Ehud Nakar, Tsvi Piran

The radio and X-rays that followed GW170817 are unlike any GRB afterglow observed before. Instead, their gradual rise resemble the radio flare predicted long ago to follow binary neutron star mergers [Nakar 2011]. Considering a blast wave moving with a Lorentz factor $\Gamma$, we show that an off-axis observer, namely an observer at $\theta_{obs}> 1/\Gamma$, sees a light curve rising faster than $F_\nu \propto t^3$. Therefore, the observed rise, $F_\nu \propto t^{0.78}$, implies that at all times we have seen on-axis emission. Namely, the emitting matter was within $\theta_{obs}<1/\Gamma$ at the time of observations (even if it was off-axis beforehand). The observations tightly constrain the Lorentz factor: $\Gamma \sim (1.5-7) (t/10 ~{\rm day})^{-0.21}$. The isotropic equivalent energy in the observed region is $E_{iso} \sim 10^{49} (t/10 ~{\rm day})^{1.3}$ erg. The energy increase can arise from a slower material moving behind the blast wave or from a matter moving at larger angles that has slowed down. While such a structure can have different origins, the only physically motivated one proposed so far, is the interaction of a relativistic jet with the ejecta and the resulting cocoon. The jet could have been choked or successful. In the latter case it has produced a short GRB pointing elsewhere (this successful jet-cocoon system is sometimes called a "structured jet"). Although circumstantial evidence disfavors a successful jet, the fate of the jet (choked or successful) cannot be decisively determined from current observations. Unfortunately, the light curve alone may not be sufficient to resolve this question, since both chocked and successful jets can lead to a gradual rise to a peak, followed by a rapid decay. Therefore, a decay of the light curve, even if observed in the near future, won't necessitate the existence of a successful jet.

Takahiro Morishima, Toshifumi Futamase

The magnetic moment of free fermions in the curved spacetime has been studied based on the general relativity. Adopting the Schwarzschild metric for the background spacetime, the effective value of the magnetic moment has been calculated up to the post-Newtonian order $O(1/c^2)$ for three cases (A) Dirac particles with g=2, (B) neutral fermions with g$\ne$2 and e=0 and (C) charged fermions with g$\ne$2 and e$\ne$0. The result shows their gravity dependence is given as $\mu_{\rm m}^{\rm eff}= (1\!+\!3\phi/c^2) \,\mu_{\rm m} $ for all of these cases in which the coupling between fermions and the electromagnetic field is essentially different. It implies that the magnetic moment is influenced by the spacetime curvature on the basis of the general relativity commonly for point-like fermions, composite fermions and spread fermions dressed with the vacuum fluctuation. The gravitational effect affects the gyro-magnetic ratio and the anomalous magnetic moment as ${\rm g}^{\rm eff} \!\simeq\! (1 \!+\! 3\phi/c^2)\,{\rm g} $, ${a}^{\rm eff} \!\simeq\! a \!+\! 3(1\!+\!a)\,\phi/c^2 $. Consequently, the anomalous magnetic moment of fermions with g$\simeq$2 measured on the Earth's surface contains the gravitational effect as $|a^{\rm eff}| \simeq 3|\phi|/c^2 \simeq 2.1\!\times\! 10^{-9}$, which implies that the gravitational anomaly of $2.1\!\times\! 10^{-9}$ is induced by the curvature of the spacetime on the basis of the general relativity in addition to the quantum radiative corrections for all fermions including electrons and muons.

Takahiro Morishima, Toshifumi Futamase, Hirohiko M. Shimizu

The general relativistic effects to the anomalous magnetic moment of the electron ${\rm g}_{\rm e}$-2 in the Earth's gravitational field have been examined. The magnetic moment of electrons to be measured on the Earth's surface is evaluated as $\mu_{\rm m}^{\rm eff} \simeq (1\!+\!3\phi/c^2)\,\mu_{\rm m}$ on the basis of the Dirac equation containing the post-Newtonian effects of the general relativity for fermions moving in the Earth's gravitational field. This implies that the anomalous magnetic moment of $10^{-9}$ appears in addition to the radiative corrections in the quantum field theory. This may seem contradictory with the fact of the 12th digit agreement between the experimental value measured on the ground level ${\rm g}_{\rm e(EXP)}$ and the theoretical value calculated in the flat spacetime ${\rm g}_{\rm e(SM)}$. In this paper, we show that the apparent contradiction can be explained consistently with the framework of the general relativity.

Takahiro Morishima, Toshifumi Futamase, Hirohiko M. Shimizu

The general relativistic effects to the anomalous magnetic moment of muons moving in the Earth's gravitational field have been examined. The Dirac equation generalized to include the general relativity suggests the magnetic moment of fermions measured on the ground level is influenced by the Earth's gravitational field as $\mu_{\rm m}^{\rm eff} \!\simeq\! (1\!+\!3\phi/c^2)\,\mu_{\rm m}$, where $\mu_{\rm m}$ is the magnetic moment in the flat spacetime and $\phi\!=\!-{G M}/{r}$ is the Earth's gravitational potential. It implies that the muon anomalous magnetic moment measured on the Earth $a_\mu\!\equiv\!{\rm g}_\mu/2\!-\!1$ contains the gravitational correction of $\left\vert a_\mu\right\vert\simeq 2.1\!\times\! 10^{-9}$ in addition to the quantum radiative corrections. The gravitationally induced anomaly may affect the comparison between the theoretical and experimental values recently reported: $a_{\mu({\rm EXP})}-a_{\mu({\rm SM})}=28.8\,\,(8.0)\times10^{-10}\,\,(3.6\,\sigma)$. In this paper, the comparison between the theory and the experiment is examined by considering the influence of the spacetime curvature to the measurement on the muon ${\rm g}_\mu\!\!-\!2$ experiment using the storage ring on the basis of the general relativity up to the post-Newtonian order of $O(1/c^2)$.

S.D. Odintsov, V.K. Oikonomou

In this work, we extend the bottom-up reconstruction framework of $F(R)$ gravity to other modified gravities, and in particular for $f(\phi)R$ and mimetic $F(R)$ gravities. We investigate which are the important conditions in order for the method to work, and we study several viable cosmological evolutions, focusing on the inflationary era. Particularly, for the $f(\phi)R$ theory case, we specify the functional form of the Hubble rate and of the scalar-to-tensor ratio as a function of the $e$-foldings number and accordingly, the rest of the physical quantities and also the slow-roll and the corresponding observational indices can be calculated. The same method is applied in the mimetic $F(R)$ gravity case, and in both cases we thoroughly analyze the resulting free parameter space, in order to show that the viability of the models presented is guaranteed and secondly that there is a wide range of values of the free parameters for which the viability of the models occurs. In addition, the reconstruction method is also studied in the context of mimetic $F(R)=R$ gravity. As we demonstrate, the resulting theory is viable, and also in this case, only the scalar-to-tensor ratio needs to be specified, since the rest follow from this condition. Finally, we discuss in brief how the reconstruction method could function for other modified gravities.