3.3 Strong Equivalence Principle: Dipolar 3 Tests of GR - 3.1 Strong Equivalence Principle: Nordtvedt

3.2 Preferred-frame effects and non-conservation of momentum 

3.2.1 Limits on tex2html_wrap_inline3085  

A non-zero tex2html_wrap_inline2821 implies that the velocity tex2html_wrap_inline3089 of a binary pulsar system (relative to a ``universal'' background reference frame given by the Cosmic Microwave Background, or CMB) will affect its orbital evolution. In a manner similar to the effects of a non-zero tex2html_wrap_inline2597, the time evolution of the eccentricity will depend on both tex2html_wrap_inline2773 and a term that tries to force the semi-major axis of the orbit to align with the projection of the system velocity onto the orbital plane.

The analysis proceeds in a similar fashion to that for tex2html_wrap_inline2597, except that the magnitude of tex2html_wrap_inline2605 is now written as [34, 18Jump To The Next Citation Point In The Article]

  equation481

where tex2html_wrap_inline3099 is the projection of the system velocity onto the orbital plane. The angle tex2html_wrap_inline3033, used in determining this projection in a manner similar to that of Equation (18Popup Equation), is now the angle between the line of sight to the pulsar and the absolute velocity of the binary system.

The figure of merit for systems used to test tex2html_wrap_inline2821 is tex2html_wrap_inline3105 . As for the tex2html_wrap_inline2597 test, the systems must be old, so that tex2html_wrap_inline3109, and tex2html_wrap_inline2773 must be larger than the rate of Galactic rotation. Examples of suitable systems are PSR J2317+1439 [27, 18] with a last published value of tex2html_wrap_inline3113 in 1996 [28Jump To The Next Citation Point In The Article], and PSR J1012+5307, with tex2html_wrap_inline3115  [84Jump To The Next Citation Point In The Article]. This latter system is especially valuable because observations of its white-dwarf component yield a radial velocity measurement [24], eliminating the need to find a lower limit on an unknown quantity. The analysis of Wex [146Jump To The Next Citation Point In The Article] yields a limit of tex2html_wrap_inline3117 . This is comparable in magnitude to the weak-field results from lunar laser ranging, but incorporates strong field effects as well.

3.2.2 Limits on tex2html_wrap_inline3119  

Tests of tex2html_wrap_inline2823 can be derived from both binary and single pulsars, using slightly different techniques. A non-zero tex2html_wrap_inline2823, which implies both a violation of local Lorentz invariance and non-conservation of momentum, will cause a rotating body to experience a self-acceleration tex2html_wrap_inline3125 in a direction orthogonal to both its spin tex2html_wrap_inline3127 and its absolute velocity tex2html_wrap_inline3089  [107]:

  equation520

Thus, the self-acceleration depends strongly on the compactness of the object, as discussed in Section  3 above.

An ensemble of single (isolated) pulsars can be used to set a limit on tex2html_wrap_inline2823 in the following manner. For any given pulsar, it is likely that some fraction of the self-acceleration will be directed along the line of sight to the Earth. Such an acceleration will contribute to the observed period derivative tex2html_wrap_inline3133 via the Doppler effect, by an amount

  equation535

where tex2html_wrap_inline3135 is a unit vector in the direction from the pulsar to the Earth. The analysis of Will [152Jump To The Next Citation Point In The Article] assumes random orientations of both the pulsar spin axes and velocities, and finds that, on average, tex2html_wrap_inline3137, independent of the pulse period. The sign of the tex2html_wrap_inline2823 contribution to tex2html_wrap_inline3133, however, may be positive or negative for any individual pulsar; thus, if there were a large contribution on average, one would expect to observe pulsars with both positive and negative total period derivatives. Young pulsars in the field of the Galaxy (pulsars in globular clusters suffer from unknown accelerations from the cluster gravitational potential and do not count toward this analysis) all show positive period derivatives, typically around tex2html_wrap_inline3143 . Thus, the maximum possible contribution from tex2html_wrap_inline2823 must also be considered to be of this size, and the limit is given by tex2html_wrap_inline3147  [152Jump To The Next Citation Point In The Article].

Bell [16Jump To The Next Citation Point In The Article] applies this test to a set of millisecond pulsars; these have much smaller period derivatives, on the order of tex2html_wrap_inline3149 s/s. Here, it is also necessary to account for the ``Shklovskii effect'' [119Jump To The Next Citation Point In The Article] in which a similar Doppler-shift addition to the period derivative results from the transverse motion of the pulsar on the sky:

  equation560

where tex2html_wrap_inline3151 is the proper motion of the pulsar and d is the distance between the Earth and the pulsar. The distance is usually poorly determined, with uncertainties of typically 30% resulting from models of the dispersive free electron density in the Galaxy [132Jump To The Next Citation Point In The Article, 30Jump To The Next Citation Point In The Article]. Nevertheless, once this correction (which is always positive) is applied to the observed period derivatives for isolated millisecond pulsars, a limit on tex2html_wrap_inline3155 on the order of tex2html_wrap_inline3157 results [16, 19Jump To The Next Citation Point In The Article].

