Radiations From Geometries Using Painlev\'e-Gullstrand Coordinates

TL;DR

This paper investigates quantum radiation phenomena in curved geometries using Painlevé-Gullstrand coordinates, revealing how different observer frames perceive radiation near black holes and their relation to flat spacetime scenarios.

Contribution

It provides a covariant analysis of quantum effects in non-singular geometries and compares perceptions of radiation between stationary and freely-falling observers.

Findings

Observer-dependent radiation detection consistent with flat spacetime analogs.

Differences in descriptions depend only on local proper accelerations.

Radiation perceptions vanish for distant observers away from the horizon.

Abstract

The principle of equivalence is used to examine covariant descriptions of quantum phenomena within the global exterior of geometries described using Painlev\'e- Gullstrand coordinates, which are everywhere non-singular away from their center. The differences between the descriptions of observers stationary within the curvilinear geometry and those in a locally-flat freely-falling system that is momentarily at rest are found to depend only on local proper accelerations, becoming vanishingly small for increasingly distant observers. Away from the horizon, the local detections of outgoing massless radiations from the ground state of the freely-falling observers are found to be consistent with those of Rindler observers accelerating in flat space-time.

Equations

$$d{s}^2=-\left(1-\frac{R_S}{r}\right)dc{t}^2+2\sqrt{\frac{R_S}{r}}dctdr+d{r}^2+r^2(d{\vartheta }^2+{\sin }^2\vartheta d{\phi }^2),$$

$$d{s}_{1+1}^2=-\left(1-\frac{R_S}{r}\right)dc{t}^2+2\sqrt{\frac{R_S}{r}}dctdr+d{r}^2.$$

$$ct=\omega ,r={\left(\frac{3}{2}\right)}^{\frac{2}{3}}{\left(R_S(\rho -\omega {)}^2\right)}^{\frac{1}{3}}.$$

$$\left(\left({\bar{g}}_{\alpha \beta }\right)\right)=\left(\begin{array}{cc}-1& 0\\ 0& \frac{R_S}{r}\end{array}\right)\underset{r⟶r(\omega ,p)}{\to }\left(\begin{array}{cc}-1& 0\\ 0& {\left(\frac{2}{3}\right)}^{\frac{2}{3}}{\left(\frac{R_S}{\rho -\omega }\right)}^{\frac{2}{3}}\end{array}\right).$$

$$\frac{d{U}^{\omega }}{d\lambda }+\frac{{\left(\frac{2}{3}\right)}^{\frac{2}{3}}{R}_S^{\frac{2}{3}}(U^{\rho }{)}^2}{3(\rho -\omega {)}^{\frac{5}{3}}}=0,\frac{d{U}^{\rho }}{d\lambda }+\frac{2U^{\omega }{U}^{\rho }-(U^{\rho }{)}^2}{3(\rho -\omega )}=0.$$

$${\omega }_{\text{flat}}(Y^0,Y^1)=Y^0+\frac{(Y^1-{\omega }_p{)}^2}{6({\omega }_p-{\rho }_p)},$$

$${\rho }_{\text{flat}}(Y^0,Y^1)={\rho }_p+{\left(\frac{3}{2}\right)}^{\frac{1}{3}}{\left(\frac{{\rho }_p-{\omega }_p}{R_S}\right)}^{\frac{1}{3}}(Y^1-{\rho }_p)+$$

$$\frac{(Y^1-{\rho }_p{)}^2}{2^{\frac{5}{3}}{3}^{\frac{1}{2}}{\left(R_S^2({\rho }_p-{\omega }_p)\right)}^{\frac{1}{3}}}-\frac{(Y^1-{\rho }_p)(Y^0-{\omega }_p)}{2^{\frac{1}{3}}{3}^{\frac{2}{3}}{\left(R_S({\rho }_p-{\omega }_p{)}^2\right)}^{\frac{1}{3}}}.$$

$$ct(Y^0,Y^1)\simeq Y^0-\sqrt{\frac{R_S}{r_p}}\frac{(Y^1-{\rho }_p{)}^2}{4r_p},$$

$$r(Y^0,Y^1)\simeq r_p+(Y^1-{\rho }_p)-\sqrt{\frac{R_S}{r_p}}(Y^0-{\omega }_p)+$$

