Abstract
General Relativity offers the possibility to model attributes of matter, like mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed only with a single field that represents its Riemannian geometry. I review this picture of ‘Geometrodynamics’ and comment on various developments after Einstein.
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Notes
- 1.
‘Es muß also entweder das dem Raume zugrude liegende Wirkliche eine diskrete Mannigfaltigkeit bilden, oder der Grund der Maßverhältnisse außerhalb, in darauf wirkenden bindenden Kräften gesucht werden.’ (Riemann 1869/1919, 20)
- 2.
Due to the standard convention that the Hamiltonian action is defined as a left action, whereas the Lie bracket on a group is defined by the commutator of left-invariant vector fields which generate right translations.
- 3.
In this article, we use ‘iff’ as abbreviation for ‘if and only if’.
- 4.
The maximal cone touches the three 2-planes λ i = 0 at the bisecting lines λ j = λ k , where i, j and k are any of the three cyclic permutations of 1, 2 and 3. Hence the cosine of the opening angle is the scalar product between \((1,1,1)\sqrt {3}\) and, say, \((1,1,0)\sqrt {2}\), which is \(\sqrt {2/3}\).
- 5.
Had we done the very same analysis in terms of π rather than K, we would have found that in eigenvalue space (now of the endomorphism \(\tilde \pi \)) the opening angle of the cone inside which the bilinear form is negative definite and outside which it is positive definite is now precisely the maximal one \(\arccos (\sqrt {2/3})\) (see previous footnote). Indeed, rewriting the bilinear form in terms of π using (12.18), it is positively proportional to \({\mbox{tr}}(\tilde \pi ^2)-\frac {1}{2}({\mbox{tr}}(\tilde \pi ))^2\). It is the one-half in front of the second term that causes this interesting coincidence.
- 6.
Note that the definition of the ADM mass (12.23) just depends on the Riemannian metric h and is independent of K. But for the theorem to hold, it is essential to require that h is such that there exists a K so that (h and K) satisfy the constraints. It is easy to write down metrics h with negative mass: Take, e.g. (12.29) with negative m for r > r ∗ > m∕2, smoothly interpolated within m∕2 < r < r ∗ to, say, the flat metric in r < m∕2. The positive mass theorem implies that for such a metric, no K can be found so that (h and K) satisfy the constraints.
- 7.
- 8.
That K is Killing is immediate, since (12.33) shows that ρ depends only on the combination X 2 − T 2 which is clearly annihilated by K.
- 9.
The \({\mathbb {R}\mathrm {P}}^3\) geon is different from the two mutually different ‘elliptic interpretations’ of the Kruskal spacetime discussed in the literature by Rindler, Gibbons, and others. In Rindler (1965) the identification map considered is J′ : (T, X, θ, φ)↦(−T, −X, θ, φ), which gives rise to singularities on the set of fixed-points (a 2-sphere) T = X = 0. Gibbons (1986) takes J″ : (T, X, θ, φ)↦(−T, −X, π − θ, φ + π), which is fixed-point free, preserves the Killing field (12.37) (which our map J does not), but does not preserve time-orientation. J″ was already considered in (Misner and Wheeler 1957, section 4.2), albeit in isotropic Schwarzschild coordinates already mentioned above, which only cover the exterior regions I and III of the Kruskal manifold.
- 10.
One uses the Killing identity
to convert the second derivatives of K
♭ into terms involving no derivatives and the Riemann tensor. - 11.
That is, linear momentum has the unit of length (like mass) and angular momentum of length-squared. They are converted into ordinary units through multiplication with c∕G.
- 12.
The distinction between electric and magnetic is conventional in Einstein-Maxwell theory without sources for the Maxwell field, since the energy-momentum tensor T for the latter is invariant under duality rotations which rotate between F and ⋆ F according to ω↦e(iφ) ω, where ω := F + i ⋆ F. Since
, where an overbar denotes complex conjugation, the invariance of T is immediate. - 13.
A classic theorem on finite abelian groups states that if pandq are integers, \({\mathbb {Z}}_{pq}\) is isomorphic to \({\mathbb {Z}}_p\oplus {\mathbb {Z}}_q\) iff p and q are coprime.
- 14.
This follows in two steps: First, one recalls \(H^2(Q,{\mathbb {Z}})\) is isomorphic to the direct sum of the free part of \(H_2(Q,{\mathbb {Z}})\) and the torsion part of \(H_1(Q,{\mathbb {Z}})\) (universal coefficient theorem). Second, one uses that \(H_1(Q,{\mathbb {Z}})\) is isomorphic to the abelianisation of the fundamental group (Hurewicz’ theorem).
- 15.
To decide this entails some subtle issues, like whether to diffeomorphisms that are homotopic (continuously connected through a one-parameter family of continuous maps) are also isotopic (continuously connected through a one-parameter family of homeomorphisms) and then also diffeotopic (continuously connected through a one-parameter family of diffeomorphims). The crucial question is whether homotopy implies isotopy, which is not at all obvious since on a homotopy the interpolating maps connecting two diffeomorphism are just required to be continuous, that is, they need not be continuously invertible as for an isotopy. For example, the inversion I(x) = −x in \({\mathbb {R}}^n\) is clearly not isotopic to the identity, but homotopic to it via ϕ t (x) = (1 − 2t)x for t ∈ [0, 1]. Then ϕ 0 = id, ϕ 1 = I and only at t = 1∕2 does the map ϕ t cease to be invertible.
- 16.
The definition of lens spaces L(p, q) in three dimensions is
, where (p, q) is a pair of positive coprime integers with p > 1, \(S^3=\{(z_1,z_2)\in {\mathbb {C}}^2\mid \vert z_1\vert ^2+\vert z_2\vert ^2=1\}\), and \((z_1,z_2)\sim (z^{\prime }_1,z^{\prime }_2)\Leftrightarrow z^{\prime }_1={\mbox{e}}(2\pi i/p)z_1\), and \(z^{\prime }_2={\mbox{e}}(2\pi i\,q/p)z_2\). One way to picture the space is to take a solid ball in \({\mathbb {R}}^3\) and identify each points on the upper hemisphere with points on the lower hemisphere after a rotation by 2πq∕p about the vertical symmetry axis. In this way, each set of p equidistant points on the equator is identified to a single point. The fundamental group of L(p, q) is \({\mathbb {Z}}_p\), i.e. independent of q, and the higher homotopy groups are those of its universal cover, S
3. This does, however, not imply that L(p, q) is homotopy equivalent, or even homeomorphic, to L(p, q). The precise relation will be stated below. - 17.
As regards the notion of chirality, an interesting refinement of this statement is that L(p, q) and L(p, q′) are orientation-preserving homeomorphic iff q′ = q ±1 (mod p) (Reidemeister 1935).
- 18.
Maxwell equations were thought of as a kind of effective theory that describes things on a coarse-grained scale, so that, e.g. the vortex knots could be approximated by point particles.
to convert the second derivatives of K
♭ into terms involving no derivatives and the Riemann tensor.
, where an overbar denotes complex conjugation, the invariance of T is immediate.
, where (p, q) is a pair of positive coprime integers with p > 1, References
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Acknowledgements
I sincerely thank the organisers of the Beyond Einstein conference at Mainz University for inviting me to this most stimulating and pleasant meeting.
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Giulini, D. (2018). Matter from Space. In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein. Einstein Studies, vol 14. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7708-6_12
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