads to a conserved quantity upon spatial integration, the angular momentum

Z Sµν = d3xM0µν (7) µν ∂tS = 0,

which, although containing both spin and orbital terms, may be separated such that one obtains only the spin part according to the relation

1 1 Sµ = √ eµνρσu S 2 −g ν ρσ (8) 1 1 Sµ = √ eµνρσg uαg g Sβγ. 2 −g να ρβ σγ Spin in External Electromagnetic Fields

In order to study the spin motion in an external magnetic ﬁeld, B, without relativistic corrections, it is most convenient to use the spin function S(t), together with the magnetic dipole moment, deﬁned similarly to the classical case as

µˆ = γSˆ (9) µ(t) = γS(t),

for some constant γ, the form of which is irrelevant for the study at hand, and ﬁnally the Hamiltonian for the problem, which takes the form

Hˆ = −µˆ · B (10) = −γB · Sˆ.

It can then be shown 6, using (2), that the spin function will behave 6 L. Landau and E. Lifshitz, Quantum as Mechanics Non-relativistic Theory, 2nd ed. (Pergamon Press, Oxford, 1974), B. Zwiebach, Two State Systems, Cam- dS = −γB × S,(11) bridge, MA, 2013 dt corresponding to the well-known Larmor precession, valid in the non-relativistic case.

For the special-relativistic case, the so-called Bargmann- Michel-Telegdi (bmt) equation may be used, which in the simplest case takes the form 7 7 J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, Inc., New DSα e h g g i York, 1999) = FαβS + − 1 uα(S Fλµu ) (12) dτ m 2 β 2 λ µ with Sα being the spin vector and Fαβ the electromagnetic tensor for the external field. Much like in the non-relativistic version we see a first term that couples the spin to the external field resulting in Larmor precession. Unlike the non-relativistic case, however, we 20 spin motion in general relativity

now ﬁnd an additional term where the external ﬁeld and the spin get further coupled to the particle’s four-velocity uα, this term accounts for a small relativistic correction, an additional precession called Thomas precession. Spin in Gravity

It is well known that classical spin, S, defined as in (3), reacts to external fields, in the simplest, non-relativistic case, according to (11), and in the special-relativistic case according to the bmt equation (12). However, S also interacts with gravitational fields leading to some general-relativistic equations of motion, of which we may now consider two versions.

The first, and simplest of them, which we may call the geodesic formalism, starts from the non-relativistic fact that a point particle not under the inﬂuence of any force will have constant velocity and spin

duα = 0 dτ (13) dSα = 0 dτ and then uses the Principle of General Covariance to arrive at the [ ] two geodesic equations 8 p.122 8 S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, 1st ed. (John Wiley Duα ∂uα & Sons, Inc., New York, 1972) = + Γα uβuγ = 0 dτ ∂τ βγ (14) DSα ∂Sα = + Γα Sβuγ = 0 dτ ∂τ βγ here the differential operator D is the α dτ where we again ﬁnd a coupling of the spin, S , and the four- operator of proper-time-derivative, velocity, uα, as in the bmt equation. deﬁned as The geodesic formalism quite simply results in the parallel trans- D = uµ∇ (15) port of the spin vector which in general should result in some preces- dτ µ sion. with ∇µ the ordinary covariant derivative (see appendix A.) The second formalism we are to consider is one that is adapted to extended bodies, it makes use instead of the spin bivector in what are 22 spin motion in general relativity

called the Mathisson-Papapetrou-Dixon (henceforth referred to simply as mpd) equations 9. They require the use of the bivector because 9 A. Papapetrou, “Spinning Test- they are meant to describe the motion of spinning massive extended Particles in General Relativity. I”, Proceedings of the Royal Society A 209, bodies and thus their angular momentum should include both the 248–258 (1951) spin and orbital parts. We can, regardless, ultimately consider only the spin by using (8). Several equivalent forms of the mpd equations exist, depending on how each of them is written, whether in terms of the 4-velocity or the 4-momentum of the particle, but the form we use in the calculations that follow will be

D D 1 muλ + u Sλµ + uπSρσRλ = 0 dτ µ dτ 2 πρσ (16) D D D Sµν + uµu Sνσ − uνu Sµσ = 0. dτ σ dτ σ dτ One ﬁnal consideration regarding the mpd equations is that they are underdetermined and require an additional constraint so as to be solvable, what is known as a spin supplementary condition (ssc). Once again, there exist several possible sscs, the discussion of which we leave for later. Sufﬁce it to say, for the time being, that the chosen ssc in what follows is the so-called Frenkel condition

αβ uαS = 0. (17) Thomas, de Sitter, and Lense-Thirring Precessions

Larmor precession was first described mathematically by Joseph Larmor in 189710, decades before the concept of spin was even intro- 10 A. S. Eddington, “Joseph Larmor, duced, in trying to explain the results of experiments described by 1857-1942”, Obituary Notices of Fellows of the Royal Society 4, 197–207 (1942) Pieter Zeeman. In 1916, Willem de Sitter calculated the precession of the Earth-moon gyroscope orbiting the sun due to general relativity, an effect which now bears his name. Only a few years later, around 191811, Josef Lense and Hans Thirring studied the weak-field approx- 11 H. Pfister, “On the history of the so- imation for a spinning spherical body of uniform density and found called Lense-Thirring effect”, General Relativity and Gravitation 39, 1735–1748 a similar precession affecting gyroscopes orbiting said body, an ef- (2007) fect that came to be named after them12. Finally, in 1927, Llewellyn 12 Pfister, in a historical review of the Thomas studied the relativistic effects on spin for flat spacetime, i.e. Lense-Thirring effect, discovered that in fact Einstein may have been first to special-relativistic effects, for use in atomic physics (the gyroscope suggest such an effect, as far back as here would be the electron orbiting the atomic nucleus), a precession 1917, to Thirring himself, and argues 13[p.1119] that it should be renamed the Einstein- effect named after him . Thirring-Lense effect. 13 C. W. Misner et al., Gravitation, 1st ed. These four account for the known precession of gyroscopes (or (W. H. Freeman and Co., San Francisco, 1973) spin, as it were) due to an external magnetic field (Larmor), special- relativistic effects (Thomas), or an external gravitational field (de Sitter and Lense-Thirring). For an extended gyroscope orbiting, say, Earth, only the lattermost three apply; Thomas precession applies when the gyroscope is close to the surface and the other two when it is placed far from the surface (the total contribution is a sum of them both, not just any one of them on its own). Specifically, Thomas precession gives the special-relativistic correction near the surface of the body, de Sitter precession accounts for the general-relativistic effect of a central mass, and Lense-Thirring precession accounts for the general-relativistic effect of the central mass itself spinning (so- called frame-dragging). Finally, it must be noted just how weak these effects can be. When considering the specific case of the precession of a gyroscope orbiting Earth, they turn out to be about, respectively, 8/3, 8, and 0.1 seconds of arc per year. All told, we can quickly see how hard these effects are to detect. 24 spin motion in general relativity

Indeed, an orbiting gyroscope away from Earth’s surface would have to stay there for some 450 years to precess one degree, and a gyroscope near the surface would have to do so for over 13000 years. Nevertheless, experiments have been carried out that conﬁrm both Thomas and de Sitter precessions, and observational data shows what would be expected from Lense-Thirring precession. Further discussion of said experiments and observations will be left for later. Part II

