3.2 Preferred-frame effects and non-conservation 3 Tests of GR - 3 Tests of GR -

3.1 Strong Equivalence Principle: Nordtvedt effect 

The possibility of direct tests of the SEP through Lunar Laser Ranging (LLR) experiments was first pointed out by Nordtvedt [104]. As the masses of Earth and the Moon contain different fractional contributions from self-gravitation, a violation of the SEP would cause them to fall differently in the Sun's gravitational field. This would result in a ``polarization'' of the orbit in the direction of the Sun. LLR tests have set a limit of tex2html_wrap_inline2943 (see, e.g., [45, 147]), where tex2html_wrap_inline2595 is a combination of PPN parameters:

  equation302

The strong-field formalism instead uses the parameter tex2html_wrap_inline2947  [41Jump To The Next Citation Point In The Article], which for object `` i '' may be written as

  eqnarray316

Pulsar-white-dwarf systems then constrain tex2html_wrap_inline2949  [41Jump To The Next Citation Point In The Article]. If the SEP is violated, the equations of motion for such a system will contain an extra acceleration tex2html_wrap_inline2951, where tex2html_wrap_inline2603 is the gravitational field of the Galaxy. As the pulsar and the white dwarf fall differently in this field, this tex2html_wrap_inline2951 term will influence the evolution of the orbit of the system. For low-eccentricity orbits, by far the largest effect will be a long-term forcing of the eccentricity toward alignment with the projection of tex2html_wrap_inline2603 onto the orbital plane of the system. Thus, the time evolution of the eccentricity vector will not only depend on the usual GR-predicted relativistic advance of periastron (tex2html_wrap_inline2773), but will also include a constant term. Damour and Schäfer [41Jump To The Next Citation Point In The Article] write the time-dependent eccentricity vector as

equation341

where tex2html_wrap_inline2607 is the tex2html_wrap_inline2773 -induced rotating eccentricity vector, and tex2html_wrap_inline2605 is the forced component. In terms of tex2html_wrap_inline2597, the magnitude of tex2html_wrap_inline2605 may be written as [41Jump To The Next Citation Point In The Article, 145Jump To The Next Citation Point In The Article]

  equation356

where tex2html_wrap_inline2971 is the projection of the gravitational field onto the orbital plane, and tex2html_wrap_inline2973 is the semi-major axis of the orbit. For small-eccentricity systems, this reduces to

  equation369

where M is the total mass of the system, and, in GR, F = 1 and G is Newton's constant.

Clearly, the primary criterion for selecting pulsars to test the SEP is for the orbital system to have a large value of tex2html_wrap_inline2981, greater than or equal to tex2html_wrap_inline2983  [145Jump To The Next Citation Point In The Article]. However, as pointed out by Damour and Schäfer [41Jump To The Next Citation Point In The Article] and Wex [145Jump To The Next Citation Point In The Article], two age-related restrictions are also needed. First of all, the pulsar must be sufficiently old that the tex2html_wrap_inline2773 -induced rotation of tex2html_wrap_inline2987 has completed many turns and tex2html_wrap_inline2607 can be assumed to be randomly oriented. This requires that the characteristic age tex2html_wrap_inline2701 be tex2html_wrap_inline2993, and thus young pulsars cannot be used. Secondly, tex2html_wrap_inline2773 itself must be larger than the rate of Galactic rotation, so that the projection of tex2html_wrap_inline2603 onto the orbit can be assumed to be constant. According to Wex [145Jump To The Next Citation Point In The Article], this holds true for pulsars with orbital periods of less than about 1000 days.

  

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Figure 4: ``Polarization'' of a nearly circular binary orbit under the influence of a forcing vector tex2html_wrap_inline2603, showing the relation between the forced eccentricity tex2html_wrap_inline2605, the eccentricity evolving under the general-relativistic advance of periastron tex2html_wrap_inline2607, and the angle tex2html_wrap_inline2609 . (After [145Jump To The Next Citation Point In The Article].)

