Ariadna ID 07/1301 AO/1-5582/07/NL/CB
Non-perturbative Effects of Rotation in Gravitationally Bound Systems
H. Balasin1 and D. Grumiller2
Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8–10/136, Vienna, A-1040, Austria
Abstract
We plan to explore in detail a possible breakdown of the Newtonian approximation to gravity on large scales in some region where grav- ity is weak and matter sources are rotating. The main insight gained from previous investigations is quite general: the Newton approxi- mation can break down in an extended (non-vacuum) region, even though it might be applicable locally everywhere. This has poten- tial consequences for galactic rotation curves, which we shall study in detail. The long-term goal is to establish a quasi-Newtonian approxi- mation scheme which takes into account this non-Newtonian behavior, but allows for simpler techniques than solving the full set of Einstein equations. Such global effects may also induce changes in the local dy- namics at smaller scales and might therefore be accessible to precision experiments.
1e-mail: hbalasin@tph.itp.tuwien.ac.at
2e-mail: grumil@hep.itp.tuwien.ac.at; temporary address: Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139
1 Introduction
On large scales gravity is the most relevant force in the Universe. Certainly, it also dominates our everyday life. For a long time Newtonian gravity was sufficient to describe all gravitational interactions, but we know now that it is not accurate for strong fields or large velocities. Rather, in these regimes the Einstein theory of General Relativity (GR) must be used, which so far has passed all experimental test with ever increasing precision [1].
In the limit of weak fields and small velocities Einstein’s equations may be approximated locally by Newton’s gravity law. This is quite convenient, because Einstein’s equations are a system of coupled non-linear partial dif- ferential equations which are prohibitively difficult to solve in full generality.
Therefore, in many applications the Newtonian limit is invoked as a first ap- proximation, with GR corrections taken into account perturbatively. How- ever, the local applicability of the Newton approximation at each point in some region of space-time need not guarantee automatically its global appli- cability in the same region. The physical system we have in mind is a galaxy, but evidently the question addressed here is of general relevance for gravitat- ing systems. In particular, the galactic rotation curves are usually described in a Newtonian way, and the discrepancy with observation [2, 3] is one of the original reasons for the prediction of dark matter, cf. e.g. [4]. More recent astrophysical observations, such as the discovery of cosmic acceleration [5, 6]
together with the data from WMAP [7], strongly suggest the presence of dark energy, which comprises nearly three quarters of the energy content of the Universe. These results may also be interpreted as an indication that at large scales some modification of the underlying gravitational theory is required.
For these reasons — as explained also in the Ariadna Call 07/1301 [8] — it is of considerable interest to understand clearly to what extent the Newtonian approximation is justified on large scales for rotating gravitationally bound systems, and how this might effect the local dynamics within such a system.
2 Physical assumptions and main goal
In order to achieve anything – whether by analytical calculations or numerical simulations – we have to employ some simplifying assumptions, both in the GR and the Newtonian calculations. We plan to focus on axially symmetric
and stationary solutions of the Einstein equations, Rab− 1
2gabR=κTab, (1)
with the energy momentum tensor of a perfect fluid
Tab =ρuaub+P(gab+uaub), (2) where ρ, P, ua are mass density, pressure and 4-velocity, respectively. All other symbols have their standard meaning (Rab, R, gab, κ are Ricci tensor, Ricci scalar, metric and gravitational coupling, respectively).
The main goal is to exhibit a discrepancy between GR and Newtonian cal- culations in the presence of rotation for realistic models, i.e., to demonstrate that there is a non-perturbative effect from “backreactions”. This addresses the first point in [8]. In the language of [8] we proposae to follow a top-down approach.3
So we want to extend and supplement the discussion in [9] (this is cited as Ref. [20] in [8]), which is the pre-cursor of this proposal. That work demanded vanishing pressure, P = 0, as well as co-rotation of the matter sources, i.e., the 4-velocity ua is proportional to the timelike Killing vector responsible for stationarity. In [9] the basic effect was pointed out; however, the model itself is not fully realistic and does not describe real galaxies. By relaxing some of the working assumptions we would like to improve the quality of the model and thereby also its credibility. In particular, we would like to allow for non-vanishing pressure and to drop the assumption of co-rotation.
However, we would like to keep stationarity and axi-symmetry in order to keep the system of equations manageable. This will allow us to exploit the generic Ansatz for a line element respecting these symmetries [10],
ds2 =−V (dt−wdφ)2 +V−1r2dφ2+ Ω2 dr2+ Λdz2
, (3)
where the functions V, w, Ω and Λ all depend on r and z, only.
3 Special case considered previously
We recall now briefly how we proceeded for the simpler case of vanishing pressure and co-rotating matter sources [9]. First, we were able to eliminate
3While the bottom-up approach mentioned in [8] certainly would provide interesting complementary insights, it appears more reasonable to us to focus on the top-down ap- proach, given the time-scale of four months.
the functions Λ and V. Second, we noticed that the function w obeys a linear second order partial differential equation, which we could solve using standard methods. Third, we exploited the standard feature [10] that once all functions in (3) but Ω are known, the latter can be determined by a line integral, up to a multiplicative integration constant. Finally, we could calcu- late the density profile ρalgebraically from all other functions. Interestingly, the function wis determined uniquely by the velocity profile, and vice versa.