The Friedmann-Lemaître- Robertson-Walker (FLRW) Metric

The Metric

The development of cosmology in the early twentieth century led to the abandonment of the theretofore prevalent assumption of "perfect" symmetry in the universe, that is, symmetry throughout both space and time. Instead, the preferred assumption, the one that was consistent with observation, became that of a spatially homogeneous and isotropic universe that would nonetheless evolve in time. In general-relativistic terms, spacetime is to be seen as having a real temporal dimension and spatial components represented by a maximally symmetric three-manifold, i.e. the spacetime manifold takes the form R × Σ, with Σ being the aforementioned maximally sym- [ ] metric three-manifold14 p.329 . We now derive the spacetime metric 14 S. M. Carroll, Spacetime and Geometry, that arises from such an assumption. 1st ed. (Pearson Education Limited, Harlow, 2014)

For simplicity’s sake we may begin by considering the more [ ] general form15 p. 403 15 S. Weinberg, Gravitation and Cosmol- ogy: Principles and Applications of the General Theory of Relativity, 1st ed. (John Wiley & Sons, Inc., New York, 1972) ds2 = −dt2 + U(r, t)dr2 + V(r, t) dθ2 + sin2 θdφ2 ,(18)

so the non-vanishing Christoffel symbols are

U˙ (r, t) V˙ (r, t) V˙ (r, t) Γt = , Γt = , Γt = sin2 θ, rr 2 θθ 2 φφ 2 U˙ (r, t) U0(r, t) Γr = Γr = , Γr = , tr rt 2U(r, t) rr 2U(r, t) V0(r, t) V0(r, t) Γr = − , Γr − sin2 θ, (19) θθ 2U(r, t) φφ 2U(r, t) V˙ (r, t) V0(r, t) Γθ = Γθ = , Γθ = Γθ = , Γθ = − sin θ cos θ, tθ θt 2V(r, t) rθ θr 2V(r, t) φφ 0 φ φ V˙ (r, t) φ φ V (r, t) φ φ Γ = Γ = , Γ = Γ = , Γ = Γ = cot θ, tφ φt 2V(r, t) rφ φr 2V(r, t) θφ φθ

∂ ∂ where the convention is a dot for ∂t and a prime for ∂r . 28 spin motion in general relativity

We may now take the following general form for a spherically symmetric homogeneous spacetime

k(u · du)2 ds2 = −g(v)dv2 + f (v) du2 + ,(20) 1 − ku2

with one v coordinate and 3 u coordinates. From here we may deﬁne the four coordinates t, r, θ, φ, as well as the function R(t) by

Z (−g(v))1/2dv ≡ t,

u1 ≡ r sin θ cos φ, u2 ≡ r sin θ sin φ, (21) u3 ≡ r cos θ, f (v) ≡ R2(t),

so that the metric takes the form

dr2 ds2 = −dt2 + R2(t) + r2dθ2 + r2 sin2 θdφ2 ,(22) 1 − kr2

2( ) U(r t) = R t V(r t) = R2(t)r2 where we can easily read off , 1−kr2 and , . Taking the partial derivatives of these and making the appropriate substitutions into (19) we arrive at the Christoffel symbols

R(t)R˙ (t) Γt = , Γt = R(t)R˙ (t)r2, Γt = R(t)R˙ (t)r2 sin2 θ, rr 1 − kr2 θθ φφ R˙ (t) Γr = Γr = , Γr = (kr2 − 1)r, Γr (kr2 − 1)r sin2 θ, tr rt R(t) θθ φφ (23) R˙ (t) 1 Γθ = Γθ = , Γθ = Γθ = , Γθ = − sin θ cos θ, tθ θt R(t) rθ θr r φφ

φ φ R˙ (t) φ φ 1 φ φ Γ = Γ = , Γ = Γ = , Γ = Γ = cot θ. tφ φt R(t) rφ φr r θφ φθ

In the special case of a ﬂat universe, where k = 0, however, it is more convenient to go back to (20) and simply consider Cartesian coordinates such that it takes the form

ds2 = −(dx0)2 + R2(t) (dx1)2 + (dx2)2 + (dx3)2 ,(24)

which leads to the far simpler Christoffel symbols the metric 29

0 ˙ 0 ˙ 0 ˙ Γ11 = R(t)R(t), Γ22 = R(t)R(t), Γ33 = R(t)R(t), R˙ (t) Γ1 = Γ1 = , 01 10 R(t) R˙ (t) (25) Γ2 = Γ2 = , 02 20 R(t) R˙ (t) Γ3 = Γ3 = . 03 30 R(t)

Point Particles in the FLRW Metric

[ ] We now consider particle kinematics in the flrw metric16 p.36 . 16 E. W. Kolb and M. S. Turner, The early The geodesic equations, as well as the constraints, that must be satis- universe, 1st ed. (CRC Press, Boca Raton, 2018) ﬁed by the velocity uµ and spin Sµ are

duλ + Γλ uµuν = 0 dτ µν µ ν ν gµνu u = uνu = −1, (26) dSλ + Γλ Sµuν = 0 dτ µν µ ν ν gµνS u = Sνu = 0.

[ ] The given magnitude for uµ17 p.37 is consistent with only a co- 17 E. W. Kolb and M. S. Turner, The early moving observer truly perceiving the universe as isotropic, it does universe, 1st ed. (CRC Press, Boca Raton, 2018) not, however, determine what each of the components is.

The easiest case is to take the comoving frame, uµ = (1, 0, 0, 0), we may thus ﬁnd for the spin’s temporal part

dS0 + Γ0 Sµuν = 0 dτ µν 0 : 0 dS i j + R˙ (t)R(t)δijS u = 0 (27) dτ S0 = const. µ 0 Sµu = 0 ⇒ S = 0,

and for the spatial part 32 spin motion in general relativity

dSi + Γi Sµuν = 0 dτ µν ! dSi R˙ (t) ¨¨* 0 + ¨S0ui + Siu0 = 0 dτ R(t) (28) dSi dR = − Si R(t) R Si = 0 Si , R(t) 0

i for some initial conditions R0 and S0. This is in particular valid whether in spherical coordinates, i ∈ {r, θ, φ}, or in Cartesian coordinates, i ∈ {1, 2, 3}.

From now on, a dot will indicate differentiation with respect to τ, while a prime will indicate differentiation with respect to t.

On the other hand we may consider a test-particle, taking only θ φ the assumption that u = u = 0 (in order for Rθφ = Rφθ = 0 to be satisﬁed according to the Einstein equations with a perfect ﬂuid as a source), the geodesic equations take the form

t r du t r r du r t r + Γ u u = 0, + 2Γtru u = 0; dτ rr dτ (29) t r 0 du du R (t) t r + R0(t)R(t)urur = 0, + 2 u u = 0. dτ dτ R(t) Now the second of these may be solved directly,

dur R0(t) dt + 2 ur = 0 dτ R(t) dτ dur R0(t) + 2 ur = 0 dτ R(t) dτ dur R0(t) + 2 ur = 0 ur dτ R(t) dur dR + 2 = 0 ur R(t) (30) Z ur du˜r Z R dR˜ = −2 r ˜r ˜ u0 u R0 R(t) ur R(t) log r = −2 log u0 R0 R 2 ur = ur 0 , 0 R(t) point particles in the flrw metric 33

r for some initial conditions u0, R0. Putting this solution into the other geodesic equation we get

dut R 4 + R0(t)R(t)ur ur 0 = 0 dτ 0 0 R(t)   t 4 dt du 0 ¨ r r R0  + R (t)¨R(¨t)u u  = 0 dτ  dτ 0 0 3  R¡4¡(t) ! dut R4 dτ ut + R0(t)ur ur 0 = 0 dτ 0 0 R3(t) dR utdut = −ur ur R4 0 0 0 R3(t) (31) Z ut Z R dR˜ u˜tdu˜t = −ur ur R4 t 0 0 0 ˜ 3 u0 R0 R (t) ! 1 1 1¡(utut − ut ut ) = 1¡ur ur R4 − ¡2 0 0 ¡2 0 0 0 2( ) 2 R t R0 ! R2 utut − ut ut = ur ur R2 0 − 1 0 0 0 0 0 R2(t) v u ! u R2 ut = tur ur R2 0 − 1 + ut ut . 0 0 0 R2(t) 0 0

We may ﬁnally use (30) and (31) to check the constraint

µ −1 = uµu µ ν −1 = gµνu u −1 = −utut + R2(t)urur ! ! ! R2 R 4 −1 = − ur ur R2 0 − 1 + ut ut + R2(t) ur ur 0 (32) 0 0 0 R2(t) 0 0 0 0 R(t) ¨4¨ 4¨ ¨R R¨ −1 = −ur u¨r 0 + ur ur R2 − ut ut + ur ur¨ 0 ¨0 0 2 0 0 0 0 0 0¨0 2 ¨ R (t) ¨ R (t) t t 2 r r −1 = −u0u0 + R0u0u0,

and we see that we may choose any ut, ur, R(t) such that the con- t r straint is satisﬁed for the initial conditions u0, u0, R0, i.e. it is enough for the initial conditions to satisfy the constraint to have it satisﬁed for any time t thereafter.

