Euler-Heisenberg black hole surrounded by perfect fluid dark matter

A generation method of new metric in the case of static spherically symmetric space-time is derived. Using this approach, we construct a metric which describes Euler-Heisenberg black hole surrounded by perfect fluid dark matter and investigate its optical and thermodynamic properties. We found that radius of shadow will increase with the increase of dark matter effect, and more strong dark matter will diminish the light intensity of accretion disk generally. Moreover, in thermodynamics, when quantum electrodynamic parameter is positive, there will be a critical value of dark matter parameter, which determine the number of black hole's critical points.


I. INTRODUCTION
Euler-Heisenberg (EH) theory is a kind of nonlinear electrodynamics, which takes quantum electrodynamics (QED) into account [1].Based on EH theory, scientists obtained charged black holes called 'EH black holes ' [2], whose optics and thermodynamics have been widely studied.EH black hole's shadow and luminosity of accretion flow aren't sensitive to QED effect, which possibly indicates one-loop correction to QED hardly works in optics [3].In contrast, QED correction will bring a transparent impact in thermodynamics compared to RN black hole.For instance, QED parameter will crucially determine the number of critical points [4, 5] and it will cause a small deviation on the critical ratio [6].In addition, high-order QED correction has been considered as well [7].On the other hand, it is believed that dark matter could potentially explain the expanding universe.As a special model of dark matter, perfect fluid dark matter (PFDM) is an ideal model that satisfies equation of state p/ρ = ϵ, leading to a static spherically space-time described by the power function or the logarithmic function [8][9][10].The possibility of PFDM in a phantom field constrained by motions of stars in spiral galaxy was studied in [11,12].Especially, PFDM inducing logarithmic form of space-time metric has attracted more interest of physicists [9,12,13].Studying the properties of black holes in the presence of PFDM will uncover the influence of dark matter on black hole physics and help to understand their mechanism in theory of gravity [8,9,13].
Optics and thermodynamics are main subjects in black hole physics and already have very mature research process.Due to black hole's huge gravitational pull, the light surrounding it will be significantly bent, leading to fascinating optical phenomena, with the shadow and image of accretion disk being most captivating [14][15][16][17][18].
In thermodynamics, since Hawking and Page firstly studied the thermodynamics of black hole in anti-de Sitter (AdS) space-time [19], there are many significant researches published [20][21][22][23].Generally speaking, the variation of cosmological constant in extended phase space will lead to more interesting results.This paper aims to take EH black hole into PFDM background and study its optical nature and thermodynamics.To obtain the solution of field equation more conveniently, this article will also provide a method to generate the new metric in static spherically symmetric space-time.Looking forward, if there are related experiments in future, our research will help to distinguish whether QED effect and dark matter work.The article is organized as follows.In Sect.II, we discuss the generation of new metric and give its corresponding conditions mathematically.In Sect.III, the metric of EH black hole surrounded by PFDM is obtained.In Sect.IV, we calculate the photon orbit and plot the image of thin accretion disk using Novikov-Thorne model.The results with different parameters and effect of Doppler shift will be shown.In Sect.V, thermodynamics of EH-AdS black hole surrounded by PFDM will be investigated, especially its criticality.The conclusion and outlook will be given in Sect.VI.We always set G = c = k = ℏ = 1.

Consider a action
where g is determinant of the metric g µν , R is scalar curvature and L (i) is Lagrangian, which represents i th effect, like electromagnetic field, dark matter, etc.We always suppose every L (i) has its corresponding static spherically symmetric solution and we only discusss static spherically symmetric space-time in this article.The coordinate system takes spherical coordinate (t, r, θ, ϕ), where t ∈ R, r > 0, θ ∈ [0, π] and ϕ ∈ R.
By varying the Eq.(II.1) with the respect to the g µν , one could get the gravitational field equation, where R µν is Ricci tensor and T µν is energy momentum tensor, which is variation of Now consider a specific circumstance that Lagrangians L (i) in Eq. (II.1) are independent.Generally, the form of L (i) is where µ describes the i th effect.We just consider the first derivative of As an example, for electromagnetic field, it is four-potential.The independence of L (i)   means that L (i) has no relationship with A (j) Theoretically, if two effects are coupling, there should be a Lagrangian describing their interaction: Only combining L (i,j) , L (i) and L (j) as one term could deal with this problem.
For our later discussion, we use g µν to raise index of Eq. (II.2): where T is independent from space-time itself.It means after solving the field equation of substituting the solution A 0(i) µ and metric only L (i) exists, , r 2 , r 2 sin θ (II.9) into T 0(i) 0 , the result should be independent from the f i (r), which means Now the solution of Eq. (II.7) could be discussed under above two requirements.

