Chaos and fast scrambling delays of dyonic Kerr-Sen-AdS₄ black hole and its ultra-spinning version

The redefinition of the coordinate $\varphi\rightarrow\varphi/\Xi$ allows us to obtain the ultra-spinning limit $a\rightarrow l$ of the dyonic Kerr-Sen-AdS₄ black hole, which now has the metric and the corresponding fields in the form of

	$\displaystyle d\hat{s}^{2}$	$\displaystyle=-\frac{\hat{\Delta}}{\hat{\Sigma}}\hat{X}^{2}+\frac{\hat{\Sigma}}{\hat{\Delta}}dr^{2}+\frac{\hat{\Sigma}}{\sin^{2}\theta}d\theta^{2}+\frac{\sin^{4}\theta}{\hat{\Sigma}}\hat{Y}^{2},$
	$\displaystyle A$	$\displaystyle=\frac{q(r+d-p^{2}/m)}{\hat{\Sigma}}\hat{X}=\frac{p\cos\theta}{\hat{\Sigma}}\hat{Y},$
	$\displaystyle B$	$\displaystyle=\frac{p(r+d-p^{2}/m)}{\hat{\Sigma}}\hat{X}+\frac{q\cos\theta}{\hat{\Sigma}}\hat{Y},$
	$\displaystyle e^{\phi}$	$\displaystyle=\frac{(r+d)^{2}+(l\cos\theta+k)^{2}}{\hat{\Sigma}},$		(14)
	$\displaystyle\chi$	$\displaystyle=2\frac{kr-dl\cos\theta}{r^{2}+(l\cos\theta+k)^{2}},$

with

	$\displaystyle\hat{X}$	$\displaystyle=dt-l\sin^{2}\theta d\varphi,$		(15)
	$\displaystyle\hat{Y}$	$\displaystyle=ldt-(r^{2}-d^{2}-k^{2}+l^{2})d\varphi,$
	$\displaystyle\hat{\Delta}(r)$	$\displaystyle=(r^{2}-d^{2}-k^{2}+l^{2})^{2}/l^{2}-2Mr+p^{2}+q^{2},$
	$\displaystyle\hat{\Sigma}$	$\displaystyle=r^{2}-d^{2}-k^{2}+l^{2}\cos^{2}\theta.$		(16)

The periodicity of $\varphi$ is now assumed to be given by $\lambda$ instead of $2\pi$. Notice that the standard dyonic Kerr-Sen-AdS₄ metric $ds^{2}$ cannot be obtained directly from the ultra-spinning metric $d\hat{s}^{2}$.
The thermodynamic quantities of the ultra-spinning dyonic Kerr-Sen-AdS₄ black hole are given by

	$\displaystyle\hat{M}$	$\displaystyle=\frac{\mu}{2\pi}m,\;\;\hat{J}=\frac{\mu}{2\pi}ml=Ml,\;\;\hat{Q}=\frac{\mu}{2\pi}q,\;\;\hat{P}=\frac{\mu}{2\pi}p,$
	$\displaystyle\hat{S}$	$\displaystyle=\frac{\mu}{2}(r_{+}^{2}-d^{2}-k^{2}+l^{2}),\;\;\hat{\Omega}_{\varphi}=\frac{l}{r_{+}^{2}-d^{2}-k^{2}+l^{2}},$
	$\displaystyle\hat{\Phi}$	$\displaystyle=\frac{q(r_{+}+d-p^{2}/m)}{r_{+}^{2}-d^{2}-k^{2}+l^{2}},\;\;\;\hat{\Psi}=\frac{p(r_{+}+d-p^{2}/2m)}{r_{+}^{2}-d^{2}-k^{2}+l^{2}},$
	$\displaystyle 2\pi\hat{T}_{H}$	$\displaystyle=2\frac{r_{+}}{l^{2}}-\frac{m}{r_{+}^{2}-d^{2}-k^{2}+l^{2}}.$		(17)

Again, $r_{+}$ is defined as the outermost horizon satisfying $\hat{\Delta}(\hat{r}_{+})=0$ and we assume that the ultra-spinning black hole is also non-extremal with $\hat{\Delta}^{\prime}(r_{+})\neq 0$. The first-law-like relation is now given by

$$d\hat{M}=\hat{T}_{H}d\hat{S}+\hat{\Omega}_{\varphi}d\hat{J}+\hat{\Phi}d\hat{Q}+\hat{\Psi}d\hat{P}+\hat{K}d\lambda,$$

