Spacetime as a Hamiltonian Orbit and Geroch’s Theorem on the Existence of Fermions

Spacetime as a Hamiltonian Orbit

and Geroch’s Theorem on the

Existence of Fermions

Viktor Bergstedt

Department of Physics and Astronomy

Kandidatprogrammet i fysik

Uppsala University, Sweden Department of Physics and Astronomy

Spacetime as a Hamiltonian Orbit

and

Geroch’s Theorem on the

Existence of Fermions

A thesis by

Viktor Bergstedt

presented for

the degree of Bachelor of Science in Physics

2020

Supervised by Reviewed by

Dr. Stephen McCormick Department of Mathematics, 𝐴𝑝𝑝𝑙𝑖𝑒𝑑 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠 𝑎𝑛𝑑 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠,

Uppsala University

Prof. Ulf Danielsson

Department of Physics and Astronomy, 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑃ℎ𝑦𝑠𝑖𝑐𝑠,

Uppsala University

Abstract

Over a century since its inception, general relativity continues to lie at the heart of some of the most researched topics in theoretical physics.

It seems likely that the coveted solutions to problems like quantum gravity are to be found in an extension of general relativity, one which may only be visible in an alternate formulation of the theory.

In this thesis we consider the possibility of casting general relativ- ity in the form of an initial value problem where spacetime is seen as the evolution of space. This evolution is shown to be constrained and of Hamiltonian type.

Not all spacetimes are physically acceptable. To be compatible with particle physics, one would like spacetime to accommodate fermi- ons. Here we can take comfort in Geroch’s theorem, which implies that any spacetime that admits a Hamiltonian formulation automatically supports the existence of fermions. We review the elements that go into the proof of this theorem.

Sammanfattning

Allmän relativitetsteori har i över hundra år legat i teoretiska fysikens framkant.Detär möjligtatt lösningarna påöppna problem som kvan- tiseringen av gravitation går att finna i en utvidgning av allmän rela- tivitetsteori – och kanske uppenbarar sig denna utvidgning bara ur en alternativ formulering av teorin.

I den här uppsatsen formuleras allmän relativitetsteori och dess Einsteinekvationer som ett begynnelsevärdesproblem, genom vilket rumtiden kan betraktas som rummets historia. Vi visar att rummets rörelseekvationer är Hamiltons ekvationer med tvångsvillkor.

Enligt partikelfysiken bör fermioner kunna finnas till i rumtiden.

Härom kan vi åberopa Gerochs sats, enligt vilken rumtider som har en Hamiltonsk formulering också medger fermioner. Vi redogör för hu- vuddragen i beviset av Gerochs sats.

Contents

PREFACE

SPACETIME AS A HAMILTONIAN ORBIT

1. THE (3+1)-SPLIT ... 1

1.1 Causality ... 1

1.2 Global hyperbolicity ... 1

1.3 Space in spacetime ... 2

1.4 Lapse ... 3

1.5 Shift ... 4

1.6 Coordinates on space ... 5

1.7 Decomposition of the metric ... 5

1.8 Decomposition of curvature ... 7

1.9 Consistency with Einstein’s equations and energy-momentum ... 8

1.10 Decomposition of the Einstein-Hilbert action ... 10

2. HAMILTONIAN EVOLUTION OF SPACE ... 11

2.1 Steps toward a Hamiltonian formulation ... 11

2.2 Evolution of spatial tensors ... 11

2.3 Canonical momenta ... 12

2.4 Einstein’s equations as Hamilton’s equations ... 13

3. APPLICATIONS ... 15

3.1 Numerical relativity ... 15

3.2 Global charges and the ADM-mass ... 16

3.3 Canonical quantization of gravity ... 17

GEROCH’S THEOREM ON THE EXISTENCE OF FERMIONS 4. FERMIONS AS SPINORS ... 20

4.1 Two types of matter ... 20

4.2 Lorentz invariance in general relativity ... 21

4.3 Spinors ... 22

4.4 Spin structures ... 23

5. EXISTENCE OF A SPIN STRUCTURE ... 24

5.1 Geroch’s theorem ... 24

5.2 Relation to global hyperbolicity ... 25

APPENDIX: A PRIMER ON SPACETIME

A.1 Manifolds ... 26

A.2 Tangent spaces ... 27

A.3 Fiber bundles ... 30

A.4 Abstract index notation ... 33

A.5 Metric tensors ... 33

A.6 Integration ... 36

A.7 Spacetime ... 37

A.8 Energy-momentum ... 40

A.9 Curvature ... 41

A.10 Einstein’s field equations ... 41

REFERENCES ... 45

Preface

This bachelor’s thesis was inspired by a pair of online lectures by Prof.

Domenico Giulini (Universität Bremen and Leibniz Universität Han- nover) on the Hamiltonian formulation of general relativity, which over the past sixty years has proven itself a fruitful approach for solving and understanding the structure of Einstein’s equations.

What is presented here is an overview this formalism. There exists by now a handful of comprehensive treatments of the subject, and it is not our ambition to add to that growing list. We will keep the exposi- tion minimal but technical, emphasizing the logic of the theory rather than derivations of complicated results.

Due to its heavy reliance on differential geometry, the thesis may be a challenging read for undergraduate students with no prior exposure.

For this reason, we have added an appendix with the optimism that no secondary sources should be required for students who have com- pleted three years of university physics. We will therefore also assume familiarity with whatever physics and mathematics students would have been covered at that level of education. This includes special rel- ativity, analytical mechanics, quantum physics, multivariable calculus, linear algebra and basic set theoretic notation and nomenclature. The appendix features no proofs and only highlights concepts which are part of the thesis. For the already initiated, definitions of some lesser- known mathematical notions have been moved to the main text as footnotes.

Conventions and notation are carefully explained in the appendix but are worth stating here too. 𝑆𝑝𝑎𝑐𝑒𝑡𝑖𝑚𝑒 always refers to a 4-dimensional, connected, smooth manifold equipped with a metric of Lorentzian sig- nature (−, +, +, +). We make systematic use of abstract index nota- tion—that is, tensors carry indices labelling the tensor type, rather than components in some basis. In keeping with this, index contraction denotes an action in some slot, rather than a sum of (products of) components. Whenever components are referred to, we will make it explicit, and spacetime indices then run over the values 0, 1, 2, 3. We employ Einstein’s summation convention and sum over any pair of repeated indices, for consistency one upper and one lower. We will work in units with 𝑐 = 𝐺 = 1 and leave ℏ dimensionful.

My heartfelt gratitude goes to my superb supervisor Steve for being a resourceful companion and enthusiast throughout this protracted but rewarding project, as well as to my subject reviewer Ulf for his helpful feedback on my thesis and its presentation. I also wish to thank exam- iner Matthias Weiszflog, who took pragmatic action in these unprece- dented times, marked by the coronavirus, by giving affected students like me a welcomed extension to duly complete our theses.

SPACETIME AS A

HAMILTONIAN ORBIT

Events in spacetime are the fundamental building blocks of relativity, whereas space and time by themselves are derived notions which may or may not make sense globally. Inthiswork,our philosophywillbeto promote space and time to central notions and view spacetime as the ‘history of space.’ Our goal is to understand this viewpoint, the conditions under which it is valid, and what it implies for the structure of spacetime and Einstein’s field equations.

