Intensification of Gravitational Wave Field Near Compact Star

TL;DR

This paper investigates how gravitational waves are lensed by compact stars, revealing that such lensing produces spherical aberration and depends on the star's mass and radius, offering new insights into wave behavior near dense objects.

Contribution

The study models the internal structure of compact stars and demonstrates the unique lensing effects on gravitational waves, including spherical aberration, using numerical simulations.

Findings

Gravitational waves are lensed by compact stars similarly to optical lensing.

Lensing effects depend significantly on the star's mass and radius.

Inside and outside the star respond differently to incoming gravitational waves.

Abstract

The gravitational waves (GWs) has been a topic of interest for its versatile capabilities of probing several aspects of cosmology and early Universe. Gravitational lensing enhances further the extent of this sort of waves and upgrade our understanding to a next level. Besides several similarities with optical waves, GWs are capable of passing through optically opaque celestial objects like stars, exoplanets unlike light waves and manifest a different kind of lensing effect. In this work we have explored the lensing action of compact objects on gravitational waves using numerical means. After modeling the internal mass distribution of the compact objects by TOV equations and tracing wavefronts using geodesic equations, we have found that the GWs are indeed lensed in a manner analogous to the optical lensing of light in presence of a thick optical lens by producing spherical aberration in the focused waves. The distance to the best focused point shows significant dependence with the mass and radius of the lensing star and unlike gravitational lensing, the region inside and outside compact objects responds differently to the incoming waves.

Equations

$$c^{2}d{\tau }^2=c^2\left(1-\frac{2GM}{c^{2}r}\right)d{t}^2-{\left(1-\frac{2GM}{c^{2}r}\right)}^{-1}d{r}^2-r^2(d{\theta }^2+{\sin }^2\theta d{\phi }^2),$$

$$\frac{d^{2}r}{d{t}^2}=\frac{2GM}{c^2{r}^2-2GMr}{\left(\frac{dr}{dt}\right)}^2+\left(r-\frac{3GM}{c^2}\right){\left(\frac{d\theta }{dt}\right)}^2,$$

$$\frac{d^2\theta }{d{t}^2}=\frac{2(3GM-c^{2}r)}{r(c^{2}r-2GM)}\frac{dr}{dt}\frac{d\theta }{dt}.$$

$$\frac{dP}{dr}=-\frac{GM}{r^2}\rho \left(1+\frac{P}{\rho c^2}\right)\left(1+\frac{4\pi r^{3}P}{M{c}^2}\right){\left(1-\frac{2GM}{c^{2}r}\right)}^{-1}.$$

$$\frac{dM}{dr}=4\pi r^2\rho .$$

$$\mathcal{I}=\frac{I}{I_{abs}},$$

$$\mathcal{D}=\dfrac{2GM_{\odot}}{c^{2}}\dfrac{9r^{2}}{c}\bigg{/}\left(\dfrac{\mathcal{M}}{M_{\odot}}\right).$$

Full text

The gravitational waves (GWs) have gained immense importance after first detection Abbott et al. (2016) for their versatile capabilities of probing several aspects of cosmology and early Universe. These newly discovered waves have been generated by some of the most violent cosmological events such as black hole - black hole mergers Abbott et al. (2016), neutron star merger Abbott et al. (2017b) etc. and carry signatures of them. Within the framework of general theory of relativity (GR), gravitational waves are in general disturbances of space-time that have many semblances with the propagation of electromagnetic waves. Beyond these similarities, the gravitational waves are also bestowed with some additional properties which make them a significant observational tool Kokkotas (2002); Thorne (1997); K. S. Thorne. In Hawking and Israel (1989).

Strong gravitational lensing Schutz (2009); Hurtado, Castaneda, and Tejeiro (2014); Wei, Chu, and Shu (2018) have turned out to be a most relevant tool in several crucial cases of astrophysical measurements Biggs et al. (1999) viz. measuring the masses of black holes Mao, Witt, and Koopmans (2001); Rusin, Keeton, and Winn (2005), estimation of density profile of galaxies Koopmans et al. (2009); Shu et al. (2017) etc. Being light-like Hobson, Efstathiou, and Lasenby (2006) in nature, GWs propagate along null geodesics Hobson, Efstathiou, and Lasenby (2006) and produce lensing phenomenon similar to the electromagnetic waves (EM waves). Unlike the EM waves GWs, which are ripples of space-time, are capable of passing through optically opaque celestial objects like stars, exoplanets and manifest a different kind of lensing effect. In the event of gravitational wave encounters such an object in the course of their propagation, the GWs suffer an additional modifications of different nature in the star and change its form. Therefore, gravitational waves may also be a tool for obtaining useful information about the deep interior of compact object.