(18)

where $\lambda$ also plays a role as the dynamical variable with a new chemical potential

$$K=m\frac{l^{2}-(r_{+}+d+q^{2}/m)(r_{+}+d-p^{2}/m)}{4\pi(r_{+}^{2}-d^{2}-k^{2}+l^{2})}.$$

(19)

Consider an equatorial ($\theta=\pi/2$) rotating null particle in the background of the dyonic Kerr-Sen-AdS₄ black hole with unit energy $\mathcal{E}=1$ and angular momentum per unit energy $\mathcal{L}$. We will then call this the rotating shock wave. Such a particle will have a geodesic $\xi^{\mu}$ observed by a stationary observer at infinity which satisfies

$$\xi^{2}=0,\;\;\;\xi\cdot\zeta_{t}=-\mathcal{E},\;\;\;\xi\cdot\zeta_{\varphi}=\mathcal{L},\;\;\;\xi^{\theta}=0.$$

(20)

The last equality is required for the equatorial orbit, which means that the particle always stays at the equator $\theta=\pi/2$. The corresponding Killing vector associated with axisymmetry is given by $\zeta_{t}=\partial_{t}-a/l^{2}\partial_{\varphi}$ and $\zeta_{\varphi}=\partial_{\varphi}$. There are two solutions to eq. (20), $\xi_{\pm}^{\mu}$, where the negative solution is obtained from reversing the axisymmetry $-\mathcal{E}\rightarrow\mathcal{E}$ and $\mathcal{L}\rightarrow-\mathcal{L}$. The geodesic solution in the background of a dyonic Kerr-Sen-AdS₄ black hole for unit energy ($\mathcal{E}=1$) is given by

$$\xi_{+}\cdot dx=dr_{*}-d\tau\equiv du,\;\;\;\xi_{-}\cdot dx=dr_{*}+d\tau\equiv dv,$$

(21)

where $\tau=(1-a\mathcal{L}/l^{2})t-\mathcal{L}\varphi$ and $r_{*}$ is the tortoise-like coordinate given by

$$r_{*}(r)=\int\frac{\tilde{f}}{\Delta}dr,$$

(22)

with $\tilde{f}$ is defined from

$$\tilde{f}^{2}=-\Delta(\mathcal{L}-a)^{2}+[\mathcal{L}a(1+(r^{2}-d^{2}-k^{2})/l^{2})-(r^{2}-d^{2}-k^{2}+a^{2})]^{2}.$$

(23)

With vanishing dilaton and axion charge, i.e. $d,k\rightarrow 0$, we recover the tortoise-like coordinate found in [6] and with vanishing angular momentum $\mathcal{L}\rightarrow 0$, we recover the standard Kruskal coordinates for rotating dyonic Kerr-Sen-AdS₄ black hole.
The Kruskal-like metric which follows the geodesic of the rotating shock wave is given by

$$ds^{2}=F(r)dudv+\tilde{h}(r)(d\varphi+\tilde{h}_{\tau}(r)d\tau)^{2},$$

(24)

with

	$\displaystyle F(r)$	$\displaystyle=\frac{\Delta(r^{2}-d^{2}-k^{2})}{\tilde{f}^{2}},\;\;\;\tilde{h}(r)=\frac{(1-a\mathcal{L}/l^{2})^{-2}}{r^{2}-d^{2}-k^{2}}\frac{\tilde{f}^{2}}{\Xi^{2}},$
	$\displaystyle\tilde{h}_{\tau}(r)$	$\displaystyle=\frac{\Xi}{\tilde{f}^{2}}\big{(}(\Delta(a-\mathcal{L})+a\big{(}a\mathcal{L}(1+(r^{2}-d^{2}-k^{2})/l^{2})-(r^{2}-d^{2}-k^{2}+a^{2})\big{)}\big{)}.$		(25)

We then shift the axial coordinate $\varphi\rightarrow\eta z+\gamma\tau$ for two reasons: one is to make sure that $\tilde{h}_{\tau}(r)+\gamma$ behaves like $\mathcal{O}(r-r_{+})$ near the horizon and to recover the black hole’s horizon area when integrating $\theta$ from $0$ to $\pi$ and $z$ from $0$ to $2\pi$. Thus, we have

$$\gamma=\frac{\Omega_{\varphi}}{1-\mu\mathcal{L}},\;\;\;\eta=\frac{1}{1-\mu\mathcal{L}}.$$

