Imprints from the global cosmological expansion to the local spacetime dynamics

Abstract

We study the general relativistic spacetime metrics surrounding massive cosmological objects, such as suns, stars, galaxies or galaxy clusters. The question addressed here is the transition of local, object-related spacetime metrics into the global, cosmological Robertson–Walker metrics. We demonstrate that the answer often quoted for this problem from the literature, the so-called Einstein–Straus vacuole, which connects a static outer Schwarzschild solution with the time-dependent Robertson–Walker universe, is inadequate to describe the local spacetime of a gravitationally bound system. Thus, we derive here an alternative model describing such bound systems by a metrics more closely tied to the fundamental problem of structure formation in the early universe and obtain a multitude of solutions characterising the time-dependence of a local scale parameter. As we can show, a specific solution out of this multitude is able to, as a by-product, surprisingly enough, explain the presently much discussed phenomenon of the PIONEER anomaly.

Appendix

A variant of the mean value theorem of integral calculus

When one is faced with an integral of the form

$$ \int_a^b f(x) \frac{dg(x)}{dx} dx, $$

(88)

where the functional form of the integrand is not completely known, one typically requires adequate approximations. However, at least in a situation where f(x) ≃ const., this integral is mostly independent of the fine structure within the interval a,b as the integration and the differentiation cancel out. Usually, an integral of this form is evaluated using the second mean value theorem of integral calculus,

$$\begin{array}{*{20}l} I &=& \int_a^b f(x) \frac{dg(x)}{dx} dx = f(\xi) \int_a^b \frac{dg(x)}{dx} dx\nonumber\\ &=& f(\xi) ( g(b\!) - g(a) ), \end{array}$$

(89)

with a ≤ c ≤ b. Now, without any more specific information, one is required to “guess” which value should be selected for c, where one possible, somewhat straightforward approximation is given by

$$ f(\xi) \simeq \frac{f(a) + f(b\!)}{2}, $$

(90)

leading to

$$ I \simeq \frac{f(a) + f(b\!)}{2} ( g(b\!) - g(a) ). \label{eq-int-average} $$

(91)

In a recent publication, (Fahr and Siewert 2006b), we have obtained precisely the same result using a different approximation, based on the decomposition

$$ f(x) = f(a) + (f(b\!) - f(a)) (c + h_s(x) + h_a(x)), \label{equ259} $$

(92)

and a similar approximation for g(x), with $c = \frac{1}{2}$, and the functions h _s (x) and h _a (x) normalised in way such that the boundary values f(a) and f(b) are reproduced. The functions h are defined using the symmetric–antisymmetric decomposition,

$$ f_{s,c}(x) = \frac{1}{2} ( f(x) + f(2d-x) ) \label{eq-def-fs-c} $$

(93)

and

$$ f_{a,c}(x) = \frac{1}{2} ( f(x) - f(2d-x) ), \label{eq-def-fa-c} $$

(94)

which fulfill the relations

$$ f_{s,c}(d + y) = f_{s,c}(d - y) $$

(95)

and

$$ f_{a,c}(d + y) = -f_{a,c}(d - y), $$

(96)

i.e. the functions are symmetric or antisymmetric with respect to x. Assuming that the decomposition defined by (92) does exist, one obtains an expression of the form

$$ I = (g(2) - g(1)) \int_a^b (f(a) + c (f(b\!) - f(a))) \left( \frac{dg_s}{dx} + \frac{dg_a}{dx} \right) + I_0(x) dx, $$

(97)

where d is the equivalent to c appearing in the function g(x), and the integral over I ₀(x) is zero on account of symmetry arguments (Fahr and Siewert 2006b), leaving us with the final result

$$ I = ((1 - c) f(a) + c f(b\!)) (g(b\!) - g(a)). $$

(98)

If we are faced with a situation where we have no specific knowledge of the numerical value of c, we may simply use c = 1/2, which then leads to Eq. 91. This relation is mostly independent of the mathematical or physical context, and it is by no means restricted to the plasma context where this relation was first applied (Fahr and Siewert 2006b). As demonstrated in the application presented in this study, this relation allows, in principle, to integrate a function which is only known numerically while still allowing to continue deriving analytical results, reducing the unknown, numerical contributions to a finite number of simple, numerical values.