Time from Space. Domenico Giulini. Parmenides Foundation München, April 30th ITP University of Hannover ZARM Bremen.

Transcription

1 ITP University of Hannover ZARM Bremen Parmenides Foundation München, April 30th /17

2 2/17

3 William Kingdon Clifford 1870 I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact: 1. That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them. 2. That this property of being curved or distorted is continually being passed from one portion of space to another after the manner of a wave. 3. That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal. 4. That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity. 3/17

4 4/17

5 Einstein s equation R µν 1 R gµν + Λ gµν = κtµν 2 Solutions are entire spacetimes (M, g). Spacetimes do not evolve; they are. No conditions as to sequential time ordering of instants, absence of closed timelike curves, causal evolution of geometry etc. are a priori specified. On the other hand, Einstein s equation is compatible with the imposition of such structures: Time orientability, ordering of spacetime as sequence of spaces, well defined initial-value formulation. 5/17

6 Intermezzo: Initial-value problems Dynamical equations in physics are usually given the form of initial-value problems. For this to make unambiguous sense, i.e. for the initial-value problem to be well posed, the equations must satisfy the following requirements. Specification. An identification and characterisation of structures that count as initial data should be possible. Existence. For each set of initial data there exists a development containing the data. Uniqueness. The development for each set of initial data is unique, possibly modulo redundancies in the formal labelling of states (i.e. gauge freedom). Stability. The map from initial data to their developments is continuous in a suitable sense; that is, small variations in the data imply small variations in their development. 6/17

7 Tame and wild spacetimes 7/17

8 Tame spacetime as space s history E t M Σ t Σ E t Σ t E t Σ t Spacetime, M, is foliated by a one-parameter family of embeddings E t of the 3-manifold Σ into M. Here t is a formal label without direct physical significance. Σ t is the image in M of Σ under E t. Each such Σ t is an instant. 8/17

9 A four-function worth of arbitrariness Σ t+dt p t αn Σ t p β For q Σ the image points p = E t(q) and p = E t+dt (q) are connected by the vector / t p whose components tangential and normal to Σ t are β (three functions) and αn (one function) respectively. 9/17

10 Timeless mechanics The law of motion for a set of n points with coordinates ( q 1,, q n) =: q under the influence of a potential V (q) can be formulated as a geodesic principle in configuration space R 3n with Riemannian metric g = (E V )g kin (Jacobi s principle): qf δ q`e V (q(λ)) T ` q(λ), q(λ) dλ = 0 (1) q i The parameter λ is irrelevant. Once initial and final positions have been specified, the duration of the physical journey (in Newtonian time) from q i to q f is determined via: s T ` q(λ), q(λ) Thus there is a map t(q i, q f ) = qf q i E V (q) dλ (2) t : R 3n R 3n R + (3) which, for given energy E, assigns to each pair of points in the configuration space the duration of the physical journey connecting them. 10/17

11 According to the evolutionary (Hamiltonian) form of Einstein s equation, spacetime is the evolution of space. This implies a large degree of redundancy (locally a four function worth of), since one and the same spacetime comprises very many different evolutions of space; as many as correspond to the freedom to waft 3-dim. space through 4-dim. spacetime. Physical configuration space is then given by the space of all geometries space can be endowed with; it is called. The dynamical law is then similar to a geodesic principle in that space: Spacetime = {sheaf of geodesics in } (4) gf δ dµq` R(g(λ)) G`g(λ), ġ(λ) ) dλ = 0 (5) g i Σ Jacobi-like principle for General Relativity (Baierlein, Sharp, and Wheeler 1962) 11/17

12 : Space of geometries (shapes and sizes) Riem(Σ) S(Σ)=Riem(Σ)/Diff(Σ) 12/17

13 h h vertical horizontal h Riem(Σ) [h] [h ] [h ] S(Σ) 13/17

14 Instants and time: partial results Often true: Two infinitesimally nearby 3-geometries, g i and g f, determine a unique maximal spacetime in which they occur as instants. Sometimes true: Two 3-geometries, g i and g f, determine a unique maximal spacetime in which they occur as instants. Their (finite) separation in time (measured by a physical observer with a clock) is fully determined. Sometimes true: Given a 3-geometry g i, its occurrence as instant (spacelike hypersurface) in a given spacetime satisfying Einstein s equation is locally, or even globally, unique. (Julian Barbour: Time is in the instant ). 14/17

15 The chronos principle (or ephemeris time) in GR - 1 Einstein s equation define on a geometric structure that, to some extent, allow to speak of distance. It is similar to the kinetic-energy metric seen above in Jacobi s principle. According to the chronos principle (Christodoulou 1974), global and local measures of physical time can be derived from global and local measures of distance on, and that local measures should coincide with global ones, so that... it is not necessary to look at the change in configuration of the entire universe in order to measure time. (D. Christodoulou 1974, p. 76). It turns out, that this principle is just realised in General Relativity, the condition for the equivalences of local times being implied by Einstein s equation (more precisely: the Hamiltonian constraint). 15/17

16 Chronos principle: The essential (simplified) picture Path length in measured by (from Einstein s equations) ds 2 = d 3 x G ab nm [g(x)]dg ab (x)dg nm(x) (6) Σ Physical time corresponding to it dτ 2 ds 2 = Σ d 3 x R(x) (7) Require it to be compatible with arbitrarily fine localisation Σ U Σ, which leads to G ab nm [g(x)] dg ab(x) dg nm(x) R[g](x) = 0 (8) dτ dτ Physical time is then given in terms of quantities by a Jacobi-like formula (compare (3) for E = 0): s gf G`ġ(λ), ġ(λ) τ(g i, g f ) = dλ (9) g i Rˆg(λ) 16/17

17 Related Thoughts Solutions to dynamical equations of motion in the form of (generalised) geodesic principles are subsets of (dynamically realised) configurations in the space of (kinematically possible) ones. These subsets are delivered to us in the form of unparametrised curves. So, even though the parameter does not matter, the structure of a one-dimensional sub-continuum remains. In particular one (or two) preferred orderings are selected. What is the significance of that? What makes us experience this solution configurations according to this order? Can we, on, characterise a function that structures it according to some definition of geometric entropy? How would its gradient flow be related to the dynamics of General Relativity? Suppose it were true that each 3-geometry had at most one isometric embedding in a spacetime satisfying Einstein s equation. What would be a better representation of the Now in an abstract physical theory? 17/17