(26)

From the coordinate transformations, we have

$$ds^{2}=F(r)dudv+h(r)(dz+h_{\tau}(r)d\tau)^{2},$$

(27)

where $h(r)=\eta^{2}\tilde{h}(r)$ and $h_{\tau}(r)\equiv\eta^{-1}(\tilde{h}_{\tau}(r)+\gamma)$.
Furthermore, we would like to work with coordinates that are affine at the horizon to generate the Dray-’t Hooft solution later on. In this case, $\{u,v\}$ coordinate is not affine, i.e. the tangent vector $\chi_{u}=\partial_{u}$ satisfy $\chi_{u}\cdot\nabla\chi_{u}^{\mu}=\mathcal{K}\chi_{u}^{\mu}$ (similar with $\chi_{v}^{\mu}$), where

$$\mathcal{K}=\frac{1}{2}\xi_{\pm}\cdot\partial F.$$

(28)

The new coordinates $\{U,V\}$ are

$$U=-e^{\kappa u},\;\;\;V=e^{\kappa v},\;\;\;\kappa\equiv\mathcal{K}|_{r=r_{+}},$$

(29)

and it can easily be shown that both $\partial_{U}$ and $\partial_{V}$ are affine coordinates at $r_{+}$. The metric in the affine coordinates is now written as

$$ds^{2}=\frac{F(r)}{\kappa^{2}UV}dUdV+h(r)(dz+h_{\tau}(r)d\tau)^{2}.$$

(30)

The value of $\kappa$ is related to the black hole’s surface gravity $\kappa_{0}$ by

$$\kappa=\frac{\kappa_{0}}{1-\mu\mathcal{L}}=\frac{2\pi T_{H}}{1-\mu\mathcal{L}}.$$

(31)

This $\kappa$ has a form similar to the standard Kerr-AdS₄ black hole with rotating shock waves. However, it differs in the value of the Hawking temperature $T_{H}$ and the angular momentum $\mu$ which now depends on the value of the dilaton and axion charge $d,k$.
Next, we would like to generate the Kruskal-like coordinates for the ultra-spinning counterpart where the metric is given by eq. (II.1). Since the ultra-spinning version of the dyonic Kerr-Sen-AdS₄ has a null $\varphi$ coordinate at the boundary $r\rightarrow\infty$ and hence the metric is non-diagonal, we need to adopt the so-called conformal completion technique to compute the conserved quantities [49, 42]. The mass and the angular momentum of the ultra-spinning dyonic Kerr-Sen-AdS₄ black hole given by eq. (17) is obtained by calculating the conserved charge $\mathcal{Q}[\zeta]$ defined in eq. (9) of [42] for a Killing vector $\zeta$. The timelike Killing vector $\zeta_{t}=\partial_{t}$ gives us the mass $\hat{M}$, while the axial Killing vector $\zeta_{\varphi}=\partial_{\varphi}$ gives us the angular momentum $\hat{J}$. Therefore, we also have to use the Killing vectors at infinity $\zeta_{t}=\partial_{t}$ and $\zeta_{\varphi}=\partial_{\varphi}$ which corresponds to the conserved energy $\mathcal{E}$ and angular momentum $\mathcal{L}$ of a rotating shock wave around the ultra-spinning black hole.
The geodesic of a rotating shock wave $\hat{\xi}^{\mu}$ in the background of an ultra-spinning dyonic Kerr-Sen-Black hole is obtained by solving eq. (20) with the ultra-spinning black hole metric in eqs (II.1) and (16) and Killing vectors $\zeta_{t}=\partial_{t},\zeta_{\varphi}=\partial_{\varphi}$. The solutions for unit energy $\mathcal{E}=1$ and an angular momentum per unit energy $\mathcal{L}$ are given in the form of

$$\hat{\xi}_{+}\cdot dx=d\hat{r}_{*}-d\hat{\tau}\equiv d\hat{u},\;\;\;\hat{\xi}_{-}\cdot dx=d\hat{r}_{*}+d\hat{\tau}\equiv d\hat{v},$$

(32)

where $\hat{\tau}=t-\mathcal{L}\varphi$ and

$$r_{*}\equiv\int\frac{\hat{\tilde{f}}dr}{\hat{\Delta}},$$

(33)

with

$$\hat{\tilde{f}}^{2}=-\hat{\Delta}(\mathcal{L}-l)^{2}+\big{(}\mathcal{L}l-(r^{2}-d^{2}-k^{2}+l^{2})\big{)}^{2}.$$