1. The (3+1)

-

1.1 Causality

Our ambition to view spacetime as the ‘history of space’ faces a serious problem. 𝐴 𝑝𝑟𝑖𝑜𝑟𝑖, all four coordinates labelling an event are treated on equal footing. However,to evolve‘space’ as an initial-value problem, we must somehow single out one coordinate globally as ‘time’ and do so inawaythatconformswiththecausal structure of spacetime.

Causality in special relativity is completely determined by the future and past light cones of Minkowski space. Any event in the future light cone of a second event, may be caused by the latter. Conversely, any event in the past light cone of a second event, may be the cause of the latter. In general relativity, causality depends not only on the events in question, but also on the curved spacetime in between. Light is no longer constrained to cones. Instead, it moves on more general null surfaces following a curved worldline.

1.2 Global hyperbolicity

Signals in a general spacetime 𝑀 can either propagate along timelike or lightlike curves. For this reason, it is convenient to define a 𝑐𝑎𝑢𝑠𝑎𝑙 𝑐𝑢𝑟𝑣𝑒 as one whose tangent is everywhere timelike 𝑜𝑟 lightlike.

A causal curve 𝜎: 𝐼 ⟶ 𝑀 is said to be 𝑖𝑛𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑏𝑙𝑒 if there is no other causal curve 𝜎^′: 𝐼^′ ⟶ 𝑀 with 𝐼 ⊂ 𝐼^′ such that 𝜎(𝐼) ⊂ 𝜎^′(𝐼).

Given a subset of events 𝑆 ⊂ 𝑀 , the 𝑓𝑢𝑡𝑢𝑟𝑒 𝐶𝑎𝑢𝑐ℎ𝑦 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑚𝑒𝑛𝑡 of 𝑆 is the set 𝐷⁺(𝑆) of all 𝑝 ∈ 𝑀 so that every inextensible past-directed causal curve through 𝑝 intersects 𝑆. In more plain language, the future Cauchy development of 𝑆 is the set of all future events that are cer- tainly caused by events in 𝑆; perfect knowledge of events in 𝑆 suffices to predict what happens in 𝐷⁺(𝑆). Analogously, we define the 𝑝𝑎𝑠𝑡 𝐶𝑎𝑢𝑐ℎ𝑦 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑚𝑒𝑛𝑡 of 𝑆 to be the set 𝐷⁻(𝑆) of all 𝑝 ∈ 𝑀 so that every inextensible future-directed causal curve through 𝑝 inter- sects 𝑆. The past Cauchy development of 𝑆 is the set of all past events that could influence nothing outside 𝑆; with perfect knowledge of events in 𝑆 it would be possible to retrodict what happened in 𝐷⁻(𝑆).

The 𝑡𝑜𝑡𝑎𝑙 𝐶𝑎𝑢𝑐ℎ𝑦 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑚𝑒𝑛𝑡 of 𝑆 is then 𝐷(𝑆) = 𝐷⁺(𝑆) ∪ 𝐷⁻(𝑆).

We say that 𝑆 is 𝑎𝑐ℎ𝑟𝑜𝑛𝑎𝑙 if no two events in 𝑆 are connected by a timelike curve. On an achronal set events cannot be put in chronolog- ical order. A closed, achronal subset whose total Cauchy development is all of spacetime is known as a 𝐶𝑎𝑢𝑐ℎ𝑦 𝑠𝑢𝑟𝑓𝑎𝑐𝑒.

Spacetimes which exhibit a Cauchy surface are coined 𝑔𝑙𝑜𝑏𝑎𝑙𝑙𝑦 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐, essentially because the scalar wave equation ∇_𝜇∇^𝜇𝜑 = 0 is well-posed as an initial value problem. A globally hyperbolic manifold 𝑀 isnecessarilyofthe form 𝑀 =ℝ × Σ fora smooth3-manifold Σ [1].¹ These spacetimes play a pivotal role in what follows.

1.3 Space in spacetime

Suppose 𝑀 is globally hyperbolic, i.e. homeomorphic to ℝ × Σ where Σ is a 3-manifold. This topological split is far from universal—plenty of spacetimes are not of this kind. Nevertheless, the requirement is sufficiently mild to encompass most spacetimes of physical interest. To this class belong Minkowski space, the Schwarzschild black hole, the Friedmann-Robertson-Walker metrics from cosmology and de Sitter space, just to name a few. There also exist important spacetimes, from both a theoretical and instructional point of view, that are not globally hyperbolic—the rotating Kerr black hole and Anti-de Sitter space be- ing famous examples.

So far, the split is a purely topological one. We must now make sure that the geometry on Σ, inherited from 𝑀 =ℝ × Σ, resembles that of physical space for any value of ℝ. Note that we need 𝑛𝑜𝑡 require that this spatial geometry be constant but rather it could (and generally it would) depend on ℝ.

1 Intuitively, a Cauchy surface provides initial data (Σ), and global hyperbolicity states that this initial data evolves (ℝ) onto the entire solution (𝑀) at or under the speed of light.

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More formally, points 𝑝 ∈ 𝑀 are of the form 𝑝 = (𝜆, 𝑞) where 𝜆 ∈ℝ and 𝑞 ∈ Σ. Now for any 𝜆₀ ∈ℝ the set Σ_𝜆0 ≔ {𝑝 ∈ 𝑀 | 𝑝 = (𝜆₀, 𝑞)} is an embedded submanifold of 𝑀 diffeomorphic to Σ. As 𝜆 ranges over ℝ the family of submanifolds {Σ_𝜆}_𝜆∈ℝ trace out 𝑀 , providing us with a smooth foliation² of spacetime into hypersurfaces Σ_𝜆. We restrict the admissible foliations by requiring that Σ_𝜆 be spacelike hypersurfaces, namely ones for which tangent vectors are spacelike with respect to 𝑔_𝜇𝜈. Now the geometry on any Σ_𝜆 is fixed by the induced metric³ 𝛾_𝑖𝑗 which, due to Σ_𝜆 being spacelike, is positive-definite. We call this foli- ation a (3+1)-𝑠𝑝𝑙𝑖𝑡 and refer to (Σ, 𝛾_𝑖𝑗) as 𝑠𝑝𝑎𝑐𝑒. Constructions intrin- sic to Σ (coordinates, vectors, covectors, tensors of higher rank…) will be called 𝑠𝑝𝑎𝑡𝑖𝑎𝑙.

Clearly, there is great redundancy in this split. In fact, diffeomorphism invariance on 𝑀 gets traded for invariance under change of foliation, and this freedom of chopping spacetime up is a key feature of the for- malism [2].

The geometry on space (the components of 𝛾_𝑖𝑗) make up only six out of ten degrees of freedom of 𝑔_𝜇𝜈. The remaining freedom (which in effect characterizes the foliation) lies in how this geometry deforms as we vary 𝜆. In passing from one leaf Σ_𝜆 to a neighboring leaf Σ_{𝜆+𝛿𝜆}, points on Σ_𝜆 are elapsed in a timelike direction (one component)and shifted in a spacelike direction (three components). This data is en- coded in the 𝑙𝑎𝑝𝑠𝑒 and 𝑠ℎ𝑖𝑓𝑡, respectively.

1.4 Lapse

Each leaf Σ_𝜆 is uniquely and smoothly labelled by its parameter 𝜆.

