Signatures of electromagnetic and sound radiation produced by gravitational waves in core-collapse supernovae

TL;DR

This paper hypothesizes that gravitational waves from core-collapse supernovae produce electromagnetic and sound radiation, which could trigger fusion re-ignition outside the core, affecting supernova remnant evolution and offering observable signatures.

Contribution

It introduces the novel idea that co-produced electromagnetic and sound radiation from gravitational waves can influence supernova evolution and be detected in remnants.

Findings

Electromagnetic and sound radiation are generated alongside gravitational waves.

Re-ignition of fusion outside the core may be triggered by this radiation.

Observable signatures in supernova remnants could confirm the process.

Abstract

This paper presents the scenario that gravitational waves, generated in core-collapse of a pre-supernova star, can produce both electromagnetic radiation and sound radiation as gravitational waves propagate outward from the collapsing core. While the energy of this co-produced electromagnetic and sound radiation is orders of magnitude smaller than the initiating gravitational radiation, the power may be sufficient to re-ignite fusion outside the collapsing core. The non-equilibrium re-ignition of fusion, in roughly the same time frame as the strongest neutrino emissions, would change the configuration of the pre-supernova star and subsequently the ejecta and the evolution of the stellar expansion of the supernova remnant. Although the co-produced electromagnetic or sound radiation could not contribute directly to the supernova explosion, the associated non-equilibrium re-ignition of fusion would alter the state outside the core leaving an observable signature in the ejecta of the supernova remnant. The aim of this paper is to argue that including this hypothesized co-produced radiation in computational models of core collapse supernovae would contribute to the evolution of the stellar expansion and consequently should be observable in the supernovae remnant, providing a confirmation of the conversion processes for gravitational radiation to electromagnetic and sound radiation.

Equations

$$\frac{1}{\sqrt{-g}}{\partial }_{\mu }\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\phi =0.$$

$$d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=-d{t}^2+d{z}^2+a^{2}d{x}^2+b^{2}d{y}^2.$$

$$\left(4F(u){\partial }_u{\partial }_v-4ikG(u){\partial }_v+H(u)({\partial }_x^2+{\partial }_y^2)\right)\phi =0,$$

$$\phi =A\frac{e^{\frac{\lambda }{k}}{\left(1-h^2{e}^{2iku}\right)}^{\frac{1}{2}\left(\frac{\lambda }{k}-1\right)}}{e^{\frac{\lambda }{k\left(1-h^2{e}^{2iku}\right)}}}{e}^{-i\lambda u}{e}^{i{p}_{v}v}{e}^{ipx}{e}^{ipy}+B.$$

$$\phi \left(t,z\right)$$

$$F_{em}=\frac{\pi c^3}{2G}{h}^4{f}^2=C{h}^4{f}^2,$$

$${\dot{E}}_{em}\left(r\right)=F_{em}4\pi r^2=4\pi C\frac{{\tilde{A}}^4}{r^2}{f}^2,$$

$$E_{em}\simeq {\dot{E}}_{em}\Delta t=4\pi C\frac{{\tilde{A}}^4}{r^2}{f}^2\Delta t,$$

$$m_{\gamma }=\frac{4\pi \hslash }{c^2}\sqrt{\frac{n{q}^2}{m}}=\hslash {\omega }_{plasma},$$

$$k=\frac{2\pi }{\lambda }\sqrt{1-\frac{{\omega }_{plasma}^2}{{\omega }^2}}.$$

$$\frac{1}{\sqrt{-\mathcal{H}}}{\partial }_{\mu }\sqrt{-\mathcal{H}}{\mathcal{H}}^{\mu \nu }{\partial }_{\nu }\Pi =0,$$

$$\frac{1}{\sqrt{-H}}{\partial }_{\mu }\sqrt{-H}{H}^{\mu \nu }{\partial }_{\nu }\Pi =0,$$

$$d{s}^2=H_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=-c_s^{2}d{t}^2+d{z}^2+a^2(t)d{x}^2+b^2(t)d{y}^2.$$

$$\left(\frac{1}{a^2}{\partial }_x^2+\frac{1}{b^2}{\partial }_y^2+\frac{1}{ab}{\partial }_z\left(ab\right){\partial }_z+{\partial }_z^2-c_s^{-2}{\partial }_t^2-c_s^{-2}\frac{1}{ab}{\partial }_t\left(ab\right){\partial }_t\right)\Pi =0.$$

$$\left({\partial }_z^2-c_s^{-2}{\partial }_t^2-c_s^{-2}\frac{1}{ab}{\partial }_t\left(ab\right){\partial }_t\right)\Pi =0.$$

$$\left({\partial }_z^2-c_s^{-2}{\partial }_t^2+\frac{2i{h}^2\omega e^{-2i\omega t}}{c_s^2(1-h^2{e}^{-2i\omega t})}{\partial }_t\right)\Pi =0.$$

$$\left({\partial }_z^2-c_s^{-2}{\partial }_t^2+\frac{2i{h}^2\omega e^{-2i\omega t}}{c_s^2}{\partial }_t\right)\Pi =0.$$