(34)

Notice that both $\hat{\tau}$ and $\hat{\tilde{f}}$ cannot be obtained by taking $a\rightarrow l$ limit of $\tilde{f}$. The Kruskal-like metric for the ultra-spinning black hole with rotating shock waves is given by

$$d\hat{s}^{2}=\hat{F}(r)d\hat{u}d\hat{v}+\hat{\tilde{h}}(r)(d\varphi+\hat{\tilde{h}}_{\hat{\tau}}(r)d\hat{\tau})^{2},$$

(35)

where now we have

	$\displaystyle\hat{F}(r)$	$\displaystyle=\frac{\hat{\Delta}(r^{2}-d^{2}-k^{2})}{\hat{\tilde{f}}^{2}},\;\;\;\hat{\tilde{h}}(r)=\frac{\tilde{f}^{2}}{r^{2}-d^{2}-k^{2}},$
	$\displaystyle\hat{\tilde{h}}_{\hat{\tau}}(r)$	$\displaystyle=\frac{-\hat{\Delta}(\mathcal{L}-l)^{2}+l(\mathcal{L}l-(r^{2}-d^{2}-k^{2}+l^{2}))}{\tilde{f}^{2}}.$		(36)

Following similar reasoning to the normal dyonic Kerr-Sen-AdS₄, we redefine the coordinate $\varphi\rightarrow\hat{\eta}\hat{z}+\hat{\gamma}\hat{\tau}$, where now

$$\hat{\eta}=\frac{1}{1-\hat{\Omega}_{\varphi}\mathcal{L}},\;\;\;\hat{\gamma}=\frac{\hat{\Omega}_{\varphi}}{1-\hat{\Omega}_{\varphi}\mathcal{L}}.$$

(37)

Again, all of the functions appear in the ultra-spinning case $\{\hat{F},\hat{\tilde{h}},\hat{\tilde{h}}_{\hat{\tau}},\hat{\eta},\hat{\gamma}\}$ cannot be directly obtained from $\{F,\tilde{h},\tilde{h}_{\tau},\eta,\gamma\}$ by taking the limit of $a\rightarrow l$ and thus they are all completely different functions. After the coordinate transformation $\varphi\rightarrow\hat{\eta}\hat{z}+\hat{\gamma}\hat{\tau}$, the Kruskal-like metric for the ultra-spinning black hole now takes the form

$$d\hat{s}^{2}=\hat{F}(r)d\hat{u}d\hat{v}+\hat{h}(r)(d\hat{z}+\hat{h}_{\hat{\tau}}(r)d\hat{\tau})^{2},$$

(38)

where $\hat{h}(r)=\hat{\eta}^{2}\hat{\tilde{h}}(r)$ and $\hat{h}_{\hat{\tau}}(r)=\hat{\eta}^{-1}(\hat{\tilde{h}}_{\hat{\tau}}+\hat{\gamma})$.
We also would like to work with affine coordinates $\{\hat{U},\hat{V}\}$ instead of $\{\hat{u},\hat{v}\}$ for the ultra-spinning black hole. Following similar reasoning as before, we obtain

$$\hat{U}=-e^{\hat{\kappa}\hat{u}},\;\;\;\hat{V}=e^{\hat{\kappa}\hat{v}},$$

(39)

where now $\hat{\kappa}$ is given by

$$\hat{\kappa}=\frac{2\pi\hat{T}_{H}}{(1-\hat{\Omega}_{\varphi}\mathcal{L})}.$$

(40)

The metric in the affine coordinates $\{\hat{U},\hat{V}\}$ is now given by

$$d\hat{s}^{2}=\frac{\hat{F}(r)}{\hat{\kappa}^{2}\hat{U}\hat{V}}d\hat{U}d\hat{V}+\hat{h}(r)(d\hat{z}+\hat{h}_{\hat{\tau}}(r)d\hat{\tau})^{2}.$$

(41)

In this coordinate system, we then generate the Dray-’t Hooft solution for the ultra-spinning dyonic Kerr-Sen-AdS₄ black hole.
After we obtain the metric in Kruskal coordinates for the dyonic Kerr-Sen-AdS₄ and its ultra-spinning counterpart, we now see how the functions $F(r)$ and $\hat{F}(r)$ behave near the horizon. Indeed, by expanding $r$ near $r_{+}$ up to the second order, we have

