Testing Chern-Simons modified gravity with orbiting superconductive gravity gradiometers: the non-dynamical formulation

Abstract

High precision superconductivity gravity gradiometers (SGG) are powerful tools for relativistic experiments. In this paper, we work out the tidal signals in non-dynamical Chern-Simons modified gravity, which could be measured by orbiting SGGs around Earth. We find that, with proper orientations of multi-axes SGGs, the tidal signals from the Chern-Simons modification can be isolated in the combined data of different axes. Furthermore, for three-axes SGGs, such combined data is the trace of the total tidal matrix, which is invariant under the rotations of SGG axes and thus free from axis pointing errors. Following nearly circular orbits, the tests of the parity-violating Chern-Simons modification and the measurements of the gravitomagnetic sector in parity-conserving metric theories can be carried out independently in the same time. A first step analysis on noise sources is also included.

Appendices

Appendix 1: The standard PPN metric

The standard PPN metric has the form [39]

$$\begin{aligned} g_{00}&= -1+2U-2\beta U^{2}-2\xi \varPhi _{W}+(2\gamma +2+\alpha _{3}+\zeta _{1}-2\xi )\varPhi _{1}\\&\quad +2(3\gamma -2\beta +1+\zeta _{2}+\xi )\varPhi _{2}+ 2(1+\zeta _{3})\varPhi _{3}+2(3\gamma +3\zeta _{4}-2\xi )\varPhi _{4}\\&\quad -(\zeta _{1}-2\xi )\mathcal {A}\!-\!(\alpha _{1}- \alpha _{2}\!-\!\alpha _{3})w^{2}U\!-\!\alpha _{2}w^{i}w^{j}U_{ij}\!+\!(2\alpha _{3}\!-\!\alpha _{1})w^{i}V_{i}+\mathcal {O}(\epsilon ^{6}),\\ g_{0i}&= -\frac{1}{2}(4\gamma +3+\alpha _{1}- \alpha _{2}+\zeta _{1}-2\xi )V_{i}-\frac{1}{2}(1+\alpha _{2}-\zeta _{1}+2\xi )W_{i}\\&\quad -\frac{1}{2}(\alpha _{1}-2\alpha _{2})w^{i}U-\alpha _{2}w^{j}U_{ij}+\mathcal {O}(\epsilon ^{5}),\\ g_{ij}&= (1+2\gamma U)\delta _{ij}+\mathcal {O}(\epsilon ^{4}), \end{aligned}$$

where the PN potentials read

$$\begin{aligned} U&= \int \frac{\rho '}{|\mathbf {x}-\mathbf {x'}|}d^{3}x',\ \ \ \ \varPhi _{1}=\int \frac{\rho 'v'^{2}}{|\mathbf {x}-\mathbf {x'}|}d^{3}x',\\ \varPhi _{2}&= \int \frac{\rho 'U'}{|\mathbf {x}-\mathbf {x'}|}d^{3}x',\ \ \ \ \varPhi _{3}=\int \frac{\rho '\varPi '}{|\mathbf {x}-\mathbf {x'}|}d^{3}x',\\ \varPhi _{4}&= \int \frac{p'}{|\mathbf {x}-\mathbf {x'}|}d^{3}x',\ \ \ \ V_{i}=\int \frac{\rho 'v'^{i}}{|\mathbf {x}-\mathbf {x'}|}d^{3}x',\\ W_{i}&= \int \frac{\rho '[\mathbf {v}'\cdot (\mathbf {x}-\mathbf {x'})](x^{i}-x'^{i})}{|\mathbf {x}-\mathbf {x'}|^{3}}d^{3}x',\\ U_{ij}&= \int \frac{\rho '(x^{i}-x'^{i})(x^{j}-x'^{j})}{|\mathbf {x}-\mathbf {x'}|^{3}}d^{3}x',\\ \mathcal {A}&= \int \frac{\rho '[\mathbf {v}'\cdot (\mathbf {x}-\mathbf {x'})]^{2}}{|\mathbf {x}-\mathbf {x'}|^{3}}d^{3}x',\\ \varPhi _{W}&= \int \frac{\rho '\rho ''(\mathbf {x}-\mathbf {x}')}{|\mathbf {x}-\mathbf {x'}|^{3}}\cdot (\frac{\mathbf {x'}-\mathbf {x}''}{|\mathbf {x'}-\mathbf {x}''|}-\frac{\mathbf {x}-\mathbf {x}''}{|\mathbf {x}-\mathbf {x}''|})d^{3}x'd^{3}x''. \end{aligned}$$