We may also perform analogous calculations in Cartesian coordinates, starting with the geodesic equations 34 spin motion in general relativity

0 i du du i i + Γ0 uiui = 0 + 2Γ u0u = 0 dτ ii dτ 0i (33) du0 dui R0(t) + R0(t)R(t)δ uiuj = 0 + 2 u0ui = 0, dτ ij dτ R(t)

then solving for the spatial components ui, all of which take the same form

dui R0(t) dx0 + 2 ui = 0 dτ R(t) dτ dui R0(t) + 2 ui = 0 dτ R(t) dτ dui R0(t) + 2 ui = 0 ui dτ R(t) dui dR + 2 = 0 (34) ui R(t) Z ui du˜i Z R dR˜ = −2 i ˜i ˜ u0 u R0 R(t) ui R(t) log = −2 log i R u0 0 R 2 ui = ui 0 , 0 R(t)

and putting this into the other geodesic equation to solve for u0

0 4 du j R + R0(t)R(t)δ ui u 0 = 0 dτ ij 0 0 R(t)   0 0 4 dx du 0 ¨¨ i j R0  + R (t)¨R(t)δiju u  = 0 dτ  dτ 0 0 3  (35) R¡4¡(t) 0 4 ! du j R dτ u0 + R0(t)δ ui u 0 = 0 dτ ij 0 0 R3(t) point particles in the flrw metric 35

j dR u0du0 = −δ ui u R4 ij 0 0 0 R3(t) Z u0 Z R ˜ ˜0 ˜0 i j 4 dR u du = −δiju u R 0 0 0 0 ˜ 3 u0 R0 R (t) ! j 1 1 1¡(u0u0 − u0u0) = 1¡δ ui u R4 − ¡2 0 0 ¡2 ij 0 0 0 2( ) 2 R t R0 2 ! j R u0u0 − u0u0 = δ ui u R2 0 − 1 0 0 ij 0 0 0 R2(t) v u 2 ! u j R u0 = tδ ui u R2 0 − 1 + u0u0. ij 0 0 0 R2(t) 0 0

We ﬁnally check the constraint

µ −1 = uµu µ ν −1 = gµνu u 0 0 2 i j −1 = −u u + R (t)δiju u 2 ! ! 4 j R j R −1 = − δ ui u R2 0 − 1 + u0u0 + R2(t)δ ui u 0 (36) ij 0 0 0 R2(t) 0 0 ij 0 0 R(t) 4 4¨ j R j j R¨ − = − i 0 + i 2 − 0 0 + i ¨ 0 1 δiju0u0 2 δiju0u0R0 u0u0 δiju0¨u0 2 R (t) ¨¨ R (t) 0 0 2 i j −1 = −u0u0 + R0δiju0u0,

where the conclusion is, similarly, that we may choose any u0, ui, 0 R(t) such that the constraint is satisﬁed for the initial conditions u0, i u0, R0.

Next we turn our attention to the spin motion of a test particle, in Cartesian coordinates with k = 0, we have the solutions, for the temporal part

dS0 + Γ0 Sµuν = 0 dτ µν dS0 + R0Rδ Siuj = 0 dτ ij 1 µ = ⇒ i j = 0 0 Sµu 0 δijS u 2 S u R (37) dS0 R0 + S0u0 = 0 dτ R dS0 dR = − S0 R R S0 = 0 S0, R 0 36 spin motion in general relativity

0 for some initial conditions R0 and S0, and for the spatial part

dSi + Γi Sµuν = 0 dτ µν dSi R0 + S0ui + Siu0 = 0 dτ R (38) dSi R0 R + 0 S0ui + Siu0 = 0 dτ R R 0 dSi R0 dR 1 + R S0ui + Si = 0, dτ R2 0 0 dτ R

where the second, inhomogeneous term might mean this is not a readily solvable differential equation. i dxi i dxi Switching between u = dτ and v = dt is possible using the relations

i i v u = q j 1 − vjv i i u v = q 39 j ( ) 1 + uju ui = q , j k 1 + hjku u

2 with the spatial metric being hij = −gij = −R (t)δij. such that (38) may be rewritten in the form

dSi dR 1 1 + R S0vi + Si = 0 dτ dτ R(t) R(t) 0 0   i i dS dR 1 1 0 u i +  R0S0 q + S  = 0 (40) dτ dτ R(t) R(t) 2 j k 1 − R (t)δjku u

i i ! dS dR 1 1 0 u i + R0S0 p + S = 0. dτ dτ R(t) R(t) 1 − R2(t)(u1u1 + u2u2 + u3u3)

Now inserting the solution we previously found for ui for a test particle, and applying the constraint on uµ to the initial conditions we have point particles in the flrw metric 37

  i i dS dR 1  1 0 u i +  R0S0 r + S  = 0 dτ dτ R(t)  R(t) R4  − 0 ( 1 1 + 2 2 + 3 3) 1 R2(t) u0u0 u0u0 u0u0   (41) i i dS dR 1  1 0 u i +  R0S0 r + S  = 0, dτ dτ R(t)  R(t) R4  − 0 (u0u0 − ) 1 R4(t) 0 0 1

and inserting the solution we found for ui once more

  i 0 2 dS dR 1  1 R0S0 i R0 i +  r u0 + S  = 0 dτ dτ R(t)  R(t) R4 R(t)  − 0 (u0u0 − ) 1 R4(t) 0 0 1   dSi dR 1 1 R3S0ui +  0 0 0 + i = (42)  3 r S  0 dτ dτ R(t)  R (t) R4  − 0 (u0u0 − ) 1 R4(t) 0 0 1   3 0 i i dR 1 R0S0u0 i dS +  q + S  = 0. R(t) R(t) 4 4 0 0 R (t) − R0(u0u0 − 1)

i 3 0 i 4 0 0 Finally deﬁning the parameters ε ≡ R0S0u0 and κ ≡ R0(u0u0 − 1) allows us to write the differential equations in the form

i ! i dR 1 ε i dS + p + S = 0, (43) R(t) R(t) R4(t) − κ

whence a solution may be found, perhaps numerically.