$$\frac{Z^{\prime \prime }}{Z}=-K^2\to Z(z)=A{e}^{\pm iKz}$$

$$\ddot{T}-2i{h}^2\omega e^{-2i\omega t}\dot{T}=-c_s^2{K}^{2}T(t)$$

$$\ddot{T}-2i{h}^2\omega e^{-2i\omega t}\dot{T}=-{\omega }^{2}T(t)$$

$$T(t)=i{e}^{-x^2/2}x\left[I_0\left(\frac{x^2}{2}\right)+I_1\left(\frac{x^2}{2}\right)\right]\mathrm{where}x\equiv h{e}^{-i\omega t},$$

$$T(t)\approx ix-i\frac{x^3}{4}\to ih{e}^{-i\omega t}=ih{e}^{-ickt}=h{e}^{-ickt+i\pi /2},$$

$$\Pi (z,t)=Z(z)T(t)=h^2{e}^{\pm iKz-ickt+i\pi /2},$$

$${\mathcal{L}}_s=\frac{1}{8\pi }{\partial }_{\mu }\Pi {\partial }^{\mu }\Pi ,$$

$$T_s^{\mu \nu }=\frac{1}{4\pi }{\partial }^{\mu }\Pi {\partial }^{\nu }\Pi -g^{\mu \nu }{\mathcal{L}}_s=\frac{1}{4\pi }{\partial }^{\mu }\Pi {\partial }^{\nu }\Pi -\frac{1}{8\pi }{g}^{\mu \nu }{\partial }_{\mu }\Pi {\partial }^{\mu }\Pi$$

$$⟨T_s^{0z}⟩=\frac{1}{2}\left(\frac{1}{4\pi }{\partial }^0{\Pi }^{\ast }{\partial }^z\Pi \right)=\frac{1}{8\pi c}{\partial }_t{\Pi }^{\ast }{\partial }_z\Pi =\frac{h^{4}kK}{2\pi },$$

$$F_s=\frac{d{E}_s}{dt}$$

$${\sigma }_{\nu }\approx 10^{-43}\left[\frac{E_{\nu }^2}{1\mathrm{Me}{\mathrm{V}}^2}\right]\mathrm{c}{\mathrm{m}}^2,$$

$$l_{\nu }\approx \frac{m_u}{\rho {\sigma }_{\nu }},$$

$$l_{\nu }\approx \left[\frac{10^{19}\frac{\mathrm{g}}{\mathrm{c}{\mathrm{m}}^3}}{\rho }\right]\left[\frac{1\mathrm{Me}{\mathrm{V}}^2}{E_{\nu }^2}\right]\mathrm{cm}.$$

Full text

I Introduction

The observation of gravitational waves has opened a new window on the Universe and our understanding of astrophysics. The coordinated international, multi-messenger observation of a kilonova ligo is a testament to the incredible potential for the future of astrophysics. Equally tantalizing is the unresolved signal above 300 Hz in the kilonova gravitational wave observation and the limitations this presents in understanding astronomical events. The detection of higher frequency gravitational waves above 300 Hz is especially important for multi-messenger studies, via gravitational waves, electromagnetic waves and neutrinos, of core-collapse supernovae (CCSNe). However, the explosive mechanism for core-collapse supernova is still not completely understood. The computer simulations of supernova often stall and the virtual star does not explode in a supernova. This problem was well described by Janka Janka17 , “The long-lasting struggle for robust neutrino-driven explosions, which should prove to be less sensitive to ‘details’ of the input physics or methods applied, still goes on.” This leaves open the possibility that the physics in the current models of core-collapse, which rely heavily on neutrinos to make the star go supernova, might not be complete - there may be additional physical mechanisms involved. The aim of this paper is to describe a previously unconsidered physical mechanism and propose the inclusion of this mechanism in future models of CCSNe.

The possibility that neutrino-driven explosion mechanisms are not the only “input physics” driving core-collapse supernova has been seriously considered in Janka17 . However, the general consensus is that the complementary or alternative mechanisms proposed so far do not contribute to the explosion. One possibility that has not been previously considered is the mechanism of conversion of gravitational waves to electromagnetic radiation and/or sound radiation coincident with the core-collapse. At the particle/Feynman diagram level the process of graviton-photon/graviton-phonon scattering, photon/phonon creation from gravitons, and graviton creation from photons/phonons, was studied at tree level for photons by Skobelev Skobelev75 (a more modern view of graviton to photon conversion can be found in Sawyer20 ). Around the same time Gibbons gibbons studied the possibility for gravitational wave backgrounds (i.e. many gravitons) to create other classical wave fields (i.e. many photons/many phonons). Gibbons found the following restriction on this conversion process: “Indeed since a ‘graviton’ presumably in some sense carries light-like momentum the creation of one or more particles with time-like or light-like momentum would violate the conservation of momentum unless the created particles were massless and precisely aligned with the momentum of the graviton”, with the emphasis added. For a long time Gibbons’ result has been taken to imply that production of an electromagnetic wave (photons) or sounds wave (phonons) from a gravitational wave background (gravitons) was forbidden. However, combining the results of Gibbons and Skobelev show that the production of electromagnetic waves from gravitational waves is possible, so long as photons/phonons are massless and the electromagnetic/sonic wave travels in the same direction as the generating gravitational wave. The Feynman diagram calculation of Skobelev indicated that this conversion of gravitons to photons was so small as to make it unlikely that these processes would be of consequence in any astrophysical process Skobelev75 , at least when looking at single photons and single gravitons.