The matter variables are the rest mass density $\rho$, pressure $p$, coordinate velocity of the matter field $v^{i}$, internal energy per unit mass $\varPi$ and the coordinate velocity of the PPN coordinate system relative to the mean rest-frame of the universe $w^{i}$. The PN orders read

$$\begin{aligned} v\sim \mathcal {O}(\epsilon ),\ \ \ \ v^{2}\sim U\sim \varPi \sim \frac{p}{\rho }\sim \mathcal {O}(\epsilon ^{2}). \end{aligned}$$

The standard PN parameters $\{\gamma ,\ \beta ,\ \xi \ ,\alpha _{1},\ \alpha _{2},\ \alpha _{3},\ \zeta _{1},\ \zeta _{2},\ \zeta _{3},\ \zeta _{4}\}$ have the following meanings. The parameters $\gamma$ and $\beta$ are the usual Eddington–Robertson–Schiff parameters used to describe the “classical” tests of GR and are in some sense the most important ones. For GR $\gamma =\beta =1$ are the only non-vanishing parameters. The parameter $\xi$ measures the preferred-location effects, $\{\alpha _{1},\ \alpha _{2},\ \alpha _{3}\}$ measure the preferred-frame effects and$\{\alpha _{3},\ \zeta _{1},\ \zeta _{2},\ \zeta _{3},\ \zeta _{4}\}$ measure the violations of global conservation laws for total momentum. The up-to-date values of these parameters are summarized in Table 1 [39].

Appendix 2: The transformation matrices and tidal matrices

From Eqs. (13, 14, 15, 16, 17), the transformation matrices between the local frame and the Earth centered PN system up to 1PN level read

$$\begin{aligned}&\mathbf {e=}\nonumber \\&\left( \begin{array}{cccc} 1+\frac{a^{2}\omega ^{2}}{2}+\frac{M}{a} &{} -a\omega \sin \varPsi &{} a\omega \cos i\cos \varPsi &{} a\omega \sin i\cos \varPsi \\ a\omega &{} -\left( 1+\frac{a^{2}\omega ^{2}}{2}-\frac{\gamma M}{a}\right) \sin \varPsi &{} \left( 1+\frac{a^{2}\omega ^{2}}{2}-\frac{\gamma M}{a}\right) \cos i\cos \varPsi &{} \left( 1+\frac{a^{2}\omega ^{2}}{2}-\frac{\gamma M}{a}\right) \sin i\sin \varPsi \\ 0 &{} \left( 1-\frac{\gamma M}{a}\right) \cos \varPsi &{} \left( 1-\frac{\gamma M}{a}\right) \cos i\sin \varPsi &{} \left( 1-\frac{\gamma M}{a}\right) \sin i\sin \varPsi \\ 0 &{} 0 &{} -\left( 1-\frac{\gamma M}{a}\right) \sin i &{} \left( 1-\frac{\gamma M}{a}\right) \cos i \end{array}\right) ,\nonumber \\ \end{aligned}$$

(27)

$$\begin{aligned}&{{\underline{\varvec{{e}}}}}=\nonumber \\&\left( \begin{array}{cccc} 1+\frac{a^{2}\omega ^{2}}{2}-\frac{M}{a} &{} -a\omega &{} 0 &{} 0\\ a\omega \sin \varPsi &{} -\left( 1+\frac{a^{2}\omega ^{2}}{2}+\frac{\gamma M}{a}\right) \sin \varPsi &{} \left( 1+\frac{\gamma M}{a}\right) \cos \varPsi &{} 0\\ -a\omega \cos i\cos \varPsi &{} \left( 1+\frac{a^{2}\omega ^{2}}{2}+\frac{\gamma M}{a}\right) \cos i\cos \varPsi &{} \left( 1+\frac{\gamma M}{a}\right) \cos i\sin \varPsi &{} -\left( 1+\frac{\gamma M}{a}\right) \sin i\\ -a\omega \sin i\cos \varPsi &{} \left( 1+\frac{a^{2}\omega ^{2}}{2}+\frac{\gamma M}{a}\right) \sin i\cos \varPsi &{} \left( 1+\frac{\gamma M}{a}\right) \sin i\sin \varPsi &{} \left( 1+\frac{\gamma M}{a}\right) \cos i \end{array}\right) . \end{aligned}$$