$$\frac{R_S}{4r_p^2}\left[(Y^1-{\rho }_p{)}^2-(Y^0-{\omega }_p{)}^2\right].$$

$$\frac{\partial Y^0}{\partial ct}=1,\frac{\partial Y^0}{\partial r}=0,\frac{\partial Y^1}{\partial ct}=\sqrt{\frac{R_S}{r_p}},\frac{\partial Y^1}{\partial r}=1,\text{and }\frac{{\partial }^2{Y}^1}{\partial c{t}^2}=\frac{R_S}{2r_p^2}.$$

$$g_{Y^0{Y}^0}\simeq -1+\frac{R_S(Y^1-{\rho }_p{)}^2}{4r_p^3},$$

$$g_{Y^0{Y}^1}=g_{Y^1{Y}^0}\simeq \frac{1}{8r_p^3}\sqrt{\frac{R_S}{r_p}}[(3r_p-R_S)(Y^1-{\rho }_p{)}^2+$$

$$-6\sqrt{r_p{R}_S}(Y^1-{\rho }_p)(Y^0-{\omega }_p)+4R_S(Y^0-{\omega }_p{)}^2],$$

$$g_{Y^1{Y}^1}\simeq 1+\frac{R_S}{4r_p^3}\left[(Y^1-{\rho }_p{)}^2-2\sqrt{\frac{R_S}{r_p}}(Y^1-{\rho }_p)(Y^0-{\omega }_p)\right].$$

$$d{s}_{1+1}^2=\left[dr-\left(1-\sqrt{\frac{R_S}{r}}\right)dct\right]\left[dr+\left(1+\sqrt{\frac{R_S}{r}}\right)dct\right].$$

$$\left(1-\sqrt{\frac{R_S}{r}}\right)\left[dct-d{r}_{\tilde{u}}\right]=0\text{and}\left(1+\sqrt{\frac{R_S}{r}}\right)\left[dct+d{r}_{\tilde{v}}\right]=0,$$

$$r_{\tilde{u}}(r)\equiv r+2\sqrt{r{R}_S}+2R_S\log \left(\sqrt{\frac{r}{R_S}}-1\right),$$

$$r_{\tilde{v}}(r)\equiv r-2\sqrt{r{R}_S}+2R_S\log \left(\sqrt{\frac{r}{R_S}}+1\right).$$

$$\tilde{u}\equiv ct-r_{\tilde{u}}\text{and}\tilde{v}\equiv ct+r_{\tilde{v}}.$$

$$\frac{d{U}^{\tilde{u}}}{d\lambda }-\frac{R_S}{2r^2}{\left(U^{\tilde{u}}\right)}^2=0\text{and}\frac{d{U}^{\tilde{v}}}{d\lambda }+\frac{R_S}{2r^2}{\left(U^{\tilde{v}}\right)}^2=0,$$

$$\Delta \tilde{u}\simeq \frac{\Delta {\tilde{\mathcal{U}}}_{\text{moving}}}{1-\sqrt{\frac{R_S}{r_p}}}+\frac{R_S}{4r_p^2}\frac{\Delta {\tilde{\mathcal{U}}}_{\text{moving}}^2}{{\left(1-\sqrt{\frac{R_S}{r_p}}\right)}^2}+\mathcal{O}[\Delta {\tilde{\mathcal{U}}}_{\text{moving}}^3],$$

$$\Delta \tilde{u}\simeq \frac{\Delta \mathcal{U}}{\sqrt{1-\frac{R_S}{r_p}}}+\frac{1}{2}\left(\frac{R_S}{2r_p}\right){\left(\frac{\Delta \mathcal{U}}{\sqrt{1-\frac{R_S}{r_p^2}}}\right)}^2,$$

$$d{s}^2=-\left[\left(1-\sqrt{\frac{R_S}{r}}\right)d\tilde{u}\right]\left[\left(1+\sqrt{\frac{R_S}{r}}\right)d\tilde{v}\right]=-dudv\simeq -d\mathcal{U}d\mathcal{V}.$$

$$\Delta u\simeq \left(1-\sqrt{\frac{R_S}{r_p}}\right)\Delta \tilde{u}\simeq \left(1-\sqrt{\frac{R_S}{r_p}}\right)\sqrt{\frac{1+\sqrt{\frac{R_S}{r_p}}}{1-\sqrt{\frac{R_S}{r_p}}}}\Delta \tilde{u}\simeq \sqrt{1-\frac{R_S}{r_p}}\Delta \tilde{u},$$

$$\Delta u\simeq \Delta \mathcal{U}+\frac{1}{2}\frac{\left(\frac{R_S}{2r_p^2}\right)}{\sqrt{1-\frac{R_S}{r_p}}}{\left(\Delta \mathcal{U}\right)}^2.$$

