Observations have found black holes spanning 10 orders of magnitude in mass across most of cosmic history. The Kerr black hole solution is, however, provisional as its behavior at infinity is incompatible with an expanding universe. Black hole models with realistic behavior at infinity predict that the gravitating mass of a black hole can increase with the expansion of the universe independently of accretion or mergers, in a manner that depends on the black hole’s interior solution. We test this prediction by considering the growth of supermassive black holes in elliptical galaxies over 0 < z ≲ 2.5. We find evidence for cosmologically coupled mass growth among these black holes, with zero cosmological coupling excluded at 99.98% confidence. The redshift dependence of the mass growth implies that, at z ≲ 7, black holes contribute an effectively constant cosmological energy density to Friedmann’s equations. The continuity equation then requires that black holes contribute cosmologically as vacuum energy. We further show that black hole production from the cosmic star formation history gives the value of ΩΛ measured by Planck while being consistent with constraints from massive compact halo objects. We thus propose that stellar remnant black holes are the astrophysical origin of dark energy, explaining the onset of accelerating expansion at z ∼ 0.7.
We show that derivation of Friedmann’s equations from the Einstein–Hilbert action, paying attention to the requirements of isotropy and homogeneity during the variation, leads to a different interpretation of pressure than what is typically adopted. Our derivation follows if we assume that the unapproximated metric and Einstein tensor have convergent perturbation series representations on a sufficiently large Robertson–Walker coordinate patch. We find the source necessarily averages all pressures, everywhere, including the interiors of compact objects. We demonstrate that our considerations apply (on appropriately restricted spacetime domains) to the Kerr solution, the Schwarzschild constant-density sphere, and the static de-Sitter sphere. From conservation of stress–energy, it follows that material contributing to the averaged pressure must shift locally in energy. We show that these cosmological energy shifts are entirely negligible for non-relativistic material. In relativistic material, however, the effect can be significant. We comment on the implications of this study for the dark energy problem.
Given a connected oriented manifold M" of dimension n and an immersion X : M" -~R "+k, there is associated to it a Gauss map g : M"~G (n, k), where G(n, k) is the Grassmann manifold of oriented n-planes in (n+k)-space. That is, there is associated to X a map g which assigns to each m ~ M the oriented n-plane tangent to X(M) at X(m). This is one way in which one may generalize the classical Gauss map for surfaces in 3-space. Suppose on the other hand, we are given a map g : M"-+G(n, k). Does there exist an immersion X: M"-.~ "+k with g as its Gauss map?Is even g locally a Gauss map and also to what extent does g uniquely determine X? In this paper, we study these questions for the case n = k= 2, however only under the additional restriction that g is supposed to be an immersion of M z into G(2, 2).The main results obtained are the following ones: In Theorem 1 we state a purely algebraic necessary condition on g for the existence of X with this g as its Gauss map. If this algebraic condition on g is fulfilled, the existence of X for g reduces-under certain "regularity" assumptions (a rank condition) -to solving a system of partial differential equations. We shall study this system if it is either of hyperbolic or of elliptic type. For the hyperbolic case we prove in Theorem 2 resp, 3 the existence resp, uniqueness of an immersion X with the given g as its Gauss map, if X is prescribed along a nowhere-characteristic curve in M (Cauchy problem). For the elliptic case we prove in Theorem 4 the existence of X with the given g as its Gauss map, if M is a closed disc, and in Theorem 5 we get for arbitrary M the uniqueness of such an X, if the values of X are prescribed along any arc in M, Some of these results are in a recent paper [1] by Aminov but his point of view and methods are very different from ours. (As a by-product of our investigations we prove in the Appendix a slight sharpening of a result of Chern and Spanier.) If~ is a vector bundle over M and m ~ M we let ~., denote the fibre over m. In the case of the tangent bundle of M, TM, we write M,. for (TM),.. IfS is a set and f: -~S is a map, we denote the restriction of f to ~,. by f,.. Also Ak({) denotes the
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