(28)

From Eqs. (12, 13, 27, 28), the 3 dimensional tidal matrix $\tilde{K}_{ij} \equiv \tilde{K}_{ij}^{N}+ \tilde{K}_{ij}^{GE}+\tilde{K}_{ij}^{GM}+\tilde{K}_{ij}^{CS}$ along the circular orbit in the Earth pointing local frame can be worked out as

$$\begin{aligned} \tilde{K}^{N}&= \frac{M}{a^{3}}\left( \begin{array}{ccc} 1 &{} 0 &{} 0\\ 0 &{} -2 &{} 0\\ 0 &{} 0 &{} 1 \end{array}\right) ,\end{aligned}$$

(29)

$$\begin{aligned} \tilde{K}^{GE}&= \frac{M}{a^{3}}\left( \begin{array}{ccc} -\frac{(2\beta +3\gamma -2)M}{a} &{} 0 &{} 0\\ 0 &{} \frac{(6\beta +5\gamma -5)M}{a}-(\gamma +2)a^{2}\omega ^{2} &{} 0\\ 0 &{} 0 &{} \frac{(-2\beta -3\gamma +2)M}{a}+(2\gamma +1)\omega ^{2}a^{2} \end{array}\right) ,\end{aligned}$$

(30)

$$\begin{aligned} \tilde{K}^{GM}&= \frac{J\omega }{a^{3}}\left( \begin{array}{ccc} 0 &{} 0 &{} -\frac{3}{2}\varDelta \sin i\cos \varPsi \\ 0 &{} 3\varDelta \cos i &{} \frac{9}{2}\varDelta \sin i\sin \varPsi \\ -\frac{3}{2}\varDelta \sin i\cos \varPsi &{} \frac{9}{2}\varDelta \sin i\sin \varPsi &{} -3\varDelta \cos i \end{array}\right) ,\end{aligned}$$

(31)

$$\begin{aligned} \tilde{K}^{CS}&= \frac{J\omega }{a^{3}}\left( \begin{array}{ccc} 0 &{} -\frac{1}{4}\chi \sin i\sin \varPsi &{} -\frac{1}{4}\chi \cos i\\ -\frac{1}{4}\chi \sin i\sin \varPsi &{} -\frac{3}{2}\chi \sin i\cos \varPsi &{} 0\\ -\frac{1}{4}\chi \cos i &{} 0 &{} \frac{1}{2}\chi \sin i\cos \varPsi \end{array}\right) . \end{aligned}$$

(32)

Here, $\tilde{K}^{N}$, $\tilde{K}^{GE}$, $\tilde{K}^{GM}$ and $\tilde{K}^{CS}$ denote the gravitational tidal matrices from the Newtonian force, the 1PN gravitoelectric force, the gravitomagnetic force and the contributions from the CS modification. Since the Earth pointing frame is rotating relative to parallel transported frames (Fermi shifted), we need to include the tidal matrix from the centrifugal force produced by such rotation

$$\begin{aligned} \tilde{K}^{\omega }=-\omega _{0}^{2}\begin{pmatrix}1\\ &{}\quad 1\\ &{} &{}\quad 0 \end{pmatrix}, \end{aligned}$$

(33)

where $\omega _{0}=\frac{d\varPsi }{dt}+\frac{1}{a}\mathcal {O}(\epsilon ^{3})$ denote the angular velocity of the rotation of the local frame.