Extended Bodies in the FLRW Metric

Presently we move on to the study of cosmological kinematics, this time for extended bodies. In order to use the mpd equation, the components of the Riemann tensor for the given metric must be calculated. Starting with the simpler k = 0 case, with Cartesian coordinates, the non-zero components of the Riemann tensor are

0 0 00 0 0 00 0 0 00 R 101 = −R 110 = R(t)R (t), R 202 = −R 220 = R(t)R (t), R 303 = −R 330 = R(t)R (t), R00(t) R1 = −R1 = , R1 = −R1 = R02(t), R1 = −R1 = R02(t), 001 010 R(t) 212 221 313 331 R00(t) (44) R2 = −R2 = , R2 = −R2 = R02(t), R2 = −R2 = R02(t), 002 020 R(t) 121 112 323 332 R00(t) R3 = −R3 = , R3 = −R3 = R02(t), R3 = −R3 = R02(t). 003 030 R(t) 131 113 232 223

At this point we may write down the mpd equations, for the comoving frame, uµ = (1, 0, 0, 0), the ﬁrst equation involving S00 (indeed we could have skipped this step and noted that, having a skew-symmetric tensor, immediately S00 = 0) is

D D 1 mu0 + u S00 + u0SρσR0 = 0 dτ 0 dτ 2 0ρσ (45) D D m − S00 = 0, dτ dτ

the ﬁrst equation involving S0i = −Si0 is 40 spin motion in general relativity

D D 1 mui + u Si0 + u0SρσRi = 0 dτ 0 dτ 2 0ρσ D D 1 − Si0 + Si0Ri = 0 dτ dτ 2 0i0 (46) D D 1 R00(t) − Si0 − Si0 = 0 dτ dτ 2 R(t) D D 1 R00(t) S0i + S0i = 0, dτ dτ 2 R(t)

while the ﬁrst equations involving Sii and Sij are identically zero (never mind the fact that Sii = 0 all the same seeing as how this is a skew-symmetric tensor.) The second equation involving S00 (once again we note that this step is ultimately unnecessary) is

D D D S00 + u0u S0σ − u0u S0σ = 0 dτ σ dτ σ dτ D D D S00 + u0u S00 − u0u S00 = 0 (47) dτ 0 dτ 0 dτ D S00 = 0, dτ

the second equation involving S0i = −Si0 is

D D D S0i + u0u Siσ − uiu S0σ = 0 dτ σ dτ σ dτ D D S0i + u0u Si0 = 0 dτ 0 dτ D D S0i − Si0 = 0 (48) dτ dτ D D S0i + S0i = 0 dτ dτ D S0i = 0, dτ

the second equation involving Sii (once again we note that this step is ultimately unnecessary; henceforth these will all be supressed) is

D ii i D iσ i D iσ S + u uσ S − u uσ S = 0 dτ dτ dτ (49) D Sii = 0, dτ

and the second equation involving Sij = −Sji is extended bodies in the flrw metric 41

D ij i D jσ j D iσ S + u uσ S − u uσ S = 0 dτ dτ dτ (50) D Sij = 0. dτ Now, (47) means (45) is trivially satisﬁed and (48) lets us write (46) as

1 R00(t) S0i = 0 2 R(t) (51) S0i = 0,

and putting everything together we get that Sµν satisﬁes two conditions

D Sµν = 0 dτ (52) S0i = 0,

explicitly

D Sµν = 0 dτ σ∇ µν = u σS 0 (53) 0 µν u ∇0S = 0 µν µ σν ν µσ ∂0S + Γ0σS + Γ0σS = 0. We now solve for S00

¨¨* 0 ¨¨* 0 00 0¨ σ0 0¨ 0σ ∂0S +¨Γ0σS +¨Γ0σS = 0 00 (54) ∂0S = 0 S00 = const.,

for S0i = −Si0

¨* 0 0i 0¨¨σi i 0σ ∂0S +¨Γ0σS + Γ0σS = 0 0i i 0i ∂0S + Γ0iS = 0 R0 ∂ S0i + S0i = 0 0 R (55) dS0i dR = − S0i R R S0i = 0 S0i, R 0 42 spin motion in general relativity

0i ii for some initial conditions R0 and S0 , for S

: 0 ii + i σi +i iσ = ∂0S Γ0σS Γ0σS 0 ii (56) ∂0S = 0 Sii = const.,

and for Sij = −Sji

ij i σj j iσ ∂0S + Γ0σS + Γ0σS = 0 ij i ij j ij ∂0S + Γ0iS + Γ0jS = 0 0 ij R ij ∂0S + 2 S = 0 R (57) dSij dR = −2 Sij R 2 R ij Sij = 0 S , R 0

ij for some initial conditions R0 and S0 . Finally, using (8) we ﬁnd

1 1 S0 = √ e00ρσg u0g g Sβγ 2 −g 00 ρβ σγ = 0 1 1 S1 = √ e10ρσg u0g g Sβγ 2 −g 00 ρβ σγ 1 1 = R4(t) S32 − S23 2 R3(t) = R(t)S32 R2 (58) = 0 S32 R(t) 0 1 1 S2 = √ e20ρσg u0g g Sβγ 2 −g 00 ρβ σγ 1 1 = R4(t) S13 − S31 2 R3(t) = R(t)S13 R2 = 0 S13 R(t) 0 extended bodies in the flrw metric 43

1 1 S3 = √ e30ρσg u0g g Sβγ 2 −g 00 ρβ σγ 1 1 = R4(t) S21 − S12 2 R3(t) = R(t)S21 R2 = 0 S21 R(t) 0

in particular this must hold for t = 0, allowing us to switch be- ij i tween the initial conditions S0 and the initial conditions S0, thus we ﬁnally write

S0 = 0 R R R (59) S1 = 0 S1 S2 = 0 S2 S3 = 0 S3 R(t) 0 R(t) 0 R(t) 0

which turn out to be exactly the same solutions we found using the geodesic formalism.

For a test particle, on the other hand, the equations don’t take quite such an easy form, we have for S0i

D D 1 mu0 + u S0i + uπSρσR0 = 0 dτ i dτ 2 πρσ D D 1 mu0 + u S0i + ujS0kδ R0 = 0 (60) dτ i dτ 2 jk j0k D D 1 mu0 + u S0i + R00(t)R(t)δ ujS0k = 0, dτ i dτ 2 jk

together with

D D D S0i + u0u Siσ − uiu S0σ = 0 dτ σ dτ σ dτ 0 i D 0i 0 D ij 1 − u u0 − u ui S + u uj S = 0 dτ dτ (61) D D 1 − uµu S0i + u0u Sij = 0 µ dτ j dτ D D 2 S0i + u0u Sij = 0, dτ j dτ

µ µ where in the last equality we recall the constraint on u , u uµ = −1, and for Sij 44 spin motion in general relativity

D D 1 mui + u Sij + uπSρσRi = 0 dτ j dτ 2 πρσ D D 1 mui + u Sij + uπSρiRi = 0 (62) dτ j dτ 2 πρi D D 1 R00(t) mui + u Sij + δ u0 − R02(t)δ uk Sρi = 0, dτ j dτ 2 R(t) 0ρ kρ

together with

D D D Sij + uiu Sjσ − uju Siσ = 0 dτ σ dτ σ dτ (63) D D D Sij − u ui Sσj + uj Siσ = 0. dτ σ dτ dτ

Putting everything together we get the following system of equations

D D 1 mu0 + u S0i + R00(t)R(t)δ ujS0k = 0, dτ i dτ 2 jk

D D 2 S0i + u0u Sij = 0, dτ j dτ (64) D D 1 R00(t) mui + u Sij + δ u0 − R02(t)δ uk Sρi = 0, dτ j dτ 2 R(t) 0ρ kρ

