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Register a new userTime delay in the strong field limit for null and timelike signals and its simple interpretation
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Gravitational lensing can happen not only for null signal but also timelike signals such as neutrinos and massive gravitational waves in some theories beyond GR. In this work we study the time delay between different relativistic images formed by signals with arbitrary asymptotic velocity $v$ in general static and spherically symmetric spacetimes. A perturbative method is used to calculate the total travel time in the strong field limit, which is found to be in quasi-series of the small parameter $a=1-b_c/b$ where $b$ is the impact parameter and $b_c$ is its critical value. The coefficients of the series are completely fixed by the behaviour of the metric functions near the particle sphere $r_c$ and only the first term of the series contains a weak logarithmic divergence. The time delay $Delta t_{n,m}$ to the leading non-trivial order was shown to equal the particle sphere circumference divided by the local signal velocity and multiplied by the winding number and the redshift factor. By assuming the Sgr A* supermassive black hole is a Hayward one, we were able to validate the quasi-series form of the total time, and reveal the effects of the spacetime parameter $l$, the signal velocity $v$ and the source/detector coordinate difference $Deltaphi_{sd}$ on the time delay. It is found that as $l$ increase from 0 to its critical value $l_c$, both $r_c$ and $Delta t_{n,m}$ decrease. The variation of $Delta t_{n+1,n}$ for $l$ from 0 to $l_c$ can be as large as $7.2times 10^1$ [s], whose measurement then can be used to constrain the value of $l$. While for ultra-relativistic neutrino or gravitational wave, the variation of $Delta t_{n,m}$ is too small to be resolved. The dependence of $Delta t_{n,-n}$ on $Delta phi_{sd}$ shows that to temporally resolve the two sequences of images from opposite sides of the lens, $|Delta phi_{sd}-pi|$ has to be larger than certain value.
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A perturbative method to compute the total travel time of both null and lightlike rays in arbitrary static spherically symmetric spacetimes in the weak field limit is proposed. The resultant total time takes a quasi-series form of the impact parameter. The coefficient of this series at a certain order $n$ is shown to be determined by the asymptotic expansion of the metric functions to the order $n+1$. To the leading order(s), the time delay, as well as the difference between the time delays of two kinds of relativistic signals, is then shown to take a universal form for all SSS spacetimes. This universal form depends on the mass $M$ and a post-Newtonian parameter $gamma$ of the spacetime. The analytical result is numerically verified using the central black hole of M87 as the gravitational lensing center.
The deflection and gravitational lensing of light and massive particles in arbitrary static, spherically symmetric and asymptotically (anti-)de Sitter spacetimes are considered in this work. We first proved that for spacetimes whose metric satisfying certain conditions, the deflection of null rays with fixed closest distance will not depend on the cosmological constant $Lambda$, while that of timelike signals and the apparent angle in gravitational lensing still depend on $Lambda$. A two-step perturbative method is then developed to compute the change of the angular coordinate and total travel time in the weak field limit. The results are quasi-series of two small quantities, with the finite distance effect of the source/detector naturally taken into account. These results are verified by applying to some known de Sitter spacetimes. Using an exact gravitational lensing equation, we solved the apparent angles of the images and time delays between them and studied the effect of $Lambda$ on them. It is found that generally, a small positive $Lambda$ will decrease the apparent angle of images from both sides of the lens and increase the time delay between them. The time delay between signals from the same side of the lens but with different energy however, will be decreased by $Lambda$.
We study the time delay between two relativistic images due to strong gravitational lensing of the light rays caused by the Kerr and Kerr-Newman black holes. Using the known form of the deflection angle in the strong deflection limit (SDL) allows us to analytically develop the formalism for the travel time of the light from the distant source winding around the black hole several times and reaching the observer. We find that the black hole with higher mass or with spin of the extreme black hole potentially have higher time delay. The effect of the charge of the black hole enhances the time delay between the images lying on the opposite side of the optical axis resulting from the light rays when one light ray is in the direct orbit and the other is in the retrograde orbit. In contrary, when both light rays travel along either direct or retrograde orbits giving the images on the same side of the optical axis, the charge effect reduces the time delay between them. We then examine the time delay observations due to the galactic and supermassive black holes respectively.
We present an idealised model of gravitational collapse, describing a collapsing rotating cylindrical shell of null dust in flat space, with the metric of a spinning cosmic string as the exterior. We find that the shell bounces before closed timelike curves can be formed. Our results also suggest slightly different definitions for the mass and angular momentum of the string.
A perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of $(1-b_c/b)$ where $b$ is the impact parameter and $b_c$ is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter $l$ and the asymptotic velocity $v$ of the signal. The BH shadow size were found to decrease slightly as $l$ increase to its critical value, and increase as $v$ decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of $l$ and decrease of $v$ will increase their values.
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