Based on our recent work on the semi-classical conversion of gravitational waves to electromagnetic radiation Jones15 ; Jones16 ; Jones17 ; Jones18 ; Gretarsson18 , we demonstrate that the electromagnetic energy produced is a small fraction of the initiating gravitational radiation energy, and is far too small to contribute directly to the explosion. On the other hand we demonstrate that the power of the co-produced electromagnetic radiation is sufficient to re-ignite fusion in the outer core layers. The production of gravitational radiation at core collapse will occur in roughly the same time frame as the emission of neutrinos. Fusion initiated by the counterpart electromagnetic radiation would alter the state of the outer core of the pre-explosion star. Since models of CCSNe are highly dependent on the initial state of the star in the model, the counterpart electromagnetic radiation could be expected to significantly impact the modeling of the explosion. After studying the production of electromagnetic radiation from gravitational waves we investigate a new mechanism whereby gravitational radiation is converted into sonic radiation, sounds waves or phonons. This new mechanism of gravitons converting to phonons is based on work Visser10 ; Sabin14 which examined gravitational waves interacting with fluids. This graviton to phonon conversion is enhanced by a factor of the ratio of the speed of light to the speed of sound in the fluid, relative to the graviton to photon process.

II Production of electromagnetic radiation from gravitational waves

We first give a rough, first order calculation of how a gravitational wave (i.e. many gravitons) can give rise to an electromagnetic wave (i.e. many photons). The aim is to show that if there is enough energy in the gravitational wave (i.e. if there are a very large number of gravitons) this can compensate for the extreme smallness of the single Feynman diagram process of gravitons going to photons, $g+g\to\gamma+\gamma$, and give rise to a physically significant amount of vacuum produced electromagnetic radiation and energy. While the outward propagating gravitational wave interacts only very weakly with the matter of the progenitor star, the co-produced electromagnetic radiation would be completely absorbed.

The luminosity of the electromagnetic radiation, vacuum produced by the gravitational wave background, has been shown to be orders of magnitude less than the gravitational wave luminosity Jones15 ; Jones16 ; Jones17 ; Jones18 ; Gretarsson18 . However, if the gravitational wave luminosity is very large, then the power radiated in the electromagnetic counterpart radiation can be large enough to be astrophysically significant, especially since electromagnetic radiation interacts much more strongly with ordinary matter as compared with gravitational radiation. In particular we will look at the possible impact this conversion of gravitational wave energy to electromagnetic wave energy might have on the explosion mechanism in core-collapse supernovae (CCSNe). In a CCSNe it is thought that neutrinos play the main role in driving the supernova explosion. However, numerical simulations which model supernova, and which incorporate only neutrinos as the mechanism behind the explosion, do not explode often enough, especially for larger mass progenitor stars Janka17 . By carrying out a rough calculation we show that it is possible that electromagnetic radiation produced by gravitational waves from the core collapse could significantly alter the state of the pre-explosion star and the subsequent processes of thermonuclear and neutrino driven supernova explosions.

We present a simple, order of magnitude calculation of the electromagnetic radiation produced by a gravitational wave background. The equation of motion for the electromagnetic field will be solved for the curved space-time of a gravitational plane wave. We start with the Lagrange density for an electromagnetic field in the Lorenz gauge which is $\mathcal{L}_{em}=-\frac{1}{2}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}$. The vector potential can be written as $A_{\mu}\left({k,\lambda,x}\right)=\epsilon_{\mu}^{(\lambda)}\phi^{(\lambda)}\left({k,x}\right)$ where $\epsilon_{\mu}^{(\lambda)}$ is the polarization. Using this vector potential in $\mathcal{L}_{em}$ leads to a scalar-like Lagrange density $\mathcal{L}_{em}=-\partial_{\mu}\varphi^{*}\partial^{\mu}\varphi$, where $\varphi=\frac{1}{\sqrt{2}}\left(\phi^{(1)}+i\phi^{(2)}\right)$. Thus the electromagnetic Lagrange density has been reduced to the form of a scalar field Lagrange density. Placing this scalar field, $\varphi(x)$, in a curved space-time background leads to the field equation

[TABLE]

This is the massless Klein-Gordon equation in a curved background with metric $g^{\mu\nu}$. We will crudely approximate the metric for the gravitational wave propagating out from the core to have the plane wave form

[TABLE]

We have set $c=1$ and taken the metric to be oscillatory with $a(u)=1+\varepsilon(u)$ and $b(u)=1-\varepsilon(u)$ where $\varepsilon(u)=he^{iku}$. Here the gravitational wave strain amplitude is $h$, the gravitational wave number is $k$, and $u=z-t$ is the standard light front coordinate for waves moving in the $+z$ direction. Substituting the metric (2) into (1) gives the field equation for the scalar field (see Jones16 ; Jones18 for details)

[TABLE]

where $F\left({u}\right)\equiv\left({1-2h^{2}e^{2iku}+h^{4}e^{4iku}}\right)$, $G\left({u}\right)\equiv\left({h^{2}e^{2iku}-h^{4}e^{4iku}}\right)$, and $H(u)\equiv\left(1+h_{+}^{2}e^{2iku}\right)$. The solution to (3) can be shown Jones16 ; Jones18 to be