D D D Sij − u ui Sσj + uj Siσ = 0, dτ σ dτ dτ

and using (A.2), (A.4), and (A.5) these become extended bodies in the flrw metric 45

j d 1 u S¨0i + u˙ + u u0Γi S˙0i + u ujujΓ0 Γ + u u0Γi S0i − u uiujΓ0 Γi − δ ukR00(t)R(t) S0j i i i 0i i jj j0 dτ i 0i i jj i0 2 jk j d − 2u ujΓ0 S˙ij − u uju0Γ0 Γ + u uju0Γ0 Γi + u ujΓ0 Sij + mu˙0 + mujujΓ0 = 0, i jj i jj 0j i jj 0i dτ i jj jj

h i ˙0i 0 i j 0 j 0i i 0 i 0j 0 ˙ij j 0 0 0 i j ij 2S + 2u Γ0i − uju u Γj0 S + uju u Γi0S + uju S − 2u Γjj − uju u Γ0i + Γ0j S = 0,

j j d j u S¨ij + u˙ + u u02Γi + Γ S˙ij + u u0u0Γi Γi + Γ + u u0Γi + Γ Sij j j j 0i 0j j 0i 0i 0j dτ j 0i 0j i k i 0 jk k 02 iρ − uju u Γ0iΓkkS + δkρu R (t)S j j ˙0i i i ˙0j − uju Γj0S + 2uju Γi0S (65) j d j j d − u uju0Γi Γ + u ujΓ S0i + u uiu0Γi Γi + Γ + u uiΓi S0j j 0i j0 dτ j j0 j i0 0i 0j dτ j i0 i 0 i i i + mu˙ + mu u Γ0i + Γi0 = 0,

˙ij 0 i j ij i ˙ jk j ˙ik S + u Γ0i + Γ0j S + uku S − uku S i k 0 0 i j k jk j k 0 0 j i k ik + u0u u Γkk + uku u Γ0j + Γ0k S − u0u u Γkk + uku u Γ0i + Γ0k S j 0i i 0j + u0u S˙ − u0u S˙ 0 j i k j k j j 0i 0 i j k i k i i 0j + u0u u Γ0i + uku u Γ0k − u Γj0 S − u0u u Γ0j + uku u Γ0k − u Γi0 S i j j i + uku u Γj0 − Γi0 = 0.

Finally, considering the Christoffel symbols we previously found we may write

R0(t) d R0(t) 1 u S¨0i + u˙ + u u0 S˙0i + δ u ujukR02(t) + u u0 S0i − δ u uiR02(t) − R00(t)R(t) ukS0j i i i R(t) jk i dτ i R(t) jk i 2 (66) d − 2δ u ukR0(t)R(t)S˙ij − δ 2u uku0R02(t) + u ukR0(t)R(t) Sij + mu˙0 + δ mujukR0(t)R(t) = 0, jk i jk i dτ i jk

R0(t) R0(t) R0(t) 2S˙0i + 2 − u uj u0 S0i + u uiu0 S0j + u u0S˙ij − 2 δ ukR0(t)R(t) − u u0u0 Sij = 0, (67) j R(t) j R(t) j jk j R(t) 46 spin motion in general relativity

" # R0(t) R0(t) 2 d R0(t) u S¨ij + u˙ + 3u u0 S˙ij + 2 u u0u0 + u u0 Sij j j j R(t) j R(t) dτ j R(t) i l 02 jk l 02 ik − δkluju u R (t)S + δklu R (t)S R0(t) R0(t) − u uj S˙0i + 2u ui S˙0j j R(t) j R(t) (68) " # " # R0(t) 2 d R0(t) R0(t) 2 d R0(t) − u uju0 + u uj S0i + 2u uiu0 + u ui S0j j R(t) dτ j R(t) j R(t) dτ j R(t) R0(t) + mu˙i + 2mu0ui = 0, R(t)

R0(t) S˙ij + 2u0 Sij + u uiS˙ jk − u ujS˙ik R(t) k k 0 0 i l 0 0 i R (t) jk j l 0 0 j R (t) ik + δklu0u u R (t)R(t) + 2uku u S − δklu0u u R (t)R(t) + 2uku u S R(t) R(t) (69) j 0i i 0j + u0u S˙ − u0u S˙ R0(t) R0(t) + u u0 + u uk − 1 uj S0i − u u0 + u uk − 1 ui S0j = 0. 0 k R(t) 0 k R(t)

We now argue that because the FLRW metric arises from considering a maximally symmetric space, every spatial component of any given tensor should be equivalent and thus we need only ﬁnd 4 solutions here, u0, ui, S0i, Sij. With this simpliﬁcation we immediately see from (69)

R0(t) S˙ij + 2u0 Sij +u uiS˙ jk −u ujS˙ik R(t) k k (((( ((( (((0 (((0 ( i l 0 ((( 0 i R (t) jk j l 0 ((( 0 j R (t) ik + δ u0u u R ((t)R(((t) + 2u u u S − δ u0u u R (t)(R((t) + 2u u u S kl ((( k ( ) kl ((( k ( ) ((( R t (((( R t j0i i0j +u0u S˙ −u0u S˙ 0 ((( 0 ((( ((R((t) ((R((t) (70) + u u0 + u(u(k (−(1 uj S0i − u u0 + u(u(k (−(1 ui S0j = 0 ((0 (( k R(t) ((0 (( k R(t) R0(t) S˙ij + 2u0 Sij = 0 R(t) R 2 Sij = 0 Sij R(t) 0

ij for some initial conditions R0, S0 . It turns out that this is exactly the same solution we found in the comoving frame. Again using the simpliﬁcation, plus the Frenkel condition (17), we may then write the remaining 3 odes as extended bodies in the flrw metric 47

0i 0i k 0 ij 0 i j 0 uiS¨ + u˙iS˙ − δjkuiu R (t)R(t)S˙ + mu˙ + δijmu u R (t)R(t) = 0, (71)

R0(t) R0(t) 2S˙0i + 2 − u uj u0 S0i + u u0S˙ij − 2 δ ukR0(t)R(t) − u u0u0 Sij = 0, (72) j R(t) j jk j R(t)

" # R0(t) 1 R0(t) 2 d R0(t) u S¨ij + u˙ + 3u u0 S˙ij + 2 δ ukR02(t) + u u0u0 + u u0 Sij j j j R(t) 2 jk j R(t) dτ j R(t) " # (73) R0(t) 2 d R0(t) R0(t) − u uju0 + u uj S0i + mu˙i + 2mu0ui = 0, j R(t) dτ j R(t) R(t)

whose form is still relatively complicated for the work at hand. Their analytical or numerical treatment is left for some future work.

Part III

The Schwarzschild Metric

The Metric

Beyond cosmological considerations, the ﬁrst kind of spacetime one might wish to study is one with a spherically symmetric gravitational ﬁeld, one caused by central masses as is generally the case with stars, planets, and black holes. This kind of spacetime, as it turns out, has both internal and external solutions. While the former are extremely important in the study of black holes—these solutions concern the space inside the event horizon—, it is the latter which interest us currently, for example for a free falling particle, or one [ ] orbiting the central mass18 p.193 . We now derive such a spacetime 18 S. M. Carroll, Spacetime and Geometry, metric. 1st ed. (Pearson Education Limited, Harlow, 2014)

We may start by considering the general form of the metric for a [ ] static isotropic spacetime19 p. 179 19 S. Weinberg, Gravitation and Cosmol- ogy: Principles and Applications of the General Theory of Relativity, 1st ed. (John Wiley & Sons, Inc., New York, 1972) ds2 = −B(r)dt2 + A(r)dr2 + r2dθ2 + r2 sin2 θdφ2,(74)

from which we may ﬁnd the only non-vanishing Christoffel symbols

B0(r) Γt = Γt = , tr rt 2B(r) B0(r) A0(r) Γr = , Γr = , tt 2A(r) rr 2A(r) r r sin2 θ Γr = − , Γr − , (75) θθ A(r) φφ A(r) 1 Γθ = Γθ = , Γθ = − sin θ cos θ, rθ θr r φφ φ φ 1 φ φ Γ = Γ = , Γ = Γ = cot θ, rφ φr r θφ φθ

and thus the only non-vanishing components of the Ricci tensor 52 spin motion in general relativity