A consequence of GR is that the presence of mass distorts the space-time fabric. We started with the most simplest form of a metric that describes the space-time curvature due to a massive object is Schwarzschild Hobson, Efstathiou, and Lasenby (2006); Schutz (2009) metric given by,

[TABLE]

where, $\tau$ denotes the proper time, $(r,\theta,\varphi)$ are the coordinates of spherical coordinate system, $M$ is the mass, which is responsible for the curvature in space time and the other notations used have their usual significance.

Since gravitational waves propagate with the speed of light, the proper time $\tau$ would vanish Schutz (2009); Hobson, Efstathiou, and Lasenby (2006) in this case. Now, with $d\tau=0$ in Eq. 1, the equation of null-geodesics in two dimensional plane can be obtained with time $t$ as the path parameter, given by

[TABLE]

[TABLE]

These two equations (Eqs. 2, 3) describe the trajectories of gravitational waves during their propagation. It is to be mentioned that, mass $M$ in Eqs. (2) and (3) is constant outside the lensing star. However the numerical value of the effective mass should change with radial vector ($r$) inside the star. The variation of mass $M$ inside a compact massive stars such as a neutron star or a quark star Witten (1984) can be evaluated by solving Tolman-Oppenheimer-Volkov equations (TOV) Oppenheimer and Volkoff (1939) numerically. The TOV equation is given by,

[TABLE]

The TOV equation relates the pressure ($P$) with the density ($\rho$) of a spherically compact symmetric star. The pressure $P$, appearing in the above equation, is related to the density ($\rho$) of the star through a polytropic relation, $P=K\rho^{\gamma}$, where $K$ is a constant and $\gamma$ is the polytropic exponent given by, $\gamma=1+\frac{1}{n}$, $n$ being the polytropic index. The variation of mass ($M$) as function of radial distance ($r$) is given by the classical relation,

[TABLE]

and describes hydrostatic equilibrium inside the star Shapiro and Teukolsky (2008). The mass distribution profile and the density profile are evaluated by solving Eqs. (4) and (5) numerically in keeping with suitable parameters. These variations are shown in the plots of Fig. 1 for an adopted neutron star core density of $2.5\times 10^{17}\>\rm{kg/m}^{3}$. In Fig. 1a we show how the density of a neutron star falls off with the increase of radial distance ($r$) from the center of its core, while in Fig. 1b the variation of mass of the star enclosed within the radial distance $r$ is plotted. For example, for $r=15$ km the mass enclosed is around $0.95\,M_{\odot}$ while mass within $8\,km$ radius from the center of the neutron star is $0.2\,M_{\odot}$ for this particular example.

In the present work, a highly compact star (core density $=2.5\times 10^{17}\>\rm{kg/m^{3}}$) is chosen as the lensing star. As mentioned earlier the rays corresponding to the gravitational waves are obtained by solving the equations of null-geodesic (Eqs. 2 and 3). The mass $M$ inside the star is obtained from the mass profile, evaluated by solving the TOV equations (Eqs. 4, 5) numerically while outside the star the total mass $\mathcal{M}$ of the star is considered.

The gravitational waves emanated by the binary mergers modify the space-time in periodic manner during its propagation. So, in order to evaluate its intensity at any particular point, its time variation needs to be taken into account. According the current observational data Abbott et al. (2016, 2017a, 2017b), the frequencies of the gravitational waves are in the range of $100\,nHz-1000\,Hz$. Consequently, the maximum phase shift between the waves can be neglected ($\lesssim\pi/50$) near the focused region in the above-mentioned case. So, for the sake of simplicity, the time dependent term is set to unity in our calculation. The effective intensity of the gravitational waves at any point is evaluated in terms of dimensionless intensity ($\mathcal{I}$) given by,

[TABLE]

where $I$ represent the measured intensity of the waves at that point and $I_{abs}$ would be the intensity if the lensing star is absent.

In Fig. 2 we furnish a plot for the trajectories of the gravitational waves in X-Y plane in presence of the chosen compact star to show the lensing effects thus produced. It can be seen from Fig. 2 that the gravitational waves are affected differently in regions inside the star compared to the way they are affected in the vicinity of the star as the propagating gravitational waves experience qualitatively different space-time distortions inside the star in contrast to its immediate vicinity outside. As a result, these gravitational waves produce dissimilar lensing effects in the presence of a compact object as shown in Fig. 2.