Hence,asmoothfunction𝑡: 𝑀 ⟶ℝtaking 𝑝 to 𝑡(𝑝) = 𝜆 where 𝑝 ∈ Σ_𝜆 exists. It is necessarily regular, d𝑡_𝜇 ≠ 0, and timelike, 𝑔^𝜇𝜈d𝑡_𝜇d𝑡_𝜈 < 0.

These facts give rise to a nonvanishing future-directed timelike vector field d𝑡^𝜇 ≔ 𝑔^𝜇𝜈d𝑡_𝜈 that is everywhere normal to the spacelike leaves.

By rescaling this vector field, we obtain a unit normal 𝑛^𝜇 ≔ −𝛼d𝑡^𝜇,

and 𝛼 = (−d𝑡_𝜇d𝑡^𝜇)^−1/2 is called the 𝑙𝑎𝑝𝑠𝑒. Because 𝑛^𝜇 is a timelike vector field it can, at any point 𝑝 ∈ 𝑀 , be regarded as the 4-velocity of an observer passing 𝑝. Borrowing terminology from fluid mechanics,

2 A smooth, 1-parameter family of embeddings Φ_𝜆: Σ → 𝑀 is said to 𝑓𝑜𝑙𝑖𝑎𝑡𝑒 a smooth manifold 𝑀 into hypersur- faces (also known as 𝑙𝑒𝑎𝑣𝑒𝑠) Σ_𝜆≔ Φ_𝜆(Σ) ≅ Σ if Φ_𝜆 is smooth, 1–1 and ⋃_𝜆∈ℝΣ_𝜆= 𝑀.

3 Clearly, 𝛾_𝑖𝑗= (Φ_𝜆^∗𝑔)_𝑖𝑗 depends on 𝜆. It is customary to suppress this dependence.

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3 The (3+1)-split

we call such an observer 𝐸𝑢𝑙𝑒𝑟𝑖𝑎𝑛. Then the lapse 𝛼(𝑝) tells us how much proper time (in units of 𝜆) elapses for a clock carried by an Eulerian observer through 𝑝 [2].

1.5 Shift

A point 𝑞 ∈ Σ has an associated worldline 𝜆 ⟼ 𝜎_𝑞(𝜆) = (𝜆, 𝑞) ∈ 𝑀 : all events that ever happened and will happen at location 𝑞. One calls 𝜎_𝑞 the 𝑡𝑖𝑚𝑒𝑙𝑖𝑛𝑒 of 𝑞. The congruence of timelines {𝜎_𝑞}_𝑞∈Σ are the in- tegral curves to a unique nonvanishing vector field 𝑡^𝜇, the 𝑡𝑖𝑚𝑒 𝑣𝑒𝑐𝑡𝑜𝑟.

In particular, the time vector 𝑡^𝜇 is the dual of d𝑡_𝜇, in the sense that 𝑡^𝜇d𝑡_𝜇 = 1. This yields the decomposition

𝑡^𝜇 = 𝛼𝑛^𝜇 + 𝛽^𝜇

where 𝛽^𝜇 ≔ 𝑡^𝜇− 𝛼𝑛^𝜇 is a spacelike vector called the 𝑠ℎ𝑖𝑓𝑡. The shift is normal to 𝑛^𝜇,

𝛽^𝜇𝑛_𝜇 = 𝑡^𝜇𝑛_𝜇− 𝛼𝑛^𝜇𝑛_𝜇 = −𝛼𝑡^𝜇d𝑡_𝜇+ 𝛼 = −𝛼 + 𝛼 = 0, and thus tangent to the spacelike leaves.

We shall find it convenient to introduce 𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑚𝑎𝑝𝑠 onto Σ, 𝑃_𝜇𝜈 ≔ 𝑔_𝜇𝜈 + 𝑛_𝜇𝑛_𝜈,

𝑃^𝜇_𝜈 ≔ 𝑔^𝜇𝜆𝑃_𝜆𝜈 = 𝛿^𝜇_𝜈+ 𝑛^𝜇𝑛_𝜈,

in terms of which we have 𝛽^𝜇 = 𝑃^𝜇_𝜈𝑡^𝜈. More generally, 𝑃^𝜇_𝜈 deletes the normal component of any 4-vector 𝑢^𝜇,

(𝑃^𝜇_𝜈𝑢^𝜈)𝑛_𝜇 = 𝛿^𝜇_𝜈𝑢^𝜈𝑛_𝜇+ 𝑢^𝜈𝑛_𝜇𝑛^𝜇𝑛_𝜈 = 𝑢^𝜇𝑛_𝜇− 𝑢^𝜈𝑛_𝜈 = 0, leaving its projection tangent to Σ.

We finally note that the time vector 𝑡^𝜇, unlike the unit normal 𝑛^𝜇, need not be timelike. It could even be spacelike, provided 𝛽_𝜇𝛽^𝜇 > 𝛼². Such ‘superluminal’ shifts are not to be feared though; they are but a coordinate artifact (sometimes a useful one) which does not represent the motion of an observer through spacetime [3].

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1.6 Coordinates on space

In principle, any system of four coordinates may be used to label points in 𝑀 . However, in a (3+1)-split some coordinates are particularly adapted to the foliation.

Local spacetime coordinates (𝑥^𝜇) = (𝑥⁰, 𝑥¹, 𝑥², 𝑥³) on 𝑀 𝑎𝑑𝑎𝑝𝑡𝑒𝑑 to Σ are formed by choosing spatial coordinates (𝑥^𝑖), 𝑖 = 1, 2, 3 on Σ and letting 𝑥⁰ = 𝑡. In adapted coordinates, some useful formulæ hold [3].

For a spacetime vector 𝑢^𝜇 tangent to Σ_𝜆, (𝑢^𝜇) = (0, 𝑢^𝑖),

𝑔_𝜇𝜈𝑢^𝜇𝑣^𝜈 = 𝑔_𝜇𝜈(𝜕𝑥^𝜇

𝜕𝑥⁰

𝜕𝑥^𝜈

𝜕𝑥⁰𝑢⁰𝑣⁰+𝜕𝑥^𝜇

𝜕𝑥^𝑖

𝜕𝑥^𝜈

𝜕𝑥^𝑗𝑢^𝑖𝑣^𝑗) =

= (𝑔_𝜇𝜈𝜕𝑥^𝜇

𝜕𝑥^𝑖

𝜕𝑥^𝜈

𝜕𝑥^𝑗) 𝑢^𝑖𝑣^𝑗 =: 𝛾_𝑖𝑗𝑢^𝑖𝑣^𝑗,

where 𝛾_𝑖𝑗 is the induced metric on Σ_𝜆 (the pullback of 𝑔_𝜇𝜈 under the coordinate map). We also have that 𝜕𝑥^𝜇/𝜕𝑥⁰ = 𝑡^𝜇 where 𝑡^𝜇 now stands for the components of the time vector in adapted coordinates.

Using 𝑃^𝜇_𝜈 it is possible to construct out of spacetime tensors (defined on 𝑀 ) corresponding spatial tensors (defined on Σ). Let 𝑇^{𝜇1,…,𝜇𝑝}_𝜈

1,…,𝜈_𝑞

be the components of a (𝑝, 𝑞)-tensor on 𝑀 , then in adapted coordinates 𝑇^{𝑖1,…,𝑖𝑝}_{𝑗1,…,𝑗𝑞} ≔ 𝑃^𝑖1_𝜇1… 𝑃^𝑖𝑝_𝜇𝑝𝑃^𝜈1_𝑗1… 𝑃^𝜈𝑞_𝑗𝑞𝑇^{𝜇1,…,𝜇𝑝}_{𝜈1,…,𝜈𝑞}

are the components of a spatial (𝑝, 𝑞)-tensor on Σ for 𝑖₁, 𝑗₁… = 1, 2, 3.