B00(r) 1 B0(r) A0(r) B0(r) 1 B0(r) R = − + + − , tt 2A(r) 4 A(r) A(r) B(r) r A(r) B00(r) 1 B0(r) A0(r) B0(r) 1 A0(r) R = − + − , rr 2B(r) 4 B(r) A(r) B(r) r A(r) (76) r A0(r) B0(r) 1 R = −1 + − + + , θθ 2A(r) A(r) B(r) A(r) r sin2 θ A0(r) B0(r) sin2 θ R = sin2 θR = − sin2 θ + − + + . φφ θθ 2A(r) A(r) B(r) A(r)

Rtt Rrr We may now consider B + A , along with the equations for empty space, Rµν = 0, to get

1 A0(r) B0(r) − + = 0, (77) rA(r) A(r) B(r)

0 0 A (r) = − B (r) hence A(r) B(r) and so we must have

1 A(r) ∝ ,(78) B(r)

indeed the fact that for r → ∞ we get gµν → ηµν guarantees that this must be an equality

1 A(r) = .(79) B(r)

We ﬁnally rewrite Rθθ using this last relation and set it to zero so as to satisfy its corresponding equation for empty space

0 Rθθ = −1 + rB (r) + B(r) = 0 d (80) rB0(r) + B(r) = rB(r) = 1, dr so that the function B(r) takes the form

C B(r) = 1 + (81) r for some constant of integration C —considering the Newtonian limit we should ﬁnd this constant to be −2M using units such that G = c = 1— and thus the metric will ﬁnally take the form

− 2M 2M 1 ds2 = − 1 − dt2 + 1 − dr2 + r2dθ2 + r2 sin2 θdφ2,(82) r r the metric 53

and (75) will look like

M Γt = Γt = − , tr rt 2Mr − r2 M(2M − r) M Γr = − , Γr = , tt r3 rr 2Mr − r2 r r 2 Γθθ = 2M − r, Γφφ = (2M − r) sin θ, (83) 1 Γθ = Γθ = , Γθ = − sin θ cos θ, rθ θr r φφ φ φ 1 φ φ Γ = Γ = , Γ = Γ = cot θ. rφ φr r θφ φθ

Point Particles in the Schwarzschild Metric

The geodesic equations, as well as the constraints, that must be satisﬁed by the velocity uµ and spin Sµ are

duλ + Γλ uµuν = 0 dτ µν µ ν ν gµνu u = uνu = −1, (84) dSλ + Γλ Sµuν = 0 dτ µν µ ν ν gµνS u = Sνu = 0.

We may now ﬁnd the 4-velocity for a test particle falling radially 20 20 inward from a distance r0 2M , that is, we solve the geodesic M. Ross, The Schwarzschild Solution equations for the 4-velocity subject to the initial conditions and Timelike Geodesics, Logan, 2016

α π x = (0, r0, , 0) 0 2 (85) α t u0 = (u0, 0, 0, 0). µ ν t Considering the constraint gµνu u = −1 we ﬁnd u0 to be

t 1 u0 = q .(86) 1 − 2M r0 We may write the geodesic equations

dut + Γt uβuγ = 0 dτ βγ (87) dut 2M − utur = 0, dτ 2Mr − r2

dur + Γr uβuγ = 0 dτ βγ (88) dur GM(2M − r) M − (ut)2 + (ur)2 + (2M − r)(uθ)2 + (2M − r) sin2 θ(uφ)2 = 0, dτ r3 2Mr − r2 56 spin motion in general relativity

θ du θ β γ + Γβγu u = 0 dτ (89) duθ 1 + uruθ − sin θ cos θ(uφ)2 = 0, dτ r and φ du φ β γ + Γβγu u = 0 dτ (90) duφ 1 + uruφ + cot θuθuφ = 0. dτ r Immediately we can integrate (87) to get

1 − 2M ! t = t r0 u u0 2M ,(91) 1 − r as well as (89) and (90) to get

uθ = 0 (92) uφ = 0. Finally, integrating (88) is not as straightforward, so we may in- µ ν stead turn back to the constraint gµνu u = −1, along with the solutions we just found, to ﬁnd

s 2M 2M ur = − .(93) r0 r

2M ! 1− q For a free falling test particle, uµ = (ut r0 , 2M − 2M , 0, 0), 0 − 2M r0 r 1 r we may ﬁnd the following solutions for the spin, for the temporal part

dSt + Γt Sµuν = 0 dτ µν dSt + Γt Stur + Γt Srut = 0 (94) dτ tr rt dSt M − Stur + Srut = 0, dτ 2Mr − r2 and for the spatial part

dSr + Γr Sµuν = 0 dτ µν dSr : 0 + ΓrSµuµ = 0 dτ µµ (95) Sr = const. µ ν r gµνS u = 0 ⇒ S = 0, point particles in the schwarzschild metric 57

dSθ + Γθ Sµuν = 0 dτ µν θ 0 dS θr :θ θ θ r θ φ :φ 0 +ΓrθS u + ΓθrS u +ΓφφS u = 0 dτ (96) dSθ 1 dr + Sθ = 0 dτ r dτ θ r θ S = S0, r0 and

φ dS φ + Γ Sµuν = 0 dτ µν φ ¨* 0 ¨¨*0 ¨¨* 0 dS φ ¨r φ φ ¨θ φ φ φ r φ ¨φ θ + Γr¨S u + Γ ¨S u + Γ rS u + Γ ¨S u = 0 dτ ¨φ ¨θφ φ ¨φθ (97) dSφ 1 dr + Sφ = 0 dτ r dτ φ r φ S = S0 . r0 We may ﬁnally return to (94), and using (95) we get

dSt M ¨¨* 0 − (Stur +¨Srut) = 0 dτ 2Mr − r2 t dS M t dr − S = 0 (98) dτ 2Mr − r2 dτ s t r(r0 − 2M) t S = S0. r0(r − 2M)

Extended Bodies in the Schwarzschild Metric

In order to use the mpd equation, the components of the Riemann tensor for the given metric must be calculated. For the Schwarzschild metric the non-zero components of the Riemann tensor are

2M M M sin2 θ Rt = −Rt = Rt = −Rt = Rt = −Rt = rrt rtr (2M − r)r2 θθt θtθ r φφt φtφ r 2M(2M − r) M M sin2 θ Rr = −Rr = Rr = −Rr = Rr = −Rr = trt ttr r4 θθr θrθ r φφr φrφ r (99) M(2M − r) M 2M sin2 θ Rθ = −Rθ = Rθ = −Rθ = Rθ = −Rθ = ttθ tθt r4 rθr rrθ (2M − r)r2 φθφ φφθ r M(2M − r) M 2M Rφ = −Rφ = Rφ = −Rφ = Rφ = −Rφ = ttφ tφt r4 rφr rrφ (2M − r)r2 θφθ θθφ r

At this point we may write down the mpd equations, for the free- µ t falling test particle, u0 = (u0, 0, 0, 0). We ﬁrst notice that the non- vanishing components of the Riemann tensor in this metric happen to be the same as in the FLRW metric—they, of course, take different values, but nevertheless we can quickly get from (64)

D t D tr 1 2M r tr mu + ur S − u S = 0, dτ dτ 2 (2M − r)r2 (100) D D 2 Sti + utu Sir = 0. dτ r dτ

Now the second of these reads for i = r 60 spin motion in general relativity

D Str = 0 dτ d Str + Γt uσSλr + Γr uσStλ = 0 dτ σλ σλ d Str + Γt urStr + Γr urStr = 0 dτ rt rr d Str + Γt + Γr urStr = 0 (101) dτ rt rr ((( d tr M ((M( r tr S + − ((((+ u S = 0 dτ ((2(Mr − r2 2Mr − r2 d Str = 0 dτ Str = const.,

and indeed putting this back into the ﬁrst equation yields Str = 0. Using this fact we further get from (64)