One could get
According to independence of L (i) , this equation is equivalent to Eq. (II.8).Assume is the solution of Eq. (II.8) and the solution of Eq. (II.7) is , r 2 , r 2 sin θ .
(II.12) Substitute g (total) µν into Eq.(II.7), nonzero term is (II.13) In the same way, where C is integration constant to be determined.
Here, it's necessary to point out that above discussion is just a mechanical deduction.In practice, if T in case of various interactions can be replaced with T 0(i) 0 when there is only one interaction, which means function of space-time itself has no influence on form of T 0(i) 0 , then one can draw a conclusion that the metric will simply meet with Eq. (II.15).Just in mathematical way, Eq. (II.5) and Eq.(II.10) can insure it.
It's reasonable to propose f i (r) in Eq. (II.9) as the influence of L (i) on the flat space-time.So, if many interactions have no coupling with each other, this impact on the space-time can get simply superimposed.Moreover, Schwawrzschild-like term C/r in Eq. (II.15) will not hinder the superimposition because C/r has no contribution to energy momentum tensor.

III. EH BLACK HOLE SURROUNDED BY PFDM
In this section, we use the method in the previous section to construct a spacetime, which describes a EH black hole surrounded by PFDM.The optical properties and thermodynamics will be investigated in Sect.IV and Sect.V.
Firstly, [2, 3] provides the line element of EH black hole, its corresponding energy momentum tensor can be found from [6], where M is the mass of the black hole, Q is the electric charge of the black hole and a is the QED parameter which is used to measure the strength of the QED correction.
Secondly, we reference the energy momentum tensor of PFDM mentioned in [13], where α is the dark matter parameter.Via substituting Eq. (III.3)into Eq.(II.13), one get the line element of space-time only PFDM exists, (III.4) Considering that dark matter won't participate with the electromagnetic interaction and it is obvious that above two energy momentum tensors are both independent from metric function f (r) to be determined.According to generation method studied in Sect.II, the line element of EH black hole surrounded by PFDM is derived (III.5) Please notice that integration constant C has been chosen as zero to insure Eq. (III.5) can degenerate to EH black hole by choosing α as zero.
For convenience, we uniformly use M = 1 in the figures in Sect.IV.

IV. OPTICAL PROPERTIES
In this section, we will study the optical properties of EH black hole surrounded by PFDM, including the photon orbit and image of thin accretion disk.