$$\frac{\bar{a}}{c^2}\equiv \frac{\left(\frac{R_S}{2r_p^2}\right)}{\sqrt{1-\frac{R_S}{r_p}}}$$

$$\frac{\bar{a}}{c^2}\Delta u\simeq \frac{\bar{a}}{c^2}\Delta \mathcal{U}+\frac{1}{2}{\left(\frac{\bar{a}}{c^2}\Delta \mathcal{U}\right)}^2.$$

$$\frac{{\tilde{a}}^{\mu }}{c^2}=\frac{\partial {\tilde{x}}^{\mu }}{\partial x^{\alpha }}\left[\frac{d^2{x}^{\alpha }}{dc{\tau }^2}+{\Gamma }_{\beta \lambda }^{\alpha }\frac{d{x}^{\beta }}{dc\tau }\frac{d{x}^{\lambda }}{dc\tau }\right]\Rightarrow {\left(a_{proper}\right)}^2={\tilde{g}}_{\mu \nu }{\tilde{a}}^{\mu }{\tilde{a}}^{\nu }.$$

$$\frac{a_{proper}}{c^2}=\frac{\frac{R_S}{2r^2}}{\sqrt{1-\frac{R_S}{r}}}.$$

$$\Delta \mathcal{U}<<\sqrt{\frac{r_p^3}{2R_S}}.$$

$$\left[\frac{\left(\frac{R_S}{2r_p^2}\right)}{\sqrt{1-\frac{R_S}{r_p}}}\right]\Delta \mathcal{U}<<1.$$

$${\xi }^0({\omega }_a,{\rho }_a)={\rho }_a\sinh \left(\frac{a{\omega }_a}{c^2}\right)\text{and}{\xi }^1({\omega }_a,{\rho }_a)={\rho }_a\cosh \left(\frac{a{\omega }_a}{c^2}\right).$$

Full text

Radiations From Geometries Using Painlevé-Gullstrand Coordinates

Pradip Kattel***e-mail address, pradip.kattel@bison.howard.edu and James Lindesay†††e-mail address, jllindesay@howard.edu

Computational Physics Laboratory

Howard University, Washington, D.C. 20059

Abstract

The principle of equivalence is used to examine covariant descriptions of quantum phenomena within the global exterior of geometries described using Painlevé-Gullstrand coordinates, which are everywhere non-singular away from their center. The differences between the descriptions of observers stationary within the curvilinear geometry and those in a locally-flat freely-falling system that is momentarily at rest are found to depend only on local proper accelerations, becoming vanishingly small for increasingly distant observers. Away from the horizon, the local detections of outgoing massless radiations from the ground state of the freely-falling observers are found to be consistent with those of Rindler observers accelerating in flat space-time.

1 Introduction

When exploring the subtleties of merging quantum mechanics with the classical theory of general relativity, it is often convenient to explore the implementation of quantum principles into stationary spherically symmetric geometries using the principle of equivalence. The Schwarzschild metric represents a global vacuum solution to Einstein’s field equations, assuming a diagonal metric form. This metric introduces a time coordinate that has a non-physical coordinate singularity. In contrast, the Painlevé-Gullstrand metric[1, 2] likewise represents a global vacuum solution that relaxes the requirement of a diagonal metric form, thereby introducing no non-physical coordinate singularities, making these coordinate convenient for describing the quantum dynamics of energy-momenta conjugate to these coordinates. Although the two metric forms can be locally related away from coordinate singularities, no coordinate transformation that is everywhere non-singular can relate these metrics. In particular, the use of the Schwarzschild time coordinates near a black hole horizon as the temporal coordinate describing the space-time dynamics transforms that coordinate singularity into a physical singularity in Einstein’s equation.

In what follows, we will implement local quantum principles in a global Painlevé-Gullstrand geometry via the principle of equivalence to examine geometric effects on local physical measurements, as well as determine the limits of applicability of local flatness. Furthermore, we will compare and contrast descriptions of physical dynamics described using Painlevé-Gullstrand coordinates with Rindler coordinates and Schwarzschild coordinates in $1+1$ dimensional space-time, particularly with regards to outgoing massless radiations.