It acts on spatial vectors 𝑢^𝑖 and spatial covectors 𝜔_𝑖 by applying (7) to (𝑢^𝜇) = (0, 𝑢^𝑖) and (𝜔_𝜇) = (0, 𝜔_𝑖) respectively.

Whenever we put Latin space indices on tensors defined on spacetime, the above identification in adapted coordinates is understood. Con- versely, whenever we put Greek spacetime indices on spatially defined tensors, we refer to the spacetime tensor which in adapted coordinates satisfies (7) 𝑎𝑛𝑑 has vanishing time components.

1.7 Decomposition of the metric

The metric 𝑔_𝜇𝜈 induces on spacelike leaves a spatial metric 𝛾_𝑖𝑗 (which implicitly depends on 𝜆). In adapted coordinates (𝑡, 𝑥^𝑖), the compo- nents of 𝑔_𝜇𝜈 are related to 𝛼, 𝛽^𝜇 and 𝛾_𝑖𝑗 as follows:

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The (3+1)-split 5

𝑔_𝜇𝜈d𝑥^𝜇 ⊗ d𝑥^𝜈 =

= 𝑔_𝜇𝜈(𝑡^𝜇d𝑡 +𝜕𝑥^𝜇

𝜕𝑥^𝑖d𝑥^𝑖) ⊗ (𝑡^𝜈d𝑡 +𝜕𝑥^𝜈

𝜕𝑥^𝑗d𝑥^𝑗) =

= 𝑔_𝜇𝜈[(𝛼𝑛^𝜇+ 𝛽^𝜇)d𝑡 +𝜕𝑥^𝜇

𝜕𝑥^𝑖 d𝑥^𝑖] ⊗ [(𝛼𝑛^𝜈 + 𝛽^𝜈)d𝑡 +𝜕𝑥^𝜈

𝜕𝑥^𝑗d𝑥^𝑗] =

= (𝛼²𝑔_𝜇𝜈𝑛^𝜇𝑛^𝜈+ 𝑔_𝜇𝜈𝛽^𝜈𝛽^𝜇)d𝑡 ⊗ d𝑡 + 𝑔_𝜇𝜈𝜕𝑥^𝜇

𝜕𝑥^𝑖𝛽^𝑖𝜕𝑥^𝜈

𝜕𝑥^𝑗d𝑡 ⊗ d𝑥^𝑗 + 𝑔_𝜇𝜈𝜕𝑥^𝜇

𝜕𝑥^𝑖

𝜕𝑥^𝜈

𝜕𝑥^𝑗𝛽^𝑗d𝑥^𝑖⊗ d𝑡 + 𝑔_𝜇𝜈𝜕𝑥^𝜇

𝜕𝑥^𝑖

𝜕𝑥^𝜈

𝜕𝑥^𝑗d𝑥^𝑖⊗ d𝑥^𝑗 =

= (−𝛼²+ 𝛾_𝑖𝑗𝛽^𝑖𝛽^𝑗)d𝑡 ⊗ d𝑡 + 𝛾_𝑖𝑗𝛽^𝑗(d𝑡 ⊗ d𝑥^𝑖+ d𝑥^𝑖⊗ d𝑡) + 𝛾_𝑖𝑗d𝑥^𝑖⊗ d𝑥^𝑗.

By the linearity of the tensor product, we can rewrite this as 𝑔_𝜇𝜈d𝑥^𝜇⊗ d𝑥^𝜈 = −𝛼²d𝑡 ⊗ d𝑡 + 𝛾_𝑖𝑗(𝛽^𝑖d𝑡 + d𝑥^𝑖) ⊗ (𝛽^𝑗d𝑡 + d𝑥^𝑗) which in matrix format reads

(𝑔_𝜇𝜈) = (−𝛼²+ 𝛽_𝑘𝛽^𝑘 𝛽_𝑗 𝛽_𝑖 𝛾_𝑖𝑗).

The matrix of the inverse metric 𝑔^𝜇𝜈 is found by inverting that of 𝑔_𝜇𝜈, and the result is

(𝑔^𝜇𝜈) =

⎝

⎜⎜

⎜⎛− 1 𝛼²

𝛽^𝑗 𝛼² 𝛽^𝑖

𝛼² 𝛾^𝑖𝑗−𝛽^𝑖𝛽^𝑗 𝛼² ⎠

⎟⎟

⎟⎞ .

It is now possible to deduce the relation of spacetime 4-volumes to spatial 3-volumes. Indeed, according to Cramer’s rule, the 00-th entry of (𝑔^𝜇𝜈) is

𝑔⁰⁰ =cof₀₀(𝑔_𝜇𝜈)

𝑔 , 𝑔 = det(𝑔_𝜇𝜈),

where the cofactor cof₀₀(𝑔_𝜇𝜈) = (−1)⁰⁺⁰𝑀₀₀(𝑔_𝜇𝜈) = 𝑀₀₀(𝑔_𝜇𝜈) is just the 00-th minor of (𝑔_𝜇𝜈), found by taking the determinant of the 3 × 3- (8)

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matrix gotten by striking out the 0-th row and 0-th column. In our case 𝑔⁰⁰= −1/𝛼² and 𝑀₀₀(𝑔_𝜇𝜈) = det(𝛾_𝑖𝑗), leading to the equality

− 1 𝛼² =𝛾

𝑔, 𝛾 = det(𝛾_𝑖𝑗), or equivalently,

√−𝑔 = 𝛼√𝛾.

This is the decomposition of the spacetime volume form. Notice that 𝛼 appears here explicitly as the rate at which a local patch of space expands in time.

1.8 Decomposition of curvature

To the spatial metric 𝛾_𝑖𝑗 corresponds a unique Levi-Civita connection 𝐷 that is torsion-free and metric compatible. It has spatial Riemann curvature tensor ℜ^𝑘_𝑙𝑖𝑗 defined by

(𝐷_𝑖𝐷_𝑗− 𝐷_𝑗𝐷_𝑖)𝑣^𝑘 = ℜ^𝑘_𝑙𝑖𝑗𝑣^𝑙

for all spatial vectors 𝑣^𝑘. From ℜ^𝑘_𝑙𝑖𝑗 the spatial Ricci tensor ℜ_𝑖𝑗 and spatial Ricci scalar ℜ are found by standard contractions. This way the metric 𝛾_𝑖𝑗 fully determines the 𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 geometry of space—those properties which can be measured inside space. In contrast, 𝑒𝑥𝑡𝑟𝑖𝑛𝑠𝑖𝑐 geometryis all about how space sits in its ambient spacetime.

If the normal 𝑛^𝜇 to a spacelike leaf Σ_𝜆 is parallel transported along Σ_𝜆, it will not in general remain normal. Using the Levi-Civita connec- tion ∇ of spacetime, the offset is governed by the 𝑒𝑥𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑐𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 𝑡𝑒𝑛𝑠𝑜𝑟

𝐾_𝜇𝜈 ≔ 𝑃_𝜇^𝛼∇_𝛼𝑛_𝜈.