	$\displaystyle F(r)=$	$\displaystyle F^{\prime}(r_{+})(r-r_{+})+$		(42)
		$\displaystyle\frac{1}{2}F^{\prime\prime}(r_{+})(r-r_{+})^{2}+...\;,$
	$\displaystyle\hat{F}(r)=$	$\displaystyle\hat{F}^{\prime}(r_{+})(r-r_{+})+$		(43)
		$\displaystyle\frac{1}{2}\hat{F}^{\prime\prime}(r_{+})(r-r_{+})^{2}+...\;,$

with all $F^{\prime}(r_{+}),F^{\prime\prime}(r_{+}),\hat{F}^{\prime}(r_{+}),\hat{F}^{\prime\prime}(r_{+})$ are non zero. Using the definitions of the Kruskal coordinates in eq. (29) and the tortoise-like coordinates in (22) for the dyonic Kerr-Sen-AdS₄ black hole, we may write that, near the horizon,

$\displaystyle(r-r_{+})F^{\prime}(r_{+})$

$\displaystyle=-\mathbb{A}UV,$

(44)

where $\mathbb{A}$ is some dimensionless proportionality constant which depends on the dyonic Kerr-Sen-AdS₄ black hole parameters.
One need to be more cautious when considering $\hat{U}\hat{V}$ near the horizon for the ultra-spinning black hole because $\hat{\kappa}$ is now equipped with $\hat{\Omega}_{\varphi}$ instead of $\mu$. However, it turns out that the value of $\hat{U}\hat{V}$ near the horizon is still linear in $r-r_{+}$ since the tortoise coordinate near the horizon behaves as

	$\displaystyle r_{*}$	$\displaystyle\approx\frac{(1-\hat{\Omega}_{\varphi}\mathcal{L})(r_{+}^{2}-d^{2}-k^{2})}{\hat{\Delta}^{\prime}(r_{+})}\ln|r-r_{+}|+C$		(45)
		$\displaystyle=\frac{1}{2\hat{\kappa}}\ln|r-r_{+}|+C,$

where $C$ is an integration constant in which the explicit value depends on the geometry of the black hole. Therefore, $\hat{U}\hat{V}\sim\mathcal{O}(r-r_{+})$ near the horizon and we may also write

$$(r-r_{+})\hat{F}^{\prime}(r_{+})=-\hat{\mathbb{A}}\hat{U}\hat{V},$$

(46)

where $\hat{\mathbb{A}}$ is a dimensionless proportionality constant that depends on the ultra-spinning black hole parameters.
The expansions of the functions $F(r)$ and $\hat{F}(r)$ near the horizon can then be extended to the second order of $r-r_{+}$. For the standard dyonic Kerr-Sen-AdS₄, we have

$$\frac{F}{\mathbb{A}UV}=-1-\frac{\mathbb{A}UV}{2}\frac{F^{\prime\prime}(r_{+})}{F^{\prime}(r_{+})}+\mathcal{O}(U^{2}V^{2}),$$

(47)

while for the ultra-spinning black hole, we have

$$\frac{\hat{F}}{\hat{\mathbb{A}}\hat{U}\hat{V}}=-1-\frac{\hat{\mathbb{A}}\hat{U}\hat{V}}{2}\frac{\hat{F}^{\prime\prime}(r_{+})}{\hat{F}^{\prime}(r_{+})}+\mathcal{O}(\hat{U}^{2}\hat{V}^{2}).$$

(48)

From the turning-point analysis, the maximum value that $\mathcal{L}$ can achieve if we need the perturbation to reach the horizon from infinity is bounded by $\mu^{-1}$ ($\mathcal{L}_{\text{max}}\leq\mu^{-1}$) and the equality holds at extremality, i.e. $\mathcal{L}_{\text{max}}=\mu^{-1}$ for $T_{H}\rightarrow 0$ [6]. For the dyonic Kerr-Sen-AdS₄ black hole, the value of $\kappa$ also survives the extremal limit. By expanding the value of $r_{+}$ near the extremal horizon radius $r_{0}$, where $\Delta(r_{0})=\Delta^{\prime}(r_{0})=0$, to the first-order approximation, we have

$$T_{H}\approx\frac{\partial T_{H}}{\partial r_{+}}\bigg{|}_{r_{0}}(r_{+}-r_{0}).$$

(49)