2 Stationary Spherically-Symmetric Geometries and the Principle of Equivalence

The global vacuum solution to Einstein’s equations for stationary, spherically-symmetric systems that will here be utilized has a metric of the form

[TABLE]

where $R_{S}\equiv\frac{2G_{N}\,M}{c^{2}}$ is the Schwarzschild radius associated with a geometry generated by mass $M$. We will in what follows examine only the 1+1 dimensional system absent any variations in the angular coordinates. The global coordinates $(ct,r)$ manifest no coordinate singularities away from the physical singularity at $r=0$ in this classical geometry, allowing their direct use in describing local dynamics everywhere. The metric exhibits an off-diagonal $dct\,dr$ term expected for radially dynamic (i.e., accreting or excreting) geometries as an analog to the $dct\,d\varphi$ term in the rotationally dynamic Kerr metric [3].

There has yet to be demonstrated any gravitating physical phenomena, either classical or quantum in nature, violating the principle of equivalence. In contrast, several generally accepted classical and quantum experiments have demonstrated the viability of this principle[4]. Thus, we consider the principle of equivalence to be an established foundational paradigm in this paper. For implementation of the principle of equivalence, a set of freely-falling coordinates will next be generated.

Radially freely-falling coordinates

We next develop a global set of freely-falling coordinates for the 1+1 dimensional geometry with metric

[TABLE]

Geodesic motion in the $(ct,r)$ coordinates have a solution set satisfying

$\left\{\frac{dct(c\tau)}{dc\tau}=1,\frac{dr(c\tau)}{dc\tau}=-\sqrt{\frac{R_{s}}{r(c\tau)}}\right\}$ in terms of the proper time $\tau$ parameterizing the freely-falling observers $r(c\tau)$. Choosing the constant of integration $\rho$ as a static space-like coordinate associated with the freely-falling system, one can develop new global coordinates given by

[TABLE]

These coordinates are similar to the Lamaître coordinates[5] developed from Schwarzschild space-time. Using these coordinates, a particular free-faller at fixed coordinate $\rho_{p}$ reaches the singular center at time $\omega=\rho_{p}$. The space-time metric (2.1) takes the diagonal form

[TABLE]

It is clear that this global metric form is non-singular everywhere away from the center $r(\omega,\rho)=0$. In 1+1 dimensions, although the Einstein tensor $G_{\mu\nu}$ vanishes everywhere in this vacuum solution, the curvature scalar $R=-\frac{8}{9(\rho-\omega)^{2}}\underset{r\rightarrow r(\omega,p)}{\Longrightarrow}-\frac{2R_{S}}{r^{3}}$ does not, making it a convenient measure of local tidal scales. The geodesic equations for four-velocities $\left(U^{\omega}\equiv\frac{d\omega}{d\lambda},U^{\rho}\equiv\frac{d\rho}{d\lambda}\right)$ are as follows:

[TABLE]

It is clear that if $U^{\rho}=0$ for $\rho\rightarrow\rho_{p}$ the system is not accelerating.

Locally flat coordinates

Next, coordinates that are locally flat for the freely falling system at $(\omega_{p},\rho_{p})$ will be developed for the metric (2.4). This local set of coordinates described as $(Y^{0},Y^{1})$ has the form

[TABLE]

One can directly verify that the metric generated by these coordinates satisfies the vacuum Einstein equation with curvature scalar $R\underset{(Y^{0},Y^{1})\rightarrow(\omega_{p},\rho_{p})}{\Longrightarrow}-\frac{2R_{S}}{r_{p}^{3}}$, demonstrating only local flatness. Expressing the accelerating coordinates $(ct,r)$ in terms of $(Y^{0},Y^{1})$ results in locally-flat freely-falling coordinates that are in relative motion to the $(ct_{p},r_{p})$ frame:

[TABLE]

The expressions are valid as long as coordinate deviations away from the local point $(\omega_{p},\rho_{p})$ are small compared to length scales associated with tidal effects, $(Y^{0}-\omega_{p})<<\sqrt{\frac{r_{p}^{3}}{2R_{S}}}$ and $(Y^{1}-\rho_{p})<<\sqrt{\frac{r_{p}^{3}}{2R_{S}}}$. These forms can be locally inverted to determine the following useful local relations for the point $(ct_{p},r_{p})$:

[TABLE]

From these expressions, it is clear that the accelerating point $(ct_{p},r_{p})$ is moving with radial Lorentz velocity $\sqrt{\frac{R_{S}}{r_{p}}}$ relative to the locally flat freely-falling coordinate frame $(Y^{0},Y^{1})$, allowing direct determination of locally freely-falling coordinates that are momentarily at rest with $(ct_{p},r_{p})$, if desired. The locally flat metric using these (moving) coordinates takes the form

[TABLE]

Thus, coordinates $(Y^{0},Y^{1})$ still generate the appropriate gravitational tidal forces (along with any local energy densities, in general) due to the metric’s quadratic form. These coordinates will be quite useful for describing local measurements implemented using the principle of equivalence in the discussion that follows.