Observe that 𝐾_𝜇𝜈𝑛^𝜈 = (𝑃_𝜇^𝛼∇_𝛼𝑛_𝜈)𝑛^𝜈 = 𝑃_𝜇^𝛼∇_𝛼(𝑛_𝜈𝑛^𝜈)/2 = 0, hence 𝐾_𝜇𝜈 may be regarded as a spatial tensor. (In fact, 𝐾_𝑖𝑗 can be defined without reference to the ambient spacetime, using the so-called 𝑊𝑒𝑖𝑛𝑔𝑎𝑟𝑡𝑒𝑛 𝑚𝑎𝑝. [2]) An important identity is

𝐾_𝜇𝜈 =1

2ℒ_𝑛𝑃_𝜇𝜈.

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The (3+1)-split 7

where ℒ_𝑛 is the Lie derivative along 𝑛^𝜇. It is easily established, ℒ_𝑛𝑃_𝜇𝜈 = 𝑛^𝛼∇_𝛼𝑃_𝜇𝜈+ 𝑃_𝜇𝛼∇_𝜈𝑛^𝛼+ 𝑃_𝜈𝛼∇_𝜇𝑛^𝛼 =

= 𝑛^𝛼∇_𝛼(𝑛_𝜇𝑛_𝜈) + 𝑔_𝜇𝛼∇_𝜈𝑛^𝛼+ 𝑔_𝜈𝛼∇_𝜇𝑛^𝛼 =

= 𝑛_𝜇𝑛^𝛼∇_𝛼𝑛_𝜈+ 𝑛_𝜈𝑛^𝛼∇_𝛼𝑛_𝜇+ ∇_𝜈𝑛_𝜇+ ∇_𝜇𝑛_𝜈 =

= (𝑃_𝜇^𝛼− 𝛿_𝜇^𝛼)∇_𝛼𝑛_𝜈+ (𝑃_𝜈^𝛼− 𝛿_𝜈^𝛼)∇_𝛼𝑛_𝜇 + ∇_𝜈𝑛_𝜇+ ∇_𝜇𝑛_𝜈 =

= (𝑃_𝜇^𝛼)∇_𝛼𝑛_𝜈+ 𝑃_𝜈^𝛼∇_𝛼𝑛_𝜇 = 2𝐾_𝜇𝜈.

In adapted coordinates this identity reduces to 𝐾_𝑖𝑗 =¹₂ℒ_𝑛𝛾_𝑖𝑗, and the extrinsic curvature is thus seen to be the ‘velocity’ of the spatial metric as measured by Eulerian observers.

It turns out that the spatial connection 𝐷 is related to the spacetime connection ∇ in the naïve way 𝐷_𝜇 = 𝑃_𝜇^𝛼∇_𝛼 [4]. This results in three fundamental projection formulæ (Ch. 2.5 in [2]). First, the spacetime Riemann tensor 𝑅^𝜌_𝜎𝜇𝜈 can be projected onto space according to the 𝐺𝑎𝑢ß–𝐶𝑜𝑑𝑎𝑧𝑧𝑖 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠

𝑃_𝜌^𝛼𝑃_𝜎^𝛽𝑃_𝜇^𝛾𝑃_𝜈^𝛿𝑅_{𝛼𝛽𝛾𝛿} = ℜ_{𝜌𝜎𝜇𝜈} + 𝐾_𝜌𝜇𝐾_𝜎𝜈− 𝐾_𝜌𝜈𝐾_𝜎𝜇. Using this result, the Ricci scalar decomposes as

𝑅 = ℜ + 𝐾_𝛼𝛽𝐾^𝛼𝛽− 𝐾²− 2∇_𝛼(𝑛^𝛽∇_𝛽𝑛^𝛼− 𝑛^𝛼∇_𝛽𝑛^𝛽) where the trace 𝐾 ≔ 𝐾_𝛼^𝛼 is called the 𝑚𝑒𝑎𝑛 𝑐𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒.

Contracting the Riemann tensor with the normal 𝑛^𝜇, and projecting it onto space, results in the equations

𝑃_𝜌^𝛼𝑃_𝜎^𝛽𝑃_𝜇^𝛾𝑛^𝛿𝑅_{𝛼𝛽𝛾𝛿} = 𝐷_𝜌𝐾_𝜎𝜇− 𝐷_𝜎𝐾_𝜌𝜇, known as the 𝐶𝑜𝑑𝑎𝑧𝑧𝑖–𝑀𝑎𝑖𝑛𝑎𝑟𝑑𝑖 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠.

1.9 Consistency with Einstein’s equations and energy-momentum We now put these equations to work. In general relativity, the geom- etry of spacetime has to obey Einstein’s field equations, 𝐺_𝜇𝜈 = 8𝜋𝑇_𝜇𝜈. This means that the left-hand sides of (16)-(18) can be replaced by suitable contractions of the energy-momentum tensor. First, observe that

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𝑃^𝛼𝛾𝑃^𝛽𝛿𝑅_{𝛼𝛽𝛾𝛿} = (𝑔^𝛼𝛾+ 𝑛^𝛼𝑛^𝛾)(𝑔^𝛽𝛿+ 𝑛^𝛽𝑛^𝛾)𝑅_{𝛼𝛽𝛾𝛿} =

= 𝑅 + 2𝑛^𝛾𝑛^𝛿𝑅_𝛾𝛿 =

= 2𝑛^𝛾𝑛^𝛿𝐺_𝛾𝛿.

But at the same time the Gauß–Codazzi equations imply 𝑃^𝛼𝛾𝑃^𝛽𝛿𝑅_{𝛼𝛽𝛾𝛿} = ℜ + 𝐾²− 𝐾_𝛼𝛽𝐾^𝛼𝛽. Hence, we obtain the geometric identity

ℜ + 𝐾²− 𝐾_𝛼𝛽𝐾^𝛼𝛽 = 2𝑛^𝛾𝑛^𝛿𝐺_𝛾𝛿,

which through Einstein’s equations 𝐺_𝛾𝛿 = 8𝜋𝑇_𝛾𝛿 becomes (in adapted coordinates)

ℜ + 𝐾²− 𝐾_𝑖𝑗𝐾^𝑖𝑗 = 16𝜋𝜌.

Here 𝜌 ≔ 𝑇_𝛾𝛿𝑛^𝛾𝑛^𝛿 is the energy density as measured by Eulerian ob- servers. This equation contains no time derivatives and so is not an evolution equation. Instead, it constitutes a constraint equation called the 𝑒𝑛𝑒𝑟𝑔𝑦 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡.

Next up, note that from 𝑃^𝜇𝛼𝑛^𝛽𝑔_𝛼𝛽 = 0,

𝑃^𝜇𝛼𝑛^𝛽𝐺_𝛼𝛽 = 𝑃^𝜇𝛼𝑛^𝛽𝑅_𝛼𝛽. But then the Codazzi–Mainardi equations yield

𝑃^𝜇𝛼𝑛^𝛽𝐺_𝛼𝛽 = 𝐷_𝛼𝐾^𝛼𝜇− 𝐷^𝜇𝐾,

which, again assuming Einstein’s equations hold, becomes (in adapted coordinates)

𝐷_𝑗(𝐾𝛾^𝑖𝑗 − 𝐾^𝑖𝑗) = 8𝜋𝑗^𝑖.

Here 𝑗^𝑖 = −𝑃^𝑖𝛼𝑇_𝛼𝛽𝑛^𝛽 is the 3-momentum density as measured by Eu- lerian observers. These constitute yet three more constraint equations, known as the 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡.