[TABLE]

$A$ and $B$ are normalization constants, $p,p_{v}$ are $\varphi$-field momenta in the $x,y$ and $v=z+t$ directions and $\lambda=\frac{p^{2}}{2p_{v}}$. The normalization constants in (4) are set to $A=-B=1$, consistent with the normalization of the Newman-Penrose scalars Teukolsky73 and the required vacuum solution for the fields in the absence of a gravitational wave, $h=0$. We now take the vacuum limit of the scalar field in the presence of the gravitational wave i.e. $h\neq 0$. We do this by taking the $\varphi$-field momenta to zero ($p,p_{v},\lambda\to 0$). In this vacuum limit of $\varphi$, we find that the scalar field does not go to zero but rather has the outgoing wave solution Jones16 ; Jones18

[TABLE]

In the last line we assumed $h$ to be small.

The relationship between the gravitational and electromagnetic luminosity or flux was developed in previous work Jones17 using the Newman-Penrose formalism Teukolsky73 . The luminosity or flux ratio for the gravitational and electromagnetic radiation in terms of the NP scalars is $\frac{dE_{em}/dt}{dE_{gw}/dt}=\frac{{\left({\frac{1}{{4\pi}}\left|{\Phi_{2}}\right|^{2}}\right)}}{{\left({\frac{1}{{16\pi k^{2}}}\left|{\Psi_{4}}\right|^{2}}\right)}}$, where $\Phi_{2}$ is the electromagnetic NP scalar and $\Psi_{4}$ is the gravitational Newman-Penrose scalar. In Jones17 it was shown that the scalar field from (5) and the metric given in (2) led to $\left|\Phi_{2}\right|^{2}=2k^{2}h^{4}$ and $\left|\Psi_{4}\right|^{2}=4k^{4}h^{2}$ which resulted in $\frac{{\dot{E}_{em}}}{{\dot{E}_{gw}}}=2h^{2}\to F_{em}=2h^{2}F_{gw}$, with $F_{em}$ and $F_{gw}$ being the electromagnetic and gravitational fluxes respectively. The gravitational wave flux associated with the metric (2) is $F_{gw}=\frac{{c^{3}}}{{16\pi G}}\left|{\dot{\varepsilon}}\right|^{2}$ and $\left|{\dot{\varepsilon}}\right|^{2}=h^{2}\omega^{2}$. Collecting all these results gives the electromagnetic flux produced from the gravitational wave, in terms of the known plane wave gravitational parameters as

[TABLE]

where $f=\frac{\omega}{2\pi}$ is the gravitational wave frequency, and $C\equiv\frac{\pi{c^{3}}}{{2G}}={6\times 10^{35}~{}\frac{{{\rm{Ws}}^{\rm{2}}}}{{{\rm{m}}^{\rm{2}}}}}={6\times 10^{42}~{}\frac{{{\rm{erg\cdot s}}}}{{{\rm{m}}^{\rm{2}}}}}$

III Counterpart production power in CCSNe

The power for the counterpart production of electromagnetic radiation from (6) is

[TABLE]

where $\tilde{A}=\left\langle{A_{20}^{E2}}\right\rangle$ is the mean quadrupole wave amplitude Muller04 which is related to the strain amplitude via $h=\frac{\tilde{A}}{r}$ and the multiplying constant. The electromagnetic energy produced in a given region of the outer core is,

[TABLE]

where $\Delta t$ is the duration of the gravitation wave signal.

Nominally the values for the the gravitational wave produced by core-collapse can be taken Muller04 to have duration of $\Delta t\sim 200~{}{\rm{ms}}$, a mean quadrupole amplitude $\tilde{A}\sim 50~{}{\rm{cm}}$, a frequency $f\sim 1~{}{\rm{kHz}}$, and an inner core radius of $r_{0}\sim 10~{}{\rm{km}}$. Substituting these values into (8) the energy produced is estimated as $E_{em}=10^{40}~{}\rm{erg}$. This $10^{40}~{}\rm{erg}$ energy of the electromagnetic radiation, co-produced by the gravitational wave, is orders of magnitude too small to contribute directly to CCSNe shock recovery and explosion; this would require energies on the order of $10^{51}~{}{\rm{erg}}$ Janka07 ; Janka12A . While the energy of the counterpart electromagnetic radiation produced by the gravitational wave is far too small to contribute directly to shock recovery, the power of the electromagnetic production is still quite large, and is of an order of magnitude for re-ignition of fusion in the star’s layers outside the iron core. This non-equilibrium ignition of fusion could aid in, or be a contributing mechanism, for shock recovery and explosion in some CCSNe. The fusion power equilibrium rate Hix99 is on the order of $10^{38}~{}\frac{{{\rm{erg}}}}{{\rm{s}}}$ and the above estimates show that the power of the co-produced electromagnetic radiation can exceed this value, which is the main point of this work.