While the waves propagate away from the lensing star, fallout in the intensity distribution starts appearing gradually. The intensity profiles of the waves in these zones are depicted in Fig. 3. For instance, in the present example, at a distance $x=20$ km from the center of the star the manifestation of lensing effect is confined in a ring-shaped region in space (Fig. 3). The extent of the ringed shaped region gradually shrinks down as the distance $x$ increases (Fig. 3) with gradual increase in intensity. Eventually the intensity of the luminous portion of the ring attains maximum at $x\sim 25$ km. The ring continuous to shrink afterwards and at a distance $x=29.5$ km we get a sharply focused point with maximum intensity. The gravitational waves are focused on a range of points located on the propagation axis (principle axis in this case), rather than a single point (Fig. 2). This phenomenon is comparable to aberration observed in thick optical lens.

The variation of the relative intensities due to lensing of gravitational waves with the distance from the center of the star are plotted in Fig. 4 for the demonstrative example considered in this work. From Fig. 4 one observes that the peak relative intensity at a distance $x=29.5$ km is $\sim 10^{4}$ (point P) which corresponds to the best focused point shown in Fig. 3f. At this distance the gravitational waves are focused to a point. Fig. 4 provides another smaller peak at Q ($\mathcal{I}\sim 100$) at a distance $x\sim 25$ km from the center of the star. It also depicts the variation of intensity inside the extended zone of focus shown in Fig. 3b. The lensing power of the star can be characterized by the distance to the best focused point (point P) which would called the characteristic distance $\mathcal{D}$ henceforth. The characteristic distance $\mathcal{D}$ of the star is analogous to the focal length of an optical lens. As the observer goes beyond $\mathcal{D}$ the maximum intensity starts falling again almost exponentially with the distance $x$ and the central focal spot completely disappear at large distance ($x\rightarrow\infty$) eventually.

For the present example $\mathcal{D}=29.6$ km. The dependence of $\mathcal{D}$ on the masses of the lensing star have been found by repeating the same calculation with other stars of different masses. In Fig. 5 the variation of $\mathcal{D}$ with the mass of the lensing stars is shown, from which it is evident that, $\mathcal{D}$ decreases as the mass of the lensing star increases. This is due to the fact that more massive the star is more deformation it would cause to its adjacent space-time that eventually increases the focusing effect. The characteristic distance $\mathcal{D}$ also changes accordingly. To examine whether there is a systematic formula that matches the dependence, we have fitted our numerical results shown in Fig. 5 with an assumed empirical relation for $\mathcal{D}$ given by,

[TABLE]

In Eq. 7, $r$ denotes the radius of the star of mass $\mathcal{M}$ which is obtained by solving the TOV equation. From the fitted graph (Fig. 5) one sees that the Eq. 7 is a good fit with ($\chi^{2}=0.0019$ 111$\chi^{2}=\sum\left(\frac{N_{D}-F_{D}}{N_{D}}\right)^{2}$, where $N_{D}$ is the numerically obtained value and $F_{D}$ is the fitted value.) our numerical results.

In this work, we have drawn an analogy between the lensing effects in gravitational waves by massive compact star and that of ordinary light by thick optical lens. For the sake of simplicity the waves are presumed to be emitted by highly red-shifted source. As a consequence, the incident waves can be considered to be plane waves, rather than spherical in nature.

In the present example, the propagating waves after suffering the lensing effect in presence of a massive star tends to be localized or focused on the other side of the lensing star instead of being focused at a single point. This indicates the aberration effects in case of gravitational lensing analogous to the optical aberration related to a thick optical lens. The relative intensity of the gravitational waves after they are subjected to the lensing effect, is found to be maximized at a distance $\mathcal{D}$, termed as characteristic distance from the center of the star. The magnitude of this distance $\mathcal{D}$ depends on the mass of the lensing star as well as its core density and this variation seems to follow an empirical relation given in Eq. 7.

In summary, we have studied the lensing effects of gravitational waves by massive lensing stars. We elaborate the similarities of the gravitational lensing phenomenon by a compact massive star with the optical lensing of light waves in the presence of a thick convex lens. Our studies demonstrate that the phenomenon of such lensing of GWs and their consequent convergence effects with enhanced intensities would help in interpreting gravitational wave signatures from deep space.

Acknowledgment

Two of the authors (S.B. and A.H.) wish to acknowledge the support received from St. Xavier’s College, kolkata Central Research Facility. A.H. also acknowledges the University Grant Commission (UGC) of the Government of India, for providing financial support, in the form of UGC-CSIR NET-JRF.