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The (3+1)-split 9

1.10 Decomposition of the Einstein-Hilbert action Recall the Einstein-Hilbert action,

𝑆_EH[𝑔_𝜇𝜈] ≔ 1

16𝜋∫ 𝑑⁴𝑥

𝑀

√−𝑔𝑅.

When extremized, it yields Einstein’s field equations for the vacuum.

In a (3+1)-split, we can insert the decompositions of the volume meas- ure and the Ricci scalar. Discarding the total divergence in (17), we obtain

𝑆_EH[𝛾_𝑖𝑗, 𝐾_𝑖𝑗] = 1

16𝜋∫ 𝑑𝑡 [𝑑³𝑥 𝛼√𝛾(ℜ + 𝐾_𝑖𝑗𝐾^𝑖𝑗− 𝐾²)]

𝑀

,

from which we identify the 𝐴𝐷𝑀 -𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑖𝑎𝑛⁴ [5] as 𝐿_ADM = ∫ 𝑑³𝑥 𝛼√𝛾(ℜ + 𝐾_𝑖𝑗𝐾^𝑖𝑗 − 𝐾²)

Σ

.

The action 𝑆_EH may now be regarded as a functional over the field configuration variables 𝛾_𝑖𝑗 and 𝐾_𝑖𝑗 on Σ. This view, however, is some- what mistaken. Notice in particular the absence of ‘time derivatives’

in the action. This does not bode well if 𝛾_𝑖𝑗 and 𝐾_𝑖𝑗 are 𝑑𝑦𝑛𝑎𝑚𝑖𝑐𝑎𝑙 variables (which by all means they are). In the following section, we define the missing time derivative and figure out how it enters the ADM-Lagrangian.

4 ‘ADM’ is short for 𝐴𝑟𝑛𝑜𝑤𝑖𝑡𝑡, 𝐷𝑒𝑠𝑒𝑟 and 𝑀𝑖𝑠𝑛𝑒𝑟—a trio of American theoretical physicists who advanced the subject of 3+1-gravitation with several landmark papers in the years 1959 to 1961. In homage to their achievements, this subject is widely referred to as the 𝐴𝐷𝑀-𝑓𝑜𝑟𝑚𝑎𝑙𝑖𝑠𝑚 of general relativity.

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2. Hamiltonian evolution of space

2.1 Steps toward a Hamiltonian formulation

In the previous section we decomposed the Einstein-Hilbert action into the one piece ℜ governing the intrinsic geometry of Σ and second piece 𝐾_𝑖𝑗𝐾^𝑖𝑗 − 𝐾² governing its extrinsic geometry. But for space to be dy- namical, “𝛾̇_𝑖𝑗” also has to enter the Lagrangian. In this section we see how to accomplish this. We then calculate the canonical momentum 𝜋^𝑖𝑗 conjugate to 𝛾_𝑖𝑗 and write down the Hamiltonian for space by means of a Legendre transform. This gives us Hamilton’s equations for (𝛾_𝑖𝑗, 𝜋^𝑖𝑗) together with constraints which effectively enforce our gauge freedom.

2.2 Evolution of spatial tensors

Suppose a spatial tensor field is defined on each slice in an open subset of the foliation. If the associated spacetime tensor field is smooth as a function of 𝜆, we can study the rate at which the spatial tensor changes. To do so we evaluate the Lie derivative of the associated spacetime tensor field along the time vector field 𝑡^𝜇, then project it onto the spatial slice,

𝑇̇^{𝜇1,…,𝜇𝑝}_𝜈

1,…,𝜈_𝑞 ≔ 𝑃^𝜇1_𝛼

1… 𝑃^𝜇𝑝_𝛼

𝑝𝑃^𝛽1_𝜈

1… 𝑃^𝛽𝑞_𝜈

𝑞ℒ_𝑡𝑇^{𝛼1,…,𝛼𝑝}_𝛽

1,…,𝛽_𝑞.

One calls 𝑇̇^{𝜇1,…,𝜇𝑝}_{𝜈1,…,𝜈𝑞} the 𝑡𝑖𝑚𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 of 𝑇^{𝜇1,…,𝜇𝑝}_{𝜈1,…,𝜈𝑞}. In adapted coordinates, the spatial components are explicitly 𝑡-dependent, and the time derivative just reduces to partial differentiation

𝑇̇^{𝑖1,…,𝑖𝑝}_𝑗

1,…,𝑗_𝑞 = 𝜕_𝑡𝑇^{𝑖1,…,𝑖𝑝}_𝑗

1,…,𝑗_𝑞.

We can now derive a neat expression for 𝛾̇_𝑖𝑗 by the following route. In general, the Lie derivative is 𝑛𝑜𝑡 𝐶^∞-linear in its lower vector argu- ment, ℒ_𝑓𝑣 ≠ 𝑓ℒ_𝑣. However, it turns out that because 𝑃_𝜇𝜈𝑛^𝜈 = 0,

ℒ_𝑓𝑛𝑃_𝜇𝜈 =

= 𝑓𝑛^𝜆∇_𝜆𝑃_𝜇𝜈 + (∇_𝜇𝑓𝑛^𝜆)𝑃_𝜆𝜈 + (∇_𝜈𝑓𝑛^𝜆)𝑃_𝜇𝜆 =

= 𝑓𝑛^𝜆∇_𝜆𝑃_𝜇𝜈 + 𝑓(∇_𝜇𝑛^𝜆)𝑃_𝜆𝜈 + (∇_𝜈𝑓𝑛^𝜆)𝑃_𝜇𝜆 =

= 𝑓ℒ_𝑛𝑃_𝜇𝜈.

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Hamiltonian evolution of space 11

Hence,

𝐾_𝜇𝜈 =1

2ℒ_𝑛𝑃_𝜇𝜈 =

= 1

2𝛼ℒ_𝛼𝑛𝑃_𝜇𝜈 = 1

2𝛼ℒ_𝑡−𝛽𝑃_𝜇𝜈 =

= 1

2𝛼ℒ_𝑡𝑃_𝜇𝜈 − 1

2𝛼ℒ_𝛽𝑃_𝜇𝜈 =

= 1

2𝛼ℒ_𝑡𝑃_𝜇𝜈− 1

𝛼𝐷_(𝜇𝛽_𝜈).

In the last step we used that 𝑃_𝜇𝜈 acts like the metric on vectors tangent to spatial slices, and that its Lie derivative can be written as a sym- metrized covariant derivative.

Now since 𝐾_𝜇𝜈 is spatial, it satisfies 𝐾_𝜇𝜈 = 𝑃_𝜇^𝜅𝑃_𝜈^𝜆𝐾_𝜅𝜆. Therefore, 𝐾_𝜇𝜈 = 1

2𝛼𝑃_𝜇^𝜅𝑃_𝜈^𝜆ℒ_𝑡𝑃_𝜅𝜆−1

𝛼𝑃_𝜇^𝜅𝑃_𝜈^𝜆𝐷_(𝜇𝛽_𝜈) =

= 1

2𝛼𝑃̇_𝜇𝜈− 1

2𝛼𝑃_𝜇^𝜅𝑃_𝜈^𝜆(𝐷_𝜅𝛽_𝜆+ 𝐷_𝜅𝛽_𝜆) =

= 1

2𝛼(𝑃̇_𝜇𝜈 − 𝐷_𝜇𝛽_𝜈 − 𝐷_𝜇𝛽_𝜈).

which in adapted coordinates reduces to 𝛾̇_𝑖𝑗 = 2𝛼𝐾_𝑖𝑗+ 2𝐷_(𝑖𝛽_𝑗).

