Negative time delay of light by a gravitational concave lens

Gravitational lens models, some of which might act like a concave lens, have been recently investigated by using a static and spherically symmetric modified spacetime metric that depends on the inverse distance to the $n$th power [T. Kitamura et al., Phys. Rev. D 87, 027501 (2013)]. We re-examine the time delay of light in a gravitational concave lens as well as a gravitational convex one. The frequency shift due to the time delay is also investigated. We show that the sign of the time delay in the lens models is the same as that of the deflection angle of light. The size of the time delay decreases with increase in the parameter $n$. We also discuss possible parameter ranges that are relevant to pulsar timing measurements in our Galaxy.

Figure 1

Frequency shift due to the gravitational time delay. There are two horizontal axes: one is corresponding to $t_S$ that is the time measured by a clock at the source of light, while the other is indicating $t_R$ that is the time measured by the receiver. The dotted vertical lines and solid ones are corresponding to the no-delay case and the time delay case, respectively. Signals of light are emitted with the same time interval, so that the intrinsic frequency of the signal can be constant. If the source is approaching the lens object, the gravitational time delay of each signal is increasing with time and hence the observed frequency is lower than the intrinsic one, If the source is receding from the lens object, on the other hand, the delay is decreasing and hence the observed frequency is higher than the intrinsic one.

Figure 2

Schematic figure for a configuration of the source (emitter) of a signal of light, the receiver of the signal, and the lens. They are denoted by $S$, $R$, and $L$, respectively. The angles corresponding to the source and the receiver are denoted by ${\mathrm{\Psi }}_S$ and ${\mathrm{\Psi }}_R$.

Figure 3

Schematic figure for a motion of the light emitter (S) with respect to the lens (L), where their positions are projected onto the celestial sphere. The solid curve denotes the motion of the lensed source projected onto the lens plane, while the unlensed linear motion is denoted by the dotted line. The closest approach of light to the lens is a function of time denoted as $r_0(t)$, and its minimum is denoted as $r_{\min }$.

Figure 4

Time delay curves for $ϵ>0$. The solid, dot-dashed, dashed, and dotted curves correspond to $n=1,2,3$, and 4, respectively. The horizontal axis denotes the time $t$ in days and the vertical axis means the time delay ${\delta }_t$ in seconds. Here, we assume $r_{\min }$ is 40 AU and $v=200\mathrm{km}/\mathrm{s}$. The lens is assumed to be a ten solar mass black hole for $n=1$ ($ϵ/r_{\min }\sim 10^{-8}$), and the parameters for the other $n$ are chosen such that the peak height of the time delay curve can remain the same as each other.

Figure 5

Time delay curves for $ϵ<0$. Here, the parameter values are the same as those in Fig. 4, except for the sign of $ϵ$.

Figure 6

Frequency shift curves for $ϵ>0$ corresponding to Fig. 4. Here, the parameter values for $n\ne 1$ are rearranged such that the peak heights of the time delay curve can remain the same as each other.

Figure 7

Frequency shift curves for $ϵ<0$ corresponding to Fig. 5. Here, the parameter values are the same as those in Fig. 6, except for the sign of $ϵ$.