3 Massless Radiations Examined by Inertial and Accelerating Observers

We next develop descriptions of massless particles in the geometry described by (2.2) in exterior regions $r>R_{S}$. The metric can be factored in the form

[TABLE]

Light-like motions can be described using null geodesics defined by

[TABLE]

where

[TABLE]

Conformal coordinates $\tilde{u}$ and $\tilde{v}$ parameterizing distinct outgoing and ingoing light-like trajectories will be defined using (3.11) in the forms

[TABLE]

Although the analytic form of $\tilde{v}-\tilde{u}$ is identical to that from the Schwarzschild metric, the form of $\tilde{v}+\tilde{u}$ is drastically different due to the non-singular near-horizon behavior of the time parameter $ct$. The geodesic equations satisfied by the four-velocities $U^{\beta}\equiv\frac{d\tilde{x}^{\beta}}{d\lambda}$ of these coordinates are given by

[TABLE]

verifying that constant $\tilde{u}$ values (vanishing $U^{\tilde{u}}$) parameterize outgoing trajectories, while constant $\tilde{v}$ values parameterize ingoing trajectories.

Relationships between the locally-flat and accelerating conformal coordinates

We next examine the relationship between the outgoing conformal coordinate $u$ of the accelerating system with that $\mathcal{U}$ of the locally-flat freely-falling frame momentarily at rest with the accelerating system. As will be motivated in the Appendix, this relationship is needed to describe the spectral nature (i.e. frequency distribution) of outgoing massless radiations. By direct substitution into the form (3.13),

[TABLE]

where $\Delta\tilde{\mathcal{U}}_{\text{moving}}\equiv(Y^{0}-\omega_{p})-(Y^{1}-\rho_{p})$ in terms of the coordinates in equations (2.6) and (2.7). A simple Lorentz transformation given by $\Delta\tilde{\mathcal{U}}_{\text{moving}}=\sqrt{\frac{1+\beta_{Y}}{1-\beta_{Y}}}\>\Delta\mathcal{U}$ relates the momentarily stationary conformal coordinate $\Delta\mathcal{U}$ to $\Delta\tilde{\mathcal{U}}_{\text{moving}}$, where $\beta_{Y}=-\sqrt{\frac{R_{S}}{r_{p}}}$. Thus, the conformal parameters are related by

[TABLE]

good to second order in deviations of the coordinate from the point of measurement.

Examining the expressions in (3.11) allows the metric to be re-expressed in the forms

[TABLE]

As defined, the conformal coordinates $(u,v)$ and $(\mathcal{U},\mathcal{V})$ are guaranteed to be quantified using the same units and measuring sticks. From the form of $d\tilde{u}=d(ct-r_{\tilde{u}})$, it is clear from the time parameter that this represents a form of the conformal coordinate at rest in the freely-falling $(\omega=ct,\rho)$ coordinate frame. This means that in the vicinity of the observation point, the boosted conformal coordinate satisfies

[TABLE]

(which can be more directly obtained using $d\tau=\sqrt{1-\frac{R_{S}}{r}}\>dt$), and similarly for $\Delta v$. Substitution of this result into the equation (3.16) defines the necessary form of the relation between the conformal coordinates consistent with the principle of equivalence:

[TABLE]

Algebraically defining an acceleration $\bar{a}$ of the form

[TABLE]

provides the second-order expression that the resulting dimensionless functions must satisfy:

[TABLE]

It only remains to physically interpret the algebraic parameter $\bar{a}$.

Local proper acceleration

We next demonstrate the form of the proper acceleration of the curvilinear observers. In flat space-time, the time-like four-velocity ($\vec{U}\equiv\frac{\vec{dY}}{dc\tau}$) invariant $\vec{U}\cdot\vec{U}=-1$ can be differentiated to demonstrate its orthogonality to the four-acceleration $\vec{A}\equiv\frac{\vec{dU}}{dc\tau}$, requiring $\vec{A}$ to be space-like. This implies that the proper-acceleration $a$ defined in the rest frame of an accelerating observer satisfies $\vec{A}\cdot\vec{A}=\frac{a_{proper}^{2}}{c^{2}}$. Affine connections $\Gamma^{\alpha}_{\beta\lambda}$ can then be defined via the principle of equivalence relating descriptions of accelerations in a locally-flat freely-falling frame to descriptions of those accelerations using curvilinear coordinates via

[TABLE]

Using the relationships between $(ct,r)$ and the freely falling coordinates $(\omega,\rho)$ from (2.3) in equation (3.22) for observers with fixed $r$ ($\frac{dr}{dc\tau}=0$) results in the form

[TABLE]

One alternatively can use the $(Y^{0},Y^{1})$ locally flat coordinates (2.7) to determine the proper acceleration of the point $(ct_{p},r_{p})$. From the expressions in (2.8), the form in (3.23) is locally verified.