2.3 Canonical momenta

The relation (28) shows that the extrinsic curvature acts, in a sense, like the generalized velocity of the spatial metric. Using this relation, we can eliminate 𝐾_𝑖𝑗 in favor of 𝛾̇_𝑖𝑗, 𝛼 and 𝛽^𝑖 in the ADM-Lagrangian.

In so doing we obtain the desired time derivative 𝛾̇_𝑖𝑗, allowing us to compute the canonical momentum 𝜋^𝑖𝑗 conjugate to the field variable 𝛾_𝑖𝑗. The calculation is straightforward (p. 67 in [2]), and the result is

𝜋^𝑖𝑗 ≔𝛿𝐿_ADM

𝛿𝛾̇_𝑖𝑗 = √𝛾(𝐾^𝑖𝑗 − 𝐾𝛾^𝑖𝑗).

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By taking the trace, we can invert this to get 𝐾_𝑖𝑗 = 1

√𝛾(𝜋_𝑖𝑗 −1

2𝛾_𝑖𝑗π) , π ≔ 𝜋_𝑖^𝑖.

Since no 𝛼̇ or 𝛽 ̇^𝑖 feature in the 𝐿_ADM, their canonical momenta vanish, Π ≔𝛿𝐿_ADM

𝛿𝛼̇ = 0, Π_𝑖 ≔ 𝛿𝐿_ADM 𝛿𝛽 ̇^𝑖 = 0.

This shows (through the Euler-Lagrange equations) that 𝛼 and 𝛽^𝑖 are 𝑛𝑜𝑡 dynamical variables, their evolution being completely arbitrary.

2.4 Einstein’s equations as Hamilton’s equations

By substituting 𝛾̇_𝑖𝑗, 𝛼 and 𝛽^𝑖 for 𝐾_𝑖𝑗 we can take (𝛾_𝑖𝑗, 𝜋^𝑖𝑗, 𝛼, Π, 𝛽^𝑖, Π_𝑖) as canonical variables for our theory. From the formula for 𝜋^𝑖𝑗 we may then perform a Legendre transform on 𝐿_ADM to obtain the 𝐴𝐷𝑀 - 𝐻𝑎𝑚𝑖𝑙𝑡𝑜𝑛𝑖𝑎𝑛

𝐻_ADM ≔ ∫ 𝑑³𝑥

Σ

(𝜋^𝑖𝑗𝛾̇_𝑖𝑗 + Π𝛼̇ + Π_𝑖𝛽 ̇^𝑖) − 𝐿_ADM =

= ∫ 𝑑³𝑥

Σ

√𝛾[−𝛼(ℜ + 𝐾²− 𝐾_𝑖𝑗𝐾^𝑖𝑗) + 2(𝐾^𝑖𝑗− 𝐾𝛾^𝑖𝑗)𝐷_𝑖𝛽_𝑗],

where we now view 𝐾_𝑖𝑗 as given by (30). Partially integrating the second term, and discarding the surface contribution, the Hamiltonian becomes

𝐻_ADM =

= − ∫ 𝑑³𝑥

Σ

√𝛾[𝛼(ℜ + 𝐾²− 𝐾_𝑖𝑗𝐾^𝑖𝑗) − 2𝛽_𝑖𝐷_𝑗(𝐾^𝑖𝑗− 𝐾𝛾^𝑖𝑗)] ≡

≡ − ∫ 𝑑³𝑥

Σ

√𝛾[𝛼ℰ + 2𝛽^𝑖ℳ_𝑖].

where we recognize

ℰ ≔ ℜ + 𝐾²− 𝐾_𝑖𝑗𝐾^𝑖𝑗 =𝛿𝐻_ADM 𝛿𝛼 , ℳ_𝑖 ≔ 𝐷^𝑗(𝐾𝛾_𝑖𝑗− 𝐾_𝑖𝑗) = 𝛿𝐻_ADM

𝛿𝛽^𝑖 ,

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Hamiltonian evolution of space 13

as the energy and momentum constraints respectively. In this fashion 𝛼 and 𝛽^𝑖 play the role of Lagrange multipliers and as such are freely specifiable. This comes as no surprise, since a free choice of lapse and shift was just diffeomorphism invariance in disguise.

Having found a Hamiltonian for space, we now complete our program by calculating the equations of motion for the dynamical variables 𝛾_𝑖𝑗 and 𝜋^𝑖𝑗,

𝛾̇_𝑖𝑗 =𝛿𝐻_ADM

𝛿𝜋^𝑖𝑗 = −2𝛼

√𝛾(𝜋_𝑖𝑗−1

2𝛾_𝑖𝑗π) + 2𝐷_(𝑖𝛽_𝑗), 𝜋̇^𝑖𝑗 = −𝛿𝐻_ADM

𝛿𝛾_𝑖𝑗 =

= −𝛼√𝛾𝔊^𝑖𝑗 + 2𝛼

√𝛾𝛾^𝑖𝑗(𝜋^𝑘𝑙𝜋_𝑘𝑙−1

2π²) − 4𝛼

√𝛾(𝜋^𝑖𝑘𝜋_𝑘^𝑗−1 2π𝜋^𝑖𝑗) + √𝛾(𝐷^𝑖𝐷^𝑗− 𝜋^𝑖𝑗𝐷_𝑘𝐷^𝑘)𝛼 + √𝛾𝐷_𝑘( 1

√𝛾𝜋^𝑖𝑗𝛽^𝑘)

− 𝜋^𝑖𝑘𝐷_𝑘𝛽^𝑗− 𝜋^𝑗𝑘𝐷_𝑘𝛽^𝑖,

where 𝔊^𝑖𝑗 ≔ ℜ^𝑖𝑗 −¹₂ℜ is the spatial Einstein curvature. The latter calculation is rather involved (pp. 142-144 in [6]) and left out here.

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3. Applications

3.1 Numerical relativity

Einstein’s field equations are a system of coupled, nonlinear partial differential equations that are notoriously difficult to solve. Only by imposing symmetries on the metric or by perturbation around a known metric can one hope to find exact solutions. If those assumptions are untenable, one could attempt to solve the equations on a supercom- puter. In recent years, 𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑖𝑡𝑦 has become a hot topic due to the emergence of faster hardware and novel computational tech- niques that improve stability and convergence [3]. In this context, the 3+1-formalism outshines its manifestly covariant counterpart.

Numerical solutions to differential equations are best done in steps.

The upshot of the 3+1-formalism is that it rephrases Einstein’s equa- tions, which are not evolution equations, as an initial value problem for space. Before evolving the geometry of space, however, a gauge choice of lapse and shift has to be made. For computational reasons, some choices are wiser than others, and the demand for good gauges (in particular, those that avoid blow-ups) has resulted in a vast cata- logue of 𝑠𝑙𝑖𝑐𝑖𝑛𝑔 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 on 𝛼 and 𝑠ℎ𝑖𝑓𝑡 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 on 𝛽^𝑖 (a thor- ough discussion can be found in Ch. 4 of [3]).