D D ir 1 M(2M − r) t M r ρi ur S + δtρu − δrρu S = 0, dτ dτ 2 r4 (2M − r)r2 (102) D D D Sij − u ui Sσj + uj Siσ = 0, dτ σ dτ dτ where once again we immediately see from the second one, taking i = θ, j = φ,

D Sθφ = 0 dτ d φ Sθφ + Γθ uσSλφ + Γ uσSθλ = 0 dτ σλ σλ d φ Str + Γθ urSθφ + Γ urSθφ = 0 dτ rθ rφ d φ Sθφ + Γθ + Γ urSθφ = 0 dτ rθ rφ d 1 1 r Sθφ + + u Sθφ = 0 (103) dτ r r d ur Sθφ = −2 Sθφ dτ r dSθφ dr = −2 Sθφ r r 2 θφ Sθφ = 0 S , r 0 r 2 φθ Sφθ = 0 S , r 0 which, once again going from the spin bivector to the spin vector using (8), yields the two solutions extended bodies in the schwarzschild metric 61

θ r θ S = S0, r0 (104) r Sφ = Sφ, r0 these being exactly the same as we previously found using the geodesic formalism. The remaining differential equations, allowing us to ﬁnd St upon applying (8), take on a more complicated form and, just as we did with the flrw metric, we choose to stop at this point, having obtained two equivalent solutions, and leave the analytical or numerical treatment of the others for future work.

Part IV

Final Remarks

On the Results from the Different Formalisms

The preceding calculations showed us that, at least as far as the cases we considered and managed to fully solve, using either formalism will result in the same motion. We are left wondering why and the answer to that goes back to the sscs, mentioned at the end of the chapter Spin in Gravity. As is also explained there, the mpd equations happen to be adapted to extended bodies, as opposed to the geodesic formalism which is specifically for point particles, the choice of ssc is then a matter of choosing the reference observer with respect to which the body’s center of mass is defined21. It turns out 21 O. Semerák, “Spinning particles that when using the Frenkel condition (also found in the literature as in vacuum spacetimes of different curvature types: Natural reference Frenkel-Mathisson-Pirani condition, or simply Pirani condition), and tetrads and massless particles”, Physical changing from the spin bivector to the spin vector with the help of Review D - Particles, Fields, Gravitation and Cosmology 92, 064036 (2015) (8), one finds that the mpd equations reduce to simply22 22 Z. Keresztes and B. Mikóczi, “Evo- lutions of spinning bodies moving in rotating black hole spacetimes”, 2019 DSα = uαa Sβ,(105) dτ β which is nothing more than the spin vector being Fermi-Walker transported. Then, considering how for negligible accelerations Fermi-Walker transport reduces to parallel transport, it is easy to see how the mpd equations, given the chosen ssc and the chosen metrics, would yield the same results as the geodesic formalism. Moreover, it is also worth noting that in the weak field approximation all the different commonly-used choices of ssc happen to be equivalent and no current experiments or observations would be able to choose amongst them23. In practice this means that the choice of 23 F. Cianfrani and G. Montani, “Spin- ssc, at least for the time being, is primarily made for computational ning particles in general relativity”, in Nuovo cimento della societa italiana di reasons. Indeed, future work to expand upon this one could consider fisica b (2007) the mpd equations for those cases deemed here too difficult to solve analytically and attempt a solution under different sscs.

De Sitter and Lense-Thirring Precessions, Revisited

As previously mentioned, de Sitter precession takes place as a result of gravitational fields due to the presence of mass while Lense-Thirring precession accounts for frame-dragging, the effect of the source of the gravitational field itself spinning. Naturally, having chosen the flrw and Schwarzschild metrics, we are only concerning ourselves with gravitational fields that result from static masses and thus the results of these, or any analogous calculations, would only correspond to de Sitter precession. On the other hand, if we were now to choose a metric for a spinning central mass and perform the calculations with respect to it, the results would contain a sum of both de Sitter and Lense-Thirring precessions. Appropriate choices for such a metric would include the one studied by Lense and Thirring themselves back in 1918, itself not an exact solution, or the Kerr metric, which is an exact solution, although it wouldn’t have been known by Lense and Thirring. It is important to note, however, that such calculations involving more complicated metrics, including those for a spinning central mass, would greatly complicate the resulting equations, which even here were not always readily solvable. One possible workaround for this, and one which has in fact been used before24, is to go into 24 O. Semerák, “Spinning particles a tetrad formalism such as the Newman-Penrose formalism, these in vacuum spacetimes of different curvature types: Natural reference being better adapted to such computations. Even after all of this, tetrads and massless particles”, Physical analytical solutions might still not be possible and one may need to Review D - Particles, Fields, Gravitation and Cosmology 92, 064036 (2015), O. use numerical methods, also an approach which has been previously Semerák and M. Šrámek, “Spinning used with great success25. particles in vacuum spacetimes of different curvature types”, Physical Review D - Particles, Fields, Gravitation Lastly we consider experiments and observations confirming de and Cosmology 92, 064032 (2015) Sitter and Lense-Thirring precessions, these happen to be of great 25 Z. Keresztes and B. Mikóczi, “Evo- importance to the theory of general relativity insofar as they serve lutions of spinning bodies moving in rotating black hole spacetimes”, as experimental confirmation for it. By comparing the data to the 2019, R. M. Plyatsko et al., “Mathisson- predictions that may be made using calculations such as the ones Papapetrou-Dixon equations in the Schwarzschild and Kerr backgrounds”, presented here, especially in the weak field approximation, we have Classical and Quantum Gravity 28, a very direct way of confirming the validity of the theory and some 195025 (2011), N. Velandia and J. M. Tejeiro, “Numerical Solutions of Mathisson-Papapetrou-Dixon equations for spinning test particles in a Kerr metric”, Momento, 60–85 (2018) 68 spin motion in general relativity