One potential criticism of the conversion of gravitational waves to electromagnetic radiation presented in section II, is that the analysis there depended crucially on the photons being massless. In fact since the gravitational waves are moving through a plasma, and since photons develop an effective mass inside a plasma given by (with factors of $\hbar$ and $c$ restored)

[TABLE]

where $n$, $q$ and $m$ are the number density, charge and mass of the component of the plasma. If there are several different components $\frac{nq^{2}}{m}$ would be replaced by $\sum_{i=1}^{n}\frac{n_{i}q_{i}^{2}}{m_{i}}$, where one simply sums over the $n$ different components. If photons in plasma have a real, effective mass then the gravitational wave production of to electromagnetic radiation/photons, described in section II would not occur. However, to understand the nature of the effective photon mass in (9) in more detail we write down the electromagnetic wave’s wave number in a plasma as

[TABLE]

From (10) one can define a momentum via $p=\hbar k$ and one can define an energy via $E=\hbar\omega$. Using this $p,E$ and the $k$ defined in (10) in the relationship $E^{2}-p^{2}c^{2}=m^{2}c^{4}$ leads to the mass given in (9) if the frequency $\omega\geq\omega_{plasma}$ Das08 ; Claro12 . If $\omega<\omega_{plasma}$ then $m^{2}_{\gamma}<0$ and $m_{\gamma}$ becomes imaginary Weaver74 ; Afanasev77 ; Dolgov17 ; Tarrant21 and the electromagnetic wave will be attenuuated and in the process rapidly heat the plasma. For interplanetary space with a charge density of $\approx 10^{7}\frac{1}{m^{3}}$ (9) will give $\omega_{plasma}\simeq 10^{5}\frac{1}{Hz}$. For higher plasma densities the plasma frequency will be larger. This is well above the frequency of 100s Hz to 1000s Hz of the gravitational waves we consider. Thus we are in the regime where $\omega<\omega_{plasma}$ then $m^{2}_{\gamma}<0$ and $m_{\gamma}$ becomes imaginary. While having a positive, real $m_{\gamma}$ would prohibit the mechanism of gravitational waves producing electromagnetic waves described in section II, having an imaginary mass should still allow this production mechanism of electromagnetic waves. A similar discussion of graviton to photon conversion in the presence of a plasma and magnetic field can be found in Dolgov17 .

IV Production of sound radiation from gravitational waves

There is another possible conversion mechanism in addition to gravitational wave (graviton) to electromagnetic wave (photon) conversion. Since the gravitational wave produced by a CCSN propagates outward in the presence of the dense plasma there exists the possibility of conversion of gravitational waves to sound waves. The presence of the plasma would be associated with an acoustic Lagrangian, $\mathcal{L}_{s}=\mathcal{L}_{s}\left(\rho,\Pi\right)$, and phonon production would occur via the relativistic hydrodynamics of the plasma in the background of the gravitational wave Visser10 ; Sabin14 ; Schutzhold18 ; Sorge18 . Here the acoustic field is represented by the scalar field, $\Pi$, and the plasma density by $\rho$. The equation of motion for this massless scalar field, $\Pi(x)$, of the acoustic Lagrangian, $\mathcal{L}_{s}$, is similar to (1), and is given by

[TABLE]

In (11) $\mathcal{H}^{\mu\nu}$ is an effective metric which takes into account both the real curved spacetime background, $g^{\mu\nu}$, plus the effects of the plasma/fluid via an acoustic metric, $\mathcal{G}^{\mu\nu}$. Following e. g. Visser10 and going to the rest frame of the plasma/fluid yields the following equations for the massless scalar field (11),

[TABLE]

but where now the new effective metric, $H^{\mu\nu}$, is

[TABLE]

Note that the components of this effective metric, $a,b$, only depend on $t$ Sabin14 ; Schutzhold18 . To determine the solution for the phonon field, $\Pi(z,t)$ we now follow the process used for photons above (more details of this procedure for photons can be found in Jones16 ). We first substitute (13) into (12) which gives the phonon field $\Pi(t,z)$ equation as

[TABLE]

The oscillatory terms from the metric are of the form $a(t)=1+\varepsilon(t)$ and $b(t)=1-\varepsilon(t)$ with $\varepsilon(t)=he^{-i\omega t}$, $h$ is the dimensionless strain amplitude and $\omega=kc$ is the frequency of the incoming gravitational wave. We take $\Pi(t,z)$ to be a function of only $t$ and $z$ and not a function of $x$ and $y$. Finally taking into account that, unlike the photon case, here $a,b$ are only a function of $t$ 111As explained in reference Sabin14 for a fluid one can move to the rest frame of the fluid whereas in vacuum and for photons the best one can do is go to the light front frame. we simplify equation (14) to,

[TABLE]

Using $ab=1-\epsilon^{2}=1-h^{2}e^{-2i\omega t}$ we can expand the last term in (15) and obtain

[TABLE]

We now do separation of variables on (16) by writing the phonon field as $\Pi(z,t)=Z(z)T(t)$. Expanding the denominator $(1-h^{2}e^{-2i\omega t})^{-1}\approx 1+h^{2}e^{-2i\omega t}+{\cal O}(h^{4})...$ gives for (16)

[TABLE]

Inserting $\Pi(z,t)=Z(z)T(t)$ into (17) and picking a separation constant $K$ gives

[TABLE]

and

[TABLE]