For slicing, a naïve choice is 𝑔𝑒𝑜𝑑𝑒𝑠𝑖𝑐 𝑠𝑙𝑖𝑐𝑖𝑛𝑔, 𝛼 = 1. (In this gauge, Eulerian observers are freely falling, hence the name.) Geodesic slicing is simple but could lead to trouble; for in a non-uniform gravitational field free observers fall at different rates, and so their geodesics have a tendency to cross, which for geodesic slicing results in a coordinate singularity. In order to fix this, one notices from (14) that the diver- gence of the 4-velocity 𝑛^𝜇 (which gives the rate of change of which 4- volumes carried by Eulerian observers) is ∇_𝜇𝑛^𝜇 = 𝐾. For geodesic slic- ing, the equations (30) and (36)-(37) imply (under a strong energy condition on 𝑇^𝜇𝜈) that 𝐾̇ > 0, which means that the 4-volumes carried by Eulerian observers must go to zero. A failsafe gauge in this regard is 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑠𝑙𝑖𝑐𝑖𝑛𝑔, which is precisely defined by 𝐾 = 𝐾̇ = 0. (The name is stems from the fact that, locally, spatial slices in this gauge have maximal volume—space is a ‘maximal hypersurface’) Maximal slicing has the advantage of avoiding physical singularities and is a standard choice among practitioners.

As for shift conditions, 𝛽^𝑖= 0 often works, but there are exceptions.

For an evolving black hole, a zero shift causes the horizon to expand as more and more Eulerian observers fall inside, eventually bringing the entire computational grid into the black hole. An outward radial shift is thus used to keep timelines from passing the horizon. Also, for

Applications 15

fast spinning objects like pulsars, frame-dragging can be so severe that the spatial metric develops ever growing shears until the simulation crashes. To counteract this, one can corotate space with a nonzero 𝛽^𝑖. There are many other interesting connections with the 3+1-formalism to numerical relativity, including questions of well-posedness and sta- bility, which are treated in detail by [3], [7] and [8].

3.2 Global charges and the ADM-mass

It is well known that spacetime does not admit a 𝑙𝑜𝑐𝑎𝑙 notion of grav- itational energy (or mass) [9]. On the other hand, one can define the 𝑡𝑜𝑡𝑎𝑙 energy of an ‘isolated system’ in a meaningful way. In fact, one energy/mass 𝑀 prominently features in the metric of a Schwarzschild black hole, through its relation to the Schwarzschild radius, 𝑟_𝑠 = 2𝑀 . But what is the actual justification for calling the parameter 𝑀 in the Schwarzschild metric the ‘mass’ of the black hole? One of several an- swers to this question was found with help of the ADM-formalism. The discussion given here is a summary of Ch. 7 in [2].

In the derivation of the ADM-Hamiltonian we discarded the gradient piece ∇_𝛼(𝑛^𝛽∇_𝛽𝑛^𝛼− 𝑛^𝛼∇_𝛽𝑛^𝛽) in the Einstein-Hilbert action on the grounds that it amounts to a surface term which vanishes upon varying the action. Let us instead integrate over a finite spacetime volume 𝒱 and keep this extra term on the boundary 𝜕𝒱. For this to make sense we need some hypotheses. We assume that spacetime is asymptotically flat⁵ and that the induced spatial boundary 𝒮_𝜆 = 𝜕𝒱 ∩ Σ_𝜆 is homeo- morphic to a 2-sphere. Then, the new ADM-Hamiltonian is (p. 106 of [2])

𝐻_ADM = − ∫ 𝑑³𝑥

Σ_𝜆^(𝑟)

√𝛾[𝛼ℰ − 2𝛽^𝑖ℳ_𝑖]

− 2 ∫ 𝑑²𝑦

𝒮_λ

√𝑞[𝛼(𝜅 − 𝜅₀) − 2𝛽^𝑖(𝐾_𝑖𝑗− 𝐾𝛾_𝑖𝑗)𝜈^𝑗].

where Σ_𝜆^(𝑟) is that part of Σ_𝜆 bounded by the radius of 𝒮_λ, 𝜅 and 𝜅₀ are traces of the extrinsic curvature of 𝒮_λ embedded in (Σ_𝜆, 𝛾_𝑖𝑗) and (Σ_𝜆, 𝛿_𝑖𝑗) respectively, 𝜈^𝑖 is the unit normal to 𝒮_λ in (Σ_𝜆, 𝛾_𝑖𝑗) pointing out toward the asymptotic region, and 𝑑²𝑦√

𝑞 is the induced surface element. For vacuum solutions to Einstein’s equations the first term is zero. If we then move the sphere out to infinity and use our gauge freedom setting 𝛼 = 1 and 𝛽^𝑖 = 0, we can define the 𝐴𝐷𝑀 -𝑚𝑎𝑠𝑠 as

5 An 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑓𝑙𝑎𝑡 spacetime is one which, in a technical sense, approaches Minkowski spacetime at ‘large distances.’ For a formal definition, see [2].

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[DOKUMENTETS RUBRIK] 2

𝑀_ADM ≔ − 1 8𝜋 lim

𝑟→∞∫ 𝑑²𝑦

𝒮_λ

√𝑞(𝜅 − 𝜅₀).

The prefactor is conventional. Evaluating 𝜅 and 𝜅₀ in asymptotically Cartesian coordinates explicitly gives

𝑀_ADM= 1 16𝜋 lim

𝑟→∞∫ 𝑑²𝑦

𝒮_λ

√𝑞𝜈^𝑘(𝛿^𝑖𝑗𝜕_𝑖𝛾_𝑗𝑘+ 𝜕_𝑘𝛾^𝑗_𝑗).

For the Schwarzschild metric, which is asymptotically flat, the integral (40) is comparatively easy to do due to the spherical symmetry. The result is 𝑀_ADM = 𝑀 (p. 110 in [2]).

For stability it is important that the ADM-mass has a lower bound, at least for physically reasonable energy-momentum distributions [9].

If the mass could take on arbitrary negative values, it would be an infinite supply of gravitational radiation. Put forward in 1960 by ADM [10], this 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑡ℎ𝑒𝑜𝑟𝑒𝑚 remained an open conjecture for almost two decades until proved in 1979 by Schoen and Yau [11] as- suming a dominant energy condition on 𝑇^𝜇𝜈.

Other global charges can be defined too. From (38) we can deduce a reasonable definition for the 𝐴𝐷𝑀 -𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚

𝑃_ADM^𝑖 ≔ 1 8𝜋 lim

𝑟→∞∫ 𝑑²𝑦

𝒮_λ

√𝑞𝜈^𝑗(𝛿^𝑖𝑘𝐾_𝑘𝑗− 𝛿^𝑖_𝑗𝐾)

and also the 𝐴𝐷𝑀 -𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝐽_ADM^𝑖 ≔ 1

8𝜋 lim

𝑟→∞∫ 𝑑²𝑦

𝒮_λ

√𝑞𝜖^𝑖𝑗𝑘𝑥_𝑗𝜈^𝑚(𝛿_𝑘^𝑙𝐾_𝑙𝑚− 𝛿_𝑘𝑚𝐾).

These quantities are indeed the Noether charges that generate spatial translations and rotations at spacelike infinity.

3.3 Canonical quantization of gravity

Quantum physics and classical general relativity are at odds. The task of reconciling the two—the problem of ‘quantum gravity’—is consid- ered by many to be the most significant open problem in theoretical physics today.

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Applications 17