In the case of a binary-pulsar-white-dwarf system, both bodies experience a self-acceleration. The combined accelerations affect both the velocity of the centre of mass of the system (an effect which may not be readily observable) and the relative motion of the two bodies [19Jump To The Next Citation Point In The Article]. The relative-motion effects break down into a term involving the coupling of the spins to the absolute motion of the centre of mass, and a second term which couples the spins to the orbital velocities of the stars. The second term induces only a very small, unobservable correction to tex2html_wrap_inline2737 and tex2html_wrap_inline2773  [19]. The first term, however, can lead to a significant test of tex2html_wrap_inline2823 . Both the compactness and the spin of the pulsar will completely dominate those of the white dwarf, making the net acceleration of the two bodies effectively

  equation574

where tex2html_wrap_inline3165 and tex2html_wrap_inline3167 denote the compactness and spin angular frequency of the pulsar, respectively, and tex2html_wrap_inline3089 is the velocity of the system. For evolutionary reasons (see, e.g., [21Jump To The Next Citation Point In The Article]), the spin axis of the pulsar may be assumed to be aligned with the orbital angular momentum of the system, hence the net effect of the acceleration will be to induce a polarization of the eccentricity vector within the orbital plane. The forced eccentricity term may be written as

  equation592

where tex2html_wrap_inline2581 is the (unknown) angle between tex2html_wrap_inline3089 and tex2html_wrap_inline3167, and P is, as usual, the spin period of the pulsar: tex2html_wrap_inline3179 .

The figure of merit for systems used to test tex2html_wrap_inline2823 is tex2html_wrap_inline3183 . The additional requirements of tex2html_wrap_inline3109 and tex2html_wrap_inline2773 being larger than the rate of Galactic rotation also hold. The 95% confidence limit derived by Wex [146] for an ensemble of binary pulsars is tex2html_wrap_inline3189, much more stringent than for the single-pulsar case.

3.2.3 Limits on tex2html_wrap_inline3191  

Another PPN parameter that predicts the non-conservation of momentum is tex2html_wrap_inline2591 . It will contribute, along with tex2html_wrap_inline2589, to an acceleration of the centre of mass of a binary system [149Jump To The Next Citation Point In The Article, 152Jump To The Next Citation Point In The Article]

  equation618

where tex2html_wrap_inline3197 is a unit vector from the centre of mass to the periastron of tex2html_wrap_inline3199 . This acceleration produces the same type of Doppler-effect contribution to a binary pulsar's tex2html_wrap_inline3133 as described in Section  3.2.2 . In a small-eccentricity system, this contribution would not be separable from the tex2html_wrap_inline3133 intrinsic to the pulsar. However, in a highly eccentric binary such as PSR B1913+16, the longitude of periastron advances significantly - for PSR B1913+16, it has advanced nearly 120 tex2html_wrap_inline3205 since the pulsar's discovery. In this case, the projection of tex2html_wrap_inline3207 along the line of sight to the Earth will change considerably over the long term, producing an effective second derivative of the pulse period. This tex2html_wrap_inline3209 is given by [149Jump To The Next Citation Point In The Article, 152Jump To The Next Citation Point In The Article]

  equation636

where tex2html_wrap_inline3211 is the mass ratio of the two stars and an average value of tex2html_wrap_inline3213 is chosen. As of 1992, the 95% confidence upper limit on tex2html_wrap_inline3209 was tex2html_wrap_inline3217  [133Jump To The Next Citation Point In The Article, 149Jump To The Next Citation Point In The Article]. This leads to an upper limit on tex2html_wrap_inline3219 of tex2html_wrap_inline3221  [149]. As tex2html_wrap_inline2589 is orders of magnitude smaller than this (see Section  3.2.2), this can be interpreted as a limit on tex2html_wrap_inline2591 alone. Although PSR B1913+16 is of course still observed, the infrequent campaign nature of the observations makes it difficult to set a much better limit on tex2html_wrap_inline3209 (J. Taylor, private communication, as cited in [75Jump To The Next Citation Point In The Article]). The other well-studied double-neutron-star binary, PSR B1534+12, yields a weaker test due to its orbital parameters and very similar component masses. A complication for this test is that an observed tex2html_wrap_inline3209 could also be interpreted as timing noise (sometimes seen in recycled pulsars [73Jump To The Next Citation Point In The Article]) or else a manifestation of profile changes due to geodetic precession [79, 75].



3.3 Strong Equivalence Principle: Dipolar 3 Tests of GR - 3.1 Strong Equivalence Principle: Nordtvedt

image Testing General Relativity with Pulsar Timing
Ingrid H. Stairs
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