From the solution in (18) one can see that $K$ is the wave number of the new wave in the fluid. Previewing what is to come we mention that later we will take the amplitude $A$ in (18) to be set by the amplitude of the gravitational wave namely $A=h$. We note that $c_{s}K=\omega_{s}$ is the frequency of this wave in the fluid. However one needs $\omega_{s}=\omega$ i.e. the frequency of the wave in the fluid equals the original GW frequency. This is the usual condition when one goes between different media (e.g. when light goes from vacuum to glass) where the speed of propagation and the wavelength/wave number change, but the frequency is the same. Therefore $c_{s}K=\omega_{s}=\omega=kc$ with $k$ the wave number of the original gravitational wave. One can use this to obtain the relationship between the wave number of the plasma and the wave number of the gravitational wave in vacuum namely $K=ck/c_{s}$. This implies $K>k$ and in turn the the wavelength of the sound wave is smaller than the wavelength of the original gravitational wave, $\lambda_{sounds}<\lambda_{gw}$. In fact if the speed of sound in the plasma is small compared to the speed of light, $c\gg c_{s}$ then $K\gg k$ and $\lambda_{sounds}\ll\lambda_{gw}$. Combining all this we find that (19) becomes

[TABLE]

This can be solved to give

[TABLE]

where $I_{n}(z)$ is the $n^{th}$ modified Bessel function of the first kind. There is another solution to (20) – the Meijer G function – but this blow up for small $x$ i.e. when $h$ is small. Expanding (21) around $x=0$ (i.e. $h=0$) gives

[TABLE]

where we have dropped higher terms in $h$ and written $\omega=kc$ in terms of the initial gravitational wave parameters, and we have moved the pre-factor of $i$ up into the exponential as a phase factor. Putting everything together gives

[TABLE]

where we have taken the amplitude of $Z(z)$ as $A=h$ so that in the limit when $c_{s}\to c$ the present solution for sound waves/phonons matches with the solution in section II for electromagnetic waves/photons. The wave number $K$ of the sound wave is very different from the original gravitational wave wave number, $k$, but this is not unexpected when one goes from one medium (vacuum) to another (plasma).

Now what we want to obtain is the sound energy flux, $F_{sound}$, from the GW energy flux $F_{gw}=\frac{dE_{gw}}{dt}$. Looking at the discussion above (6) we used the square of the gravitational Newman-Penrose scalar, $\left|\Psi_{4}\right|^{2}=4k^{4}h^{2}$ , to get $F_{gw}=\frac{c^{3}h^{2}\omega^{2}}{16\pi G}$. Then we used the ratio of electromagnetic flux to gravitational wave flux, namely $\frac{dE_{em}/dt}{dE_{gw}/dt}=\frac{{\left({\frac{1}{{4\pi}}\left|{\Phi_{2}}\right|^{2}}\right)}}{{\left({\frac{1}{{16\pi k^{2}}}\left|{\Psi_{4}}\right|^{2}}\right)}}$, where $\Phi_{2}$ is the electromagnetic Newman-Penrose scalar. In the case of sound we do not know of a Newman-Penrose scalar for calculating the energy flux due to sound. Therefore in this case we will calculate the sound energy flux using the sound Lagrange density to calculate the energy-momentum tensor for sound and then from this obtain the sound energy flux. The Lagrange density for sound can be written as

[TABLE]

which is the scalar field equivalent of the vector, electromagnetic Lagrange density ${\cal L}_{EM}=\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}$. Using the standard procedure the energy-momentum tensor from ${\cal L}_{s}$ is

[TABLE]

The energy flux in the $i^{th}$ direction is now given by $T^{0i}_{s}$ and since the wave is traveling in the $z$-direction we want $T^{0z}_{s}=\frac{1}{4\pi}\partial^{0}\Pi\partial^{z}\Pi$. Actually we want the time averaged energy flux which is

[TABLE]

where we have used $\Pi(z,t)$ from (23) and the fact that $\partial^{0}=\frac{1}{c}\partial_{t}$. We use $\langle T^{0z}_{s}\rangle=\frac{h^{4}kK}{2\pi}$ to replace $\frac{1}{4\pi}|\Phi_{2}|^{2}$ when we calculate the ratio of sound energy flux to gravitational wave energy flux so that $\frac{dE_{s}/dt}{dE_{gw}/dt}=\frac{\langle T^{0z}\rangle}{\left({\frac{1}{16\pi k^{2}}\left|{\Psi_{4}}\right|^{2}}\right)}=\frac{2h^{2}K}{k}$. Using all this the flux of sound energy can be written

[TABLE]

where we have used $F_{gw}=\left(\frac{c^{3}h^{2}\omega^{2}}{16\pi G}\right)$ from section II, $\omega=2\pi f$, and $\frac{K}{k}=\frac{c}{c_{s}}$. The sound energy flux in (27) is similar to the electromagnetic flux in (6) except for an enhancement factor of $\frac{c}{c_{s}}$. For example, if the speed of sound where $10^{4}$ smaller than the speed of light (i.e. $\frac{c}{c_{s}}=10^{4}$) then this enhancement factor would increase the energy pumped into the system by the conversion of GWs to sound/phonons by a factor of $10^{4}$, or $10^{44}$ ergs which is still several orders of magnitude too small to directly lead to shock recovery, but as in the photon case of section II it could pump enough energy in the system to re-ignite fusion.