On the other hand, the value of $(1-\mu\mathcal{L})$ can also be expanded near the extremal value $r_{0}$,

$$1-\mu\mathcal{L}\approx\bigg{(}\frac{1}{\mu}\frac{\partial\mu}{\partial r_{+}}\bigg{)}_{r_{0}}(r_{+}-r_{0}).$$

(50)

Therefore, by taking the extremal limit $r_{+}\rightarrow r_{0}$, we obtain the extremal value of $\kappa$,

$$\kappa_{\text{ext}}=\frac{2\pi(\partial T_{H}/\partial r_{+})_{r_{0}}}{(\mu^{-1}\partial\mu/\partial r_{+})_{r_{0}}}=-2\pi T_{L}\mu_{\text{ext}},$$

(51)

where $\mu_{\text{ext}}$ is given by eq. (12) with $r_{+}$ is replaced by $r_{0}$ and $\kappa_{\text{ext}}$ is related by the definition of the left-moving Frolov-Thorne temperature found by the Kerr/CFT correspondence for the extremal dyonic Kerr-Sen-AdS₄ black hole $T_{L}\equiv-\frac{(\partial T_{H}/\partial r_{+})_{r_{0}}}{(\partial\mu/\partial r_{+})_{r_{0}}}$ derived in [40, 41].
For the dyonic Kerr-Sen-AdS₄ black hole, we have

$$\kappa_{\text{ext}}=\frac{(1+(r_{0}^{2}-d^{2}-k^{2})/l^{2})(1+a^{2}/l^{2}+(6r_{0}^{2}-2d^{2}-2k^{2})/l^{2})}{2r_{0}\Xi},$$

(52)

while for the ultra-spinning counterpart (following the same procedure), we have

	$\displaystyle\hat{\kappa}_{\text{ext}}$	$\displaystyle=-2\pi\hat{T}_{L}\hat{\Omega}^{\text{ext}}_{\varphi}$		(53)
		$\displaystyle=\frac{(6r_{+}^{2}-2d^{2}-2k^{2}+2l^{2})}{2r_{+}l^{2}}.$		(54)

Both $\kappa_{\text{ext}}$ and $\hat{\kappa}_{\text{ext}}$ scales with the left-moving Frolov-Thorne temperature calculated by the Kerr/CFT. The difference lies in the type of angular momentum which appear in both $\kappa_{\text{ext}}$ and $\hat{\kappa}_{\text{ext}}$ where it is given by $\mu_{\text{ext}}$ for the former and $\hat{\Omega}^{\text{ext}}_{\varphi}$ for the latter (instead of $\hat{\mu}_{\text{ext}}$). This difference lies in the definition of angular velocities. For the standard black hole, $\mu$ is the difference between the horizon’s angular velocity and the angular velocity of a stationary observer at infinity. For the ultra-spinning black hole, however, we do not have a notion of the angular velocity of a stationary observer at infinity because the $\varphi$ coordinate becomes null. Instead, what appears in $1-\hat{\Omega}_{\varphi}\mathcal{L}$ is the horizon’s angular velocity $\hat{\Omega}_{\varphi}$ which is the chemical potential for $\hat{J}$. Up until this point, all of the hatted functions for the ultra-spinning black hole do not depend on the new dynamical variable $\lambda$.

After we obtain the Kruskal-like coordinates for rotating null geodesic for both standard dyonic Kerr-Sen-AdS₄ black hole and its ultra-spinning counterpart, we are now able to generate the Dray-’t Hooft solution. For a Kruskal-like metric perturbed by an in-falling rotating shock wave at the equator located at $U_{0}$ for the standard black hole and at $\hat{U}_{0}$ for the ultra-spinning black hole, the metric is given by the shift for $U>U_{0}$,

$$V\rightarrow\tilde{V}=V+\alpha\Theta(U-U_{0}),$$

(55)

for the dyonic Kerr-Sen-AdS₄, and

$$\hat{V}\rightarrow\hat{\tilde{V}}=\hat{V}+\hat{\alpha}\Theta(\hat{U}-\hat{U}_{0}),$$

(56)

for its ultra-spinning version. It is known that the shift should contain a function $f(\theta)$ which captures the perturbation away from the equator. However, when we only consider $\theta=\pi/2$, we may normalize the function such that $f(\pi/2)=1$. In this section, we will calculate the Dray-’t Hooft solution and the strength of the shock waves $\alpha$ for the standard dyonic Kerr-Sen-AdS₄ first, then followed by its ultra-spinning counterpart later on.