of its assertions, specifically some that have no counterpart in Newto- nian gravity. The first tests of de Sitter precession26 were made for the Earth- 26 B. Bertotti et al., “New test of gen- Moon gyroscope orbiting the sun, mostly using Lunar Laser Ranging eral relativity: Measurement of de sitter geodetic precession rate for lunar (llr), that is, using retroreflectors planted on the moon during the perigee”, Physical Review Letters 58, Apollo missions. First in 1987 by Bertotti, Ciufolini, and Bender using 1062 (1987), I. I. Shapiro et al., “Mea- surement of the de sitter precession llr, as well as Very Long Baseline Interferometry (vlbi), to obtain of the moon: A relativistic three-body data with an accuracy of the order of 10%; next in 1988 by Shapiro, effect”, Physical Review Letters 61, 2643 Reasenberg, Chandler, and Babcock, and in 1989 by Dickey, Newhall, (1988), J. O. Dickey et al., “Investigating relativity using lunar laser ranging: and Williams, both teams using llr and obtaining data with an Geodetic precession and the Nordtvedt accuracy of about 2%; finally in 1991 by Müller, Schneider, Soffel, and effect”, Advances in Space Research 9, 75–78 (1989), J. Mueller et al., “Testing Ruder, and in 1994 by Dickey, Bender, Faller, Newhall, Ricklefs, Ries, Einstein’s theory of gravity by analyz- Shelus, Veillet, Whipple, Wiant, Williams, and Yoder, both teams once ing Lunar Laser Ranging data”, The again using llr and obtaining data with an accuracy of about 1%. Astrophysical Journal 382,L101 (1991), J. O. Dickey et al., “Lunar laser rang- 27 As for Lense-Thirring precession, the first proposed experiments ing: A continuing legacy of the apollo came in 1977 from Cugusi and Proverbio and involved using the program”, Science 265, 482–492 (1994) 27 Laser Geodynamics Satellite (lageos) launched by nasa in 1976. L. Cugusi and E. Proverbio, “Rela- tivistic Effects on the Motion of Earth’s 28 A different method was proposed in 1986 by Ciufolini using two Artificial Satellites”, Astronomy and satellites and the first tests were performed in 1996 by Ciufolini, Luc- Astrophysics 69, 321–325 (1978) 28 chesi, Vespe, and Mandiello using lageos, together with lageos 2 I. Ciufolini, “Measurement of the Lense-Thirring drag on high-altitude, launched in 1992. Tests using these two satellites, albeit a somewhat laser-ranged artificial satellites”, Phys- different method, were once again performed29 in 2004 by Ciufolini ical Review Letters 56, 278 (1986), 2006 I. Ciufolini et al., “Measurement of and Pavlis, and in by Ciufolini, Pavlis, and Peron. Finally, simi- dragging of inertial frames and grav- lar experiments are being performed by the Italian Space Agency un- itomagnetic field using laser-ranged der the direction of Ciufolini30, the Laser Relativity Satellite (lares) satellites”, Il Nuovo Cimento A 109, 575–590 (1996) launched in 2012 with the aim of measuring the effect with an accu- 29 I. Ciufolini and E. C. Pavlis, “A racy of 1%, and the lares 2 is expected to launch in December of confirmation of the general relativis- this year, hopefully being able to improve the accuracy to 0.2%. It tic prediction of the Lense-Thirring effect”, Nature 431, 958–960 (2004), should be noted that the accuracy of all experiments performed so I. Ciufolini et al., “Determination of far is nevertheless still up for debate31. One final possible future ex- frame-dragging using Earth gravity models from CHAMP and GRACE”, periment worth mentioning would use data from the Juno spacecraft New Astronomy 11, 527–550 (2006) 32 launched in 2011 , finally testing the effect with a source different 30 I. Ciufolini et al., “Towards a one per- from Earth. cent measurement of frame dragging Lastly, an experiment that managed to test both effects was Grav- by spin with satellite laser ranging to LAGEOS, LAGEOS 2 and LARES and ity Probe B, launched in 2004 and collecting data for about a year GRACE gravity models”, Space Science with an error of about 19% and the predicted values lying at the Reviews 148, 71–104 (2009), I. Ciufolini 33 et al., “A new laser-ranged satellite for center of the confidence interval . General Relativity and space geodesy: I. An introduction to the LARES2 space experiment”, European Physical Journal Plus 132, 336 (2017) 31 G. Renzetti, “Some reflections on the Lageos frame-dragging experiment in view of recent data analyses”, New Astronomy 29, 25–27 (2014) 32 L. Iorio, “Juno, the angular momentum of Jupiter and the Lense-Thirring effect”, New Astronomy 15, 554–560 (2010) 33 C. W. Everitt et al., “Gravity probe B: Final results of a space experiment to test general relativity”, Physical Review Letters 106, 221101 (2011) Conclusion

As we set out to do in the introduction, the precession of a gyroscope due to both a central mass and a rotating central mass was briefly explained as motivation, followed by the calculations of spin motion in the flrw metric and the Schwarzschild metric, both of them using the geodesic formalism as well as the mpd equations with the Frenkel condition, returning finally to the two types of precession, how they serve as confirmation of the general theory of relativity, and the experiments which have been carried out to test them. As far as the calculations made, although a number of papers have somewhat similar treatments for the Schwarzschild metric and the Kerr metric, we tried here to provide something novel with the extended particle kinematics in the flrw metric, including a distinction between a particle part of the cosmological flow and a test particle separate from it. The results from the calculations were not always able to be presented in full, at times arriving only at simplified differential equations which may or may not have analytical solutions, and the treatment of which, analytical or numerical, is left for future work. Where solutions were found they turned out to be equivalent in both formalisms, this being finally explained by the choice of ssc which in certain specific cases, and in the weak field approximation, reduce the mpd equations to simple parallel transport. Moreover, we explained how in the weak field approximation all choices of ssc are equivalent and thus the choice is ultimately to be made for ease of computation, future treatments of this subject could attempt to find those solutions here deemed too difficult by trying different sscs. Something that was ultimately missing from all the calculations was Lense-Thirring precession, having chosen metrics where the source of the gravitational field is not rotating. Future work could attempt calculations similar to the ones here for metrics that in- corporate a rotating central mass such as the one studied by Lense and Thirring or the complete Kerr solution. Where these treatments might need to differ is in the use of a tetrad formalism such as the 70 spin motion in general relativity

Newman-Penrose formalism to ease the computations; this may ultimately expand, not only this work, but also that of Daniel Rozo carried out in 2019 under the supervision of Professor Pedro Bargueño, Ph.D.34, in which he provided an overview of these formalisms and 34 D. F. Rozo Oviedo, Introducción a how they can become a powerful computational tool in some cases la formulación de Newman-Penrose y al formalismo de tétradas con aplicación en for which they are well adapted. cálculos en Relatividad General, 2019

Overall, what is presented here is an overview of the motion of spin in external gravitational ﬁelds, the importance of studying it as far as experimental conﬁrmation of general relativity is concerned, and different tools that may be used to do so, along with some seem- ingly new treatment in the extended particle kinematics. It certainly leaves a lot open for future work on the subject, taking different ap- proaches and considering different cases of great interest. Bibliography

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Appendix A: Operator of Proper-Time-Derivative

D The differential operator dτ is the operator of proper-time- derivative, deﬁned as

D = uµ∇ , (A.1) dτ µ with ∇µ the ordinary covariant derivative. When we have this operator acting on a 4-vector, for instance, we have

D ν µ ν A = u ∇µ A dτ (A.2) dxµ = ∂ Aν + Γν Aλ . dτ µ µλ The RHS in particular may not be applied to position, xµ, since it is not a 4-vector in curved space-time (however displacement, dxµ, is) but the operator by itself applied to position actually speciﬁes µ Dxµ 4-velocity, u ≡ Dτ , and likewise a second application speciﬁes µ Duµ 4-acceleration, a ≡ Dτ , in this case taking the RHS yields

D dxµ uν = ∂ uν + Γν uλ dτ dτ µ µλ dxµ ∂ dxν dxµ = + Γν uλ (A.3) dτ ∂xµ dτ dτ µλ d2xν = + Γν uµuλ, dτ2 µλ where we see how imposing aµ = 0 leads directly to the geodesic equations—as it should be, since a free-falling body in a gravitational ﬁeld must follow a geodesic.

The operator may also act on (2,0) tensors as follows

D Aµν = uλ∇ Aµν dτ λ λ (A.4) dx µ = ∂ Aµν + Γ Aρν + Γν Aµρ , dτ λ λρ λρ 76 spin motion in general relativity

which, in addition to the application given here for spin, ﬁnds use in certain applications in electrodynamics.

Finally, as may be seen from the ﬁrst of the mpd equations, we D D λµ would like to be able to readily expand terms of the form dτ uµ dτ S . We then have for arbitrary Aµν

D D u Aµν = uρ∇ (u uσ∇ Aµν) dτ ν dτ ρ ν σ µ = uρ∂ (u uσ∇ Aµν) + Γ u uρuσ∇ Aαν ρ ν σ ρα ν σ (A.5) ρ σ µν µ σ βν ν ρ µβ = u ∂ρ uνu ∂σ A + Γσβuνu A + Γσβuνu A µ ρ σ αν µ α ρ σ βν µ ν ρ σ αβ + Γραuνu u ∂σ A + ΓραΓσβuνu u A + ΓραΓσβuνu u A ,

thus yielding second order derivatives that make the mpd equations considerably harder to solve than geodesic equations.