This phonon production in the relativistic hydrodynamics of the plasma is consistent with calculations of dispersion for gravitational waves in the Einstein equations including the stress energy of the plasma Madore72 ; Sacchetti79 ; Barta18 .

V Comments on the Computational CCSNe models

In this section we discuss several aspects of the computational models of CCSNe as they relate - pro or con - to the above proposed counterpart electromagnetic/sound radiation production from gravitational waves.

V.1 Energetics in one dimensional models

Since fusion in the iron core is endothermic, the electromagnetic production would have to occur in the silicon layer to contribute to the explosion mechanism through re-ignition of fusion. This proposed re-ignition of fusion outside the collapsing core could complement or help supplant the neutrino-driven explosion mechanism. However, the relation of the fusion re-ignition to a neutrino-driven explosion would be highly model dependent due to the inverse square relation of the radius of the power of the co-produced electromagnetic/sound radiation as given in Sections III and IV. The relative significance of this contribution to the explosion could only be determined by including this proposed effect in computer models of core-collapse supernova. One could also expect that the relative significance of this proposed re-ignition of silicon fusion would be highly dependent on the progenitor star which can vary greatly by mass, metallicity, and spin; as observed by Janka et al. Janka12 , “The predicted signals of stellar explosions depend on the complex interplay of a wide variety of input physics and require knowledge of the initial condition in the progenitor stars.”.

The timing of when the counterpart electromagnetic or sound radiation is produced is also critical in seeing if this gravitational wave co-produced electromagnetic or sound radiation is potentially important in CCSNe. In this regard one finds the favorable hint that counterpart electromagnetic and sound radiation are very nearly coincident with accepted models of stall and shock recovery Muller04 ; Janka07 ; Janka12 . The coincidence in the timing of the production of counterpart electromagnetic or sound radiation from gravitational waves with the possibility of fusion re-ignition just outside the iron core, is a positive point for the possibility that these mechanisms for re-ignition of fusion could be important in models of core-collapse supernova. Because of the wide variation of initial conditions of the progenitor star Ott16 , and the coincidence with the stall and recovery, it is not possible to determine a priori the relative importance of the counterpart electromagnetic or sound radiation production and other “input physics” to stellar explosion.

Because of the fourth order dependence of counterpart production on ${\tilde{A}}$ in Section III, one finds that the mean quadrupole wave amplitude will strongly influence the effect of counterpart radiation production in the supernova recovery and explosion. Models of gravitational wave quadrupole amplitudes give amplitudes on the order of centimeters, but can vary from $\sim 5$ cm to $\sim 50$ cm Muller04 ; Radice19 . This one order of magnitude change in the quadrupole wave amplitude can result in large changes in counterpart electromagnetic or sound power production by up to four orders of magnitude. The inclusion of counterpart production in CCSNe simulations would result in models which are very sensitive to the magnitude of the gravitational wave produced in core collapse. Such sensitivity to the gravitational waves coming from core collapse are not found in current models. Current models assume that the gravitational wave propagates freely out from the core and contributes only marginally to the supernova energetics. However, with the strong, fourth order dependence of counterpart electromagnetic and sound radiation production on the gravitational wave quadrupole amplitude, the inclusion of this co-produced radiation in models of stall recovery and explosion would offer a sensitive test for models of gravitational wave production in CCSNe which could be compared with observations.

There are other researchers who have proposed the idea that re-ignition of fusion could play a role in CCSNe. In an early work Burbidge57 the proposal was made that thermonuclear fusion through gravitational compression could play a role in CCSNe. This proposal was famously abandoned Janka12A because the energy produced in the stellar layers is insufficient. The possibility of this conventional thermonuclear mechanism producing a stellar explosion in core-collapse has been recently revived Kushnir15 ; Blum16 , where the model has produce supernova explosions in 1-dimensional models with highly tuned initial conditions. However, in Kushnir15 the lower layers of silicon and oxygen are assumed to contribute negligibly to re-ignition of fusion burning. Including the co-produced electromagnetic or sound radiation in these models could significantly effect the burning in the lower layers and hence the evolution of the models such as those proposed in Kushnir15 ; Blum16 .

V.2 Assumptions of three dimensional models

While inclusion of counterpart electromagnetic and/or sound production in symmetric, 1-dimensional models of core collapse is interesting, the more important applications are likely to be in asymmetric, 3-dimensional models. Re-ignition of fusion in the outer core of the progenitor star, and therefore the equation of state of the stellar material (i.e. temperature, pressure, density) will be asymmetric during the collapse. This lack of spherical symmetry during collapse, increases the efficiency of the neutrino driven explosion mechanism, helping to facilitate explosion Ott16 . Asymmetric models have added support to the neutrino mechanism but have not “convincingly established or even proven” Janka16 the neutrino driven mechanism. Co-produced radiation can re-ignite nuclear burning where the outgoing gravitational wave is strong. This re-ignition of nuclear fusion in the outer core of the star can greatly influence the local equation of state and in particular the local density. In turn the local density has a strong effect on the effectiveness of the neutrino driven mechanism for CCSNe explosion.