This allowed us to set up an inverse problem in a simple way: take some galactic rotation curve as input, determine w and Ω (up to a constant), pre- dict from this input the density profile ρ and compare with experiment. It turned out that the density profile was not fully realistic, but neither was it completely unrealistic, so there is a good chance that an improved version of our algorithm provides better results. Two natural ways to improve the model is to drop the assumptions of vanishing pressure and of co-rotation, as planned in the proposed project.
However, the main point in [9] was not the construction of a fully realistic model of a galaxy, but rather to point out an important difference to purely Newtonian considerations. To this end we took the same velocity profile as input in a GR and a Newtonian calculation and derived the density profile in the galactic plane in both approaches. We found that locally we can always achieve agreement between the GR and the Newtonian calculation, which is hardly surprising given that velocities are small and gravitational fields are weak. The same was true globally, i.e., for any value of the radial coordinate r, if the velocity profile did not change its features abruptly – e.g. for a velocity profile which is linear in r or constant for all values of r.
Surprisingly, we found that the global equivalence ceases to hold if there is some change in the velocity profile. Then we obtained for the ratio between the GR prediction ρ and the Newtonian prediction ρN the simple result
ρ ρN
=β
1 + r2(V0)2 V2+ 2rV V0
, (4)
whereβ is an adjustable integration constant and V(r) is the velocity profile in the galactic plane. The most relevant case here is a velocity profile which is linearly rising in the central region (around r = 0) and then flattens at some critical radius r0. For that case we found that Newtonian calculations overestimate the amount of matter required to produce this velocity profile by 30% as compared to the GR calculations.
4 Project implementation and main questions
It is not completely straightforward to implement the algorithm outlined above for the more general case, P 6= 0 and in the absence of co-rotation.
In particular, it is very likely that we are going to need a computer algebra system such as Maple together with GRTensor [11] to deal with the increased complexity of the Einstein equations. Therefore, we plan to collaborate with a junior sub-contractor who has good experience with the software required and with GR, namely Nicolas Yunes, currently at Penn State University.
One of us (DG) has been collaborating with him since May 2006, with a first publication coming out soon [12]. Since DG will remain temporarily at MIT the collaboration with Nicolas Yunes will be quasi-local, with mutual visits to be sponsored from other resources. Therefore, no additional costs will arise from the participation of Nicolas Yunes. As most of the computer work will be required in the early stages of the project, the timing for DGs return to Vienna in Spring 2008 is advantageous for the project. After DGs return the collaboration between the main participants of the proposal (HB, DG and our junior contractor in Vienna) will become more intense, but we stress that even before we shall stay in permanent contact by e-mail to expedite the development of the project. The temporary split of the senior collaborators, HB and DG, is actually a bonus, because each of us can focus on the respective junior collaborator.
Once the equations are sufficiently tamed by the computer algebra system we should be capable to answer the following main questions:
1. Is there again a discrepancy (at the 30%-level) between GR and New- tonian calculations?
2. Can this discrepancy be isolated, in the sense that a quasi-Newtonian calculation can be employed instead of the full GR calculation?
3. Is it again possible to provide a velocity profile as input and to derive a matter density profile as output?
4. If so, does that density profile lead to predictions which agree with observations?
5. If not, which of the remaining assumptions (axi-symmetry, stationarity, perfect fluid matter sources) should be relaxed?
While the last three points are somewhat specific to galactic physics, the first one, if answered affirmative, together with the second point are of broader appplicability and relevance. In particular, numerical simulations which em- ploy the Newtonian approximation could then be improved in their quality without becoming too unwieldy. However, the detailed study of such appli- cations clearly goes beyond the scope of the current project.
5 Supplementary considerations
An additional task of pedagogical nature is to clarify the mutually contradict- ing claims that exist in the recent literature on similar issues. In particular, the technically wrong results presented in [13] inspired numerous comments and counter-comments cf. e.g. [13–27] (many of these papers are cited in [8]).
Also, some claims that this effect may replace dark matter could not be substantiated by our own study [9], and seem unlikely on general grounds, because dark matter can be inferred also from other observations, such as structure formation, gravitational lensing, clusters, etc. However, it appears to be true that such an effect slightly reduces the amount of dark matter required to explain the galactic rotation curves. As astrophysical measure- ments reach unprecedented accuracy, these corrections could be of relevance for the precise determination of the cosmological parameters. We would like to investigate whether this remains true if more realistic assumptions are considered, such as the ones explained above. This addresses the second issue in [8].
Conceptually similar problems arise in cosmology, where the averaging over inhomogeneous matter distributions leads to relevant corrections to the averaged Einstein equations and their Newtonian limit (there is a lot of recent literature available on this subject; for an overview cf. e.g. [28–31]).
Where applicable, a connection to the effects observed in rotating matter distributions will be explored, and the relation to these other approaches should be clarified. This addresses partially the third issue in [8].
Finally, we want to clarify to what extent these effects play any role for solar system tests. We are slightly pessimistic concerning this point, because on solar system scales the gravitational sources are essentially isolated and therefore approximated quite well by point particles. On the other hand, there is an abundance of very accurate data, so we cannot completely exclude the possibility of some effects which might be accessible to high precision
experiments. This also addresses partially the third issue in [8].
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