Converting Equation (16Popup Equation) to a limit on tex2html_wrap_inline2597 requires some statistical arguments to deal with the unknowns in the problem. First is the actual component of the observed eccentricity vector (or upper limit) along a given direction. Damour and Schäfer [41Jump To The Next Citation Point In The Article] assume the worst case of possible cancellation between the two components of tex2html_wrap_inline2987, namely that tex2html_wrap_inline3011 . With an angle tex2html_wrap_inline2609 between tex2html_wrap_inline2971 and tex2html_wrap_inline3017 (see Figure  4), they write tex2html_wrap_inline3019 . Wex [145Jump To The Next Citation Point In The Article, 146Jump To The Next Citation Point In The Article] corrects this slightly and uses the inequality

  equation420

where tex2html_wrap_inline3021 . In both cases, tex2html_wrap_inline2609 is assumed to have a uniform probability distribution between 0 and tex2html_wrap_inline3025 .

Next comes the task of estimating the projection of tex2html_wrap_inline2603 onto the orbital plane. The projection can be written as

  equation433

where i is the inclination angle of the orbital plane relative to the line of sight, tex2html_wrap_inline3031 is the line of nodes, and tex2html_wrap_inline3033 is the angle between the line of sight to the pulsar and tex2html_wrap_inline2603  [41Jump To The Next Citation Point In The Article]. The values of tex2html_wrap_inline3033 and tex2html_wrap_inline3039 can be determined from models of the Galactic potential (see, e.g., [83Jump To The Next Citation Point In The Article, 1Jump To The Next Citation Point In The Article]). The inclination angle i can be estimated if even crude estimates of the neutron star and companion masses are available, from statistics of NS masses (see, e.g., [136Jump To The Next Citation Point In The Article]) and/or a relation between the size of the orbit and the WD companion mass (see, e.g., [114]). However, the angle tex2html_wrap_inline3031 is also usually unknown and also must be assumed to be uniformly distributed between 0 and tex2html_wrap_inline3025 .

Damour and Schäfer [41Jump To The Next Citation Point In The Article] use the PSR B1953+29 system and integrate over the angles tex2html_wrap_inline2609 and tex2html_wrap_inline3031 to determine a 90% confidence upper limit of tex2html_wrap_inline3051 . Wex [145] uses an ensemble of pulsars, calculating for each system the probability (fractional area in tex2html_wrap_inline2609 - tex2html_wrap_inline3031 space) that tex2html_wrap_inline2597 is less than a given value, and then deriving a cumulative probability for each value of tex2html_wrap_inline2597 . In this way he derives tex2html_wrap_inline3061 at 95% confidence. However, this method may be vulnerable to selection effects; perhaps the observed systems are not representative of the true population. Wex [146Jump To The Next Citation Point In The Article] later overcomes this problem by inverting the question. Given a value of tex2html_wrap_inline2597, an upper limit on tex2html_wrap_inline3065 is obtained from Equation (17Popup Equation). A Monte Carlo simulation of the expected pulsar population (assuming a range of masses based on evolutionary models and a random orientation of tex2html_wrap_inline3031) then yields a certain fraction of the population that agree with this limit on tex2html_wrap_inline3065 . The collection of pulsars ultimately gives a limit of tex2html_wrap_inline3071 at 95% confidence. This is slightly weaker than Wex's previous limit but derived in a more rigorous manner.

Prospects for improving the limits come from the discovery of new suitable pulsars, and from better limits on eccentricity from long-term timing of the current set of pulsars. In principle, measurement of the full orbital orientation (i.e., tex2html_wrap_inline3031 and i) for certain systems could reduce the dependence on statistical arguments. However, the possibility of cancellation between tex2html_wrap_inline3077 and tex2html_wrap_inline3079 will always remain. Thus, even though the required angles have in fact been measured for the millisecond pulsar J0437-4715 [139Jump To The Next Citation Point In The Article], its comparatively large observed eccentricity of tex2html_wrap_inline3083 and short orbital period mean it will not significantly affect the current limits.



3.2 Preferred-frame effects and non-conservation 3 Tests of GR - 3 Tests of GR -

image Testing General Relativity with Pulsar Timing
Ingrid H. Stairs
http://www.livingreviews.org/lrr-2003-5
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