To investigate the potential influence of this change in local density we briefly review the interaction of the outgoing neutrino radiation with the material of the supernova progenitor. This will demonstrate the potential importance of counterpart electromagnetic or sound production and re-ignition of fusion in the standard neutrino mechanism. The order of magnitude of the neutrino cross section in CCSN processes is Bethe85

[TABLE]

where $E_{\nu}$ is the energy of the neutrinos in MeV. Using this cross section the neutrino mean free path length is

[TABLE]

where $m_{u}\approx 10^{-24}~{}\rm{g}$ and $\rho$ is the local density. Collecting terms the neutrino mean free path can be written as

[TABLE]

We see in (30) that the interaction of the outgoing neutrino radiation with the outer core is determined by the local nucleon density and the neutrino energy. In approximating the neutrino mean free path in (30) for the outer core, we will assume a CCSN neutrino energy, $E_{\nu}$, on the order of $10~{}\rm{MeV}$. The local nucleon density in the outer core is estimated to be on the order of $10^{6}-10^{12}~{}\rm{\frac{g}{cm^{3}}}$ Janka12 ; Bethe85 resulting in a neutrino mean free path on the order of $10^{5}-10^{11}~{}\rm{cm}$. The equation of state, and therefore the local density, is strongly dependent on nuclear burning. Thus the re-ignition of fusion by the local co-produced radiation would have a strong effect on the local density. This in turn would strongly influence the neutrino mean free path and therefore the effectiveness of neutrinos in exploding the star. Thus, including the effects of the co-produced radiation would make neutrino driven CCSNe highly dependent on the local power production of the co-produced radiation. The extent of the effect of this co-produced radiation on neutrino driven explosions can only be determined through the inclusion of this previously unconsidered physics in core collapse simulations.

V.3 Electrodynamic Assumptions in Existing Computational Models

Over the past few decades computer simulations of CCSNe have been greatly improved, but still do not “explode” often enough to be consistent with observations of CCSNe Ott16 ; Janka17 . The physical processes producing CCSNe are theoretically described by, general relativity, neutrino energetics, nuclear physics, hydrodynamics, and electrodynamics. The simulations of core-collapse are computationally demanding and some physical assumptions and approximations must be made. One method of simplification is to exclude some of the physics of CCSNe in the simulations. For example, many simulations include general relativity, neutrino energetics, nuclear physics, and hydrodynamics, but exclude electrodynamics. Including electrodynamics in the simulation can be computationally demanding. There are three cases generally considered for numerical simulations of CCSNe that include electrodynamics Bucciantini13 ; Palenzuela13 . The first is the isotropic and resistive case, $e^{\mu}=\eta j^{\mu}$, which is essentially the 4-vector form of usual Ohm’s Law, ${\bf J}=\sigma{\bf E}$. Here $e^{\mu}=F^{\mu\nu}u_{\nu}$ is the “electric” field of an observer co-moving with respect to the fluid. The second case is the inclusion of a mean field dynamo, $e^{\mu}=\xi b^{\mu}+\eta j^{\mu}$ which is a 4-vector generalization of Ohm’s Law to include magnetic fields. Here $b^{\mu}={\cal F}^{\mu\nu}u_{\nu}$ is the “magnetic” field of an observer co-moving with respect to the fluid. The third and final case is the ideal magnetohydrodynamic limit $e^{\mu}=0$ which in 3-vector form becomes $E^{i}=-\epsilon^{ijk}v_{j}B_{k}$.

If existing simulations of the CCSNe did include more of the physics – in particular if they included gravitational wave production and full electrodynamics – the energetics associated with counterpart electromagnetic radiation production would be realized in the computer simulations. However, to date, the most realistic simulations of CCSNe, that include magnetohydrodynamics, impose the ideal magnetohydrodynamic limit, $e^{\mu}=0$ Font08 . This assumption eliminates the possibility that these simulations will take into account electromagnetic radiation, especially the counterpart electromagnetic radiation proposed here.

VI Conclusions

Our rough, first order calculation, of the contribution to supernova explosion from electromagnetic and sonic radiation co-produced by gravitational waves from core-collapse, cannot conclusively demonstrate the significance of these phenomena in stellar explosion. However, the converse is equally clear, that models of core-collapse supernova may be missing a potentially important or even critical pieces of physics. The point of this work is that computer simulations of core-collapse supernova should consider this process of converting gravitational wave energy into electromagnetic/sound wave energy, along with more well known mechanisms.

Interestingly the ideas presented here of the heating of the material in a supernova bear a resemblance to mechanisms proposed for heating the solar corona solar-corona . It has been known for some time that the solar corona, which is several solar radii above the Sun’s surface (i.e. the photosphere) is heated to several millions of degrees Kelvin while the photosphere is a much lower temperature of about 6000 K. One of the explanations for this is that sound waves carry energy from the photosphere to the corona and then deposit their energy in the corona as the corona gets too diffuse to propagate the sound waves and their energy further. Here, one of the mechanisms for pumping extra energy into the material of the supernova is for the gravitational waves to produce phonons/sonic radiation which then reheats the material to the point where fusion can be re-ignited.

Acknowledgment: PJ would like to thank Brennan Hughey for helpful discussions and Quentin Bailey for asking the right questions about existing CCSNe computational models. DS is supported by a 2023-2024 KITP Fellows Award. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.