Appendix 3: The explicit form of the gradients readouts

For the two-axes SGG discussed in Sect. 2.3, see Eq.(18) and Fig. 2, the readouts along $\hat{\mathbf {p}}$ and $\hat{\mathbf {q}}$ are

$$\begin{aligned} \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{\omega }&= \tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{\omega }=-\omega _{0}^{2}\ \ \ \ \ \ \ \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{N}=\tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{N}=-\frac{M}{2a^{3}},\\ \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{GE}&= \tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{GE}=\frac{(4\beta +2\gamma -3)M^{2}}{2a^{4}}-\frac{(\gamma +2)M\omega ^{2}}{2a},\\ \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{GM}&= \tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{GM}=\frac{3\varDelta J\omega \cos i}{2a^{3}},\\ \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{CS}&= -\frac{\chi J\omega \sin i(3\cos \varPsi +\sin \varPsi )}{4a^{3}},\ \ \ \ \ \tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{CS}=-\frac{\chi J\omega \sin i(3\cos \varPsi -\sin \varPsi )}{4a^{3}}. \end{aligned}$$

For the three-axes SGG in Sect. 2.3, see Eqs. (21, 22, 23) and Fig. (3), the readouts along $\hat{\mathbf {n}}$, $\hat{\mathbf {p}}$ and $\hat{\mathbf {q}}$ are

$$\begin{aligned} \tilde{K}_{\hat{\mathbf {n}}\hat{\mathbf {n}}}^{\omega }&= -\omega _{0}^{2},\quad \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{\omega }=\tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{\omega }=-\frac{\omega _{0}^{2}}{2},\\ \tilde{K}_{\hat{\mathbf {n}}\hat{\mathbf {n}}}^{N}&= -\frac{M(3\cos 2\phi +1)}{2a^{3}},\quad \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{N}=\tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{N}=\frac{M(3\cos 2\phi +1)}{4a^{3}},\\ \tilde{K}_{\hat{\mathbf {n}}\hat{\mathbf {n}}}^{GE}&= \frac{\left( (6\beta +5\gamma -5)\cos ^{2}\phi -(2\beta +3\gamma -2)\sin ^{2}\phi \right) M^{2}}{a^{4}}-\frac{(\gamma +2)\cos ^{2}\phi M\omega ^{2}}{a},\\ \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{GE}&= \tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{GE}=-\frac{((4\gamma -1)+(8\beta +8\gamma -7)\cos 2\phi )M^{2}}{4a^{4}}+\frac{((\gamma +2)\cos 2\phi +3\gamma )M\omega ^{2}}{4a},\\ \tilde{K}_{\hat{\mathbf {n}}\hat{\mathbf {n}}}^{GM}&= \frac{3\varDelta J\omega \cos i\cos ^{2}\phi }{a^{3}},\\ \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{GM}&= -\frac{3\varDelta J\omega \left( \sin i(3\sin \phi \sin \varPsi -\cos \phi \cos \varPsi )+\cos i\cos ^{2}\phi \right) }{2a^{3}},\\ \tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{GM}&= -\frac{3\varDelta J\omega \left( \sin i(-3\sin \phi \sin \varPsi +\cos \phi \cos \varPsi )+\cos i\cos ^{2}\phi \right) }{2a^{3}},\\ \tilde{K}_{\hat{\mathbf {n}}\hat{\mathbf {n}}}^{CS}&= \frac{J\chi \omega \sin i\cos \phi (\sin \phi \sin \varPsi -3\cos \phi \cos \varPsi )}{2a^{3}},\\ \tilde{K}_{\hat{\mathbf {p}}\hat{\mathbf {p}}}^{CS}&= \frac{J\chi \omega \left( \sin i\left( (1-3\sin ^{2}\phi )\cos \varPsi -\sin \phi \cos \phi \sin \varPsi \right) +\cos i\cos \phi \right) }{4a^{3}},\\ \tilde{K}_{\hat{\mathbf {q}}\hat{\mathbf {q}}}^{CS}&= \frac{J\chi \omega \left( \sin i\left( (1-3\sin ^{2}\phi )\cos \varPsi -\sin \phi \cos \phi \sin \varPsi \right) -\cos i\cos \phi \right) }{4a^{3}}. \end{aligned}$$