A. Geodesic equation and photon orbit
Consider a particle moving in this space-time where ẋµ = dx µ /dλ is four-velocity of particle, λ is proper time τ for physical particle and affine parameter for photon.For simplicity, we choose θ = π/2, θ = 0, which means particle only moves in equatorial plane.Euler-Lagrange equation gives two conserved quantities, where E and L represent the energy and angular momentum of particle.According to Eq. (IV.2),Eq. (IV.3) and Eq.(IV.4), If use Eq.(IV.4) and Eq.(IV.5) to eliminate parameter λ, dr dϕ Now consider the motion of photon.For photon, L = 0, Eq. (IV.6) becomes With the transform u = 1/r, Eq. (IV.7) is rewritten as where For photon, parameter b is the distance from the black hole to the asymptotic line of photon orbit at infinity.
We first focus on the circular orbit of photon, which satisfies   disk, the effects of friction, collisions, and compression will result in intense thermal radiation, which can be observed by external observers.
Thin accretion disk means that accretion disk can be simply abstracted into a plane with negligible thickness, and the influence of accretion disk on space-time can be ignored.
Consider an observer at infinity to ensure he will receive parallel rays, which will simplify our discussion.Coordinate system is set in (IV.12) If fix r, one will get a curve expressed by b = b (β) in xOy plane, which is precisely the image of circular orbit with a radius of r on the disk.Because the photon in this case will pass through the disk for n times, this image is called 'n th order image'.Now introduce a crucial concept: the inner edge r in of accretion disk.The inner edge r in of the accretion disk actually means the innermost stable circular orbit, because circular orbits within this range will be unstable, such that the particle will fall into the black hole or escape to flee farther when it is perturbed.r in should satisfy [17,18] It's worth mentioning that according to [14], n th order image is negligible when n ≥ 3.So we plot the first and second order images of accretion disk for different azimuth angles θ of observer.
As seen in Fig. IV.5, when the observer is in direction of North Pole, a series of circles will appear, which is consistent with the symmetry of space-time.As θ increases, the first order image gradually takes on a hat-like shape.
To obtain the image with brightness, the Novikov-Thorne model is chosen [24,25], where F (r) is radiation flux, M is black hole's accretion rate, g is metric determinant, and r in represents the inner edge of accretion disk.E, Ω and L are energy, angular velocity and angular momentum of physical particle moving on the disk.
Consider a particle moving in a circular orbit, use L = −1/2, one get Ω by particle's geodesic equation, Signs indicate two directions of rotation and have no essential distinctions.We choose positive in future calculations.Energy and angular momentum are given by Eq. (IV.3) and Eq.(IV.4): . (IV.17 Due to the different gravitational fields at disk and observer and their relative motion, the frequency shift will happen.According to [14], the radiation flux observer received will be where z is redshift factor defined as E em and E obs are the the emitted light intensity at the light source and the light intensity received by the observer, respectively.According to [15], where p µ = g µν ẋν is photon's four-momentum and u µ is particle's four-velocity.The (ii) Comparing to the first order image, the second order image already became very weak.It's foreseeable that for n ≥ 3, the image will be negligible; (iv) In general, the brightness of disk when α = −0.4 is more strong than α = −0.5.This conforms to the result in Fig. IV.6, showing that the more intense dark matter effect will significantly reduce the brightness of disk.Although at a long distance, PFDM will increase the radiation flux slowly, the radiation flux rapidly tends to zero, making it appear negligible in the image.Of note is that the when θ = π/3, the first image for α = −0.5 is more intense than α = −0.4,which is possible due to Doppler effect because redshift factor is also related to α.

V. THERMODYNAMICS
In this section, we investigate the thermodynamics of EH black hole surrounded by PFDM in AdS space-time.Research for black hole thermodynamics in extended phase space requires us to regard cosmological constant Λ as a thermodynamic variable.It seems to have contradiction, but there are also some good reasons why the variation of Λ should be included in considerations [26].And it's clear to see the physical meaning in [22], which shows that Λ is regarded as pressure, A. EH-AdS black hole surrounded by PFDM One can regard the cosmological constant Λ as an effect of special Lagrangian written as whose corresponding energy momentum tensor is which caused field equation with cosmological constant.
Anti-de Sitter space-time is which apparently is just a constant without f (r).And further, EH theory and model of PFDM have nothing to do with cosmological constant.That is to say, these three effects have no interactions.
According to the discussion in Sect.II, when take the EH black hole surrounded by PFDM into AdS space-time, the line element will convert to

B. Thermodynamic functions and equation of state
The mass of black hole satisfies g (r h ) = 0, The Hawking temperature is defined by its surface gravity [27], The entropy is Thermodynamic first law is Here M is mass of black hole, which is also interpreted as enthalpy.T is temperature, V is thermodynamic volume, Φ is electric potential and µ can be treated as dark matter potential.V , Φ and µ can be calculated form Eq. (V.10), It's crucial to verify whether T = T H because T ̸ = T H is possible [28].If T ̸ = T H , revising thermodynamic first law becomes necessary.

C. Phase transition and critical point
In the extended thermodynamic framework, the temperature T is rewritten as a function of pressure P and radius of horizon r h , or entropy We mainly pay attention to criticality, the condition of critical point is mentioned in [29], And one can find the condition of second-order phase transition which is rewritten as following formula according to Eq. (V.15), ∂T ∂r h P,α,a,Q = 0, (V.21) whose solution is Substitute this solution into Eq.V.18, one could get the temperature when secondorder phase transition occurs T p : With the help of Appendix.A, critical point meets with In our subsequent discussion, 'phase transition' we talk about default to 'secondorder phase transition'.
One can find the phase transition curves in Figure .V.1.One could find critical points CP 1 and CP 2 on the phase transition curves.One can get α c by substituting Eq. (V.30) into Eq.(V.25).

VI. CONCLUSION AND OUTLOOK
Firstly, we derived a method to generate a new metric for static spherically symmetric space-time, which showed if multiple interactions have no coupling and energy momentum tensor T 0 0 is independent from metric to be determined, the solution of field equation will conform to the linear superposition.Using the this method, we obtained the metric of EH black hole surrounded by PFDM and study its optics and thermodynamics.It is found that shadow will increase when dark matter become intense.More strong dark matter will diminish the light intensity of accretion disk and there is exception when considering Doppler effect.In thermodynamics, one can see that PFDM will drastically affect the number of critical points when QED parameter is positive.
Looking forward, due to the conservation of angular momentum, black holes that are born in the collapse of stars are rarely non-rotating.Therefore, it will be more realistic to consider rotation.In thermodynamics, there are more properties to be researched, like critical exponents [26,28].It is meaningful to study if thermodynamics of this kind of black holes is similar to Van der Waals system [20,21,23].And there is Joule-Thomson expansion, which is corresponding to isenthalpic process.It is meaningful to investigate Joule-Thomson expansion of this black hole compared to existed researches [30][31][32].
Record the solution as r = r ph , b = b c .r ph is the radius of circular orbit and b c = r ph / g (r ph ) is impact parameter of the photon in circular orbit.Please notice that A ′ represents dA/dr for any scalar function A. Next, we will show how the value of b influences the light's motion.Take M = Q = a = 1, α = −0.5 for an example, the function V ef f (r) for different b is plotted in Fig. IV.1.As seen in the figure: For b > b c , there is a maximum positive root r m (b) of V ef f .The photon comes from infinity and will never enter the region r < r m .Instead, it will reach the point r m and then return to infinity along a path with the same shape as the incoming trajectory; For b = b c , the photon will arrive r = r ph and then perpetually undergo circular motion; For b < b c , the photon will continuously approach the black hole until falling into the horizon.Consider an observer at infinity to ensure he will only receive parallel light, then b c is precisely the radius of the black hole shadow he observes, because of the reversibility of light path.
Figure IV.1:Variation of V ef f with respect to r.The red, blue and orange lines

yFigure
Figure IV.2:(a) Graph of function φ (b).(b) Photon orbits for different values of b.
5: The first order images (red lines) and second order images (orange lines) of circular orbit r = constant.on accretion disk.The values of r are chosen from r in to 50M equidistantly.θ from left to right are 0, π/6, π/4 and π/3

)Fig. IV. 6
Fig. IV.6 shows how PFDM affects the radiation flux of accretion disk.At short

Figure IV. 6 :
Figure IV.6:Graphs of relative radiation flux for different α.I em α is the radiation flux F (r).I 0 α=−0.5 is the maximum value of radiation flux for α = −0.5.We set

Figure IV. 7 :
Figure IV.7:First and second order images of accretion disk for different θ. θ from

Figure IV. 8 :
Figure IV.8:First and second order images of accretion disk for different θ. θ from

Figure V. 1 :
Figure V.1: The solid lines in the figure represent second-order phase transition

TFigure V. 3 :
Figure V.2: The solid lines in figure represent critical point curve T cp , the other