Therefore, we see that the algebraically defined dimensional acceleration from expression (3.21) is the same as the local proper acceleration of the stationary observer using the curvilinear coordinates, $\bar{a}=a_{proper}$. This means that any physical effects implemented into the curved space-time via the principle of equivalence anywhere within the (exterior) geometry necessarily involve only local proper accelerations.

Flatness scales

To insure local flatness, the locally freely falling frame momentarily at rest with the accelerating frame satisfying Eqn. (3.19) is only valid for coordinate variations small compared to lengths associated with curvatures:

[TABLE]

This means that any micro-physical measurement scales must satisfy this criterion. Likewise, the form in Eqn. (3.19) can reliably represent an expansion of a function only if

[TABLE]

These relations imply that any functional form can be reliably tested using the principle of equivalence for r_{p}\mathrel{\mathchoice{\lower 3.0pt\vbox{\halign{\mathsurround 0pt\displaystyle\hfil#\hfil\cr>\crcr\sim\crcr}}}{\lower 3.0pt\vbox{\halign{\mathsurround 0pt\textstyle\hfil#\hfil\cr>\crcr\sim\crcr}}}{\lower 3.0pt\vbox{\halign{\mathsurround 0pt\scriptstyle\hfil#\hfil\cr>\crcr\sim\crcr}}}{\lower 3.0pt\vbox{\halign{\mathsurround 0pt\scriptscriptstyle\hfil#\hfil\cr>\crcr\sim\crcr}}}}\frac{9}{8}R_{S}. For local measurements nearer than this, testable functional dependencies are not reliable. Furthermore, an expression of the form $y\simeq x+\frac{1}{2}x^{2}$ as in equation (3.21) could represent $y=-log(1+x)$ (which results in a “thermal” distribution), the geometric series $y=\frac{x}{1-\frac{x}{2}}$, or numerous alternative functional forms. What has been established is the use of the local proper acceleration in the power series expansion of the dimensionless variables that represents the functional form.

4 Comparisons of Accelerating Observers

In this section we will compare descriptions of accelerating observers in various geometries. Observers in flat space-time who undergo a constant local proper acceleration can construct a (Rindler) coordinate patch $(\omega_{a},\rho_{a})$ that relates to the global Minkowski coordinates $(\xi^{0},\xi^{1})$ via

[TABLE]

The outgoing horizon $\xi^{0}-\xi^{1}=0$, which is an asymptote of the accelerating trajectory, has coordinate $\rho_{a}=0$. Furthermore, Rindler coordinates $(ct,x)\equiv(\omega_{a},\>\frac{c^{2}}{a}log\left(\frac{a\,\rho_{a}}{c^{2}}\right))$ conducive for developing conformal coordinates can be utilized, resulting in the following forms for the metric:

[TABLE]

From this expression, one can directly construct conformal inertial coordinates $(\mathcal{U}\equiv\xi^{0}-\xi^{1},\mathcal{V}\equiv\xi^{0}+\xi^{1})$ and accelerating coordinates $(\tilde{u}\equiv ct-x,\tilde{v}\equiv ct+x)$. The coordinate $x$ maps the horizon $\rho_{a}=0$ to $x_{H}=-\infty$.

The near-horizon form of the 1+1 dimensional Schwarzschild metric

[TABLE]

shares a coordinate singularity (with $t_{S}\rightarrow\infty$) at the horizon $r_{H}=R_{S}$ with the Rindler metric at $\rho_{aH}=0$. The near-horizon identifications $\rho_{a}\simeq 2\sqrt{R_{S}(r-R_{S})}$ and $d\omega_{a}\simeq\frac{dct_{S}}{2R_{S}}$ for $\left(\frac{r-R_{S}}{R_{S}}\right)<<1$ are used to assert an “effective acceleration” $a_{effective}=\frac{c^{2}}{2R_{S}}$ resulting in a thermal temperature associated with the horizon by asymptotic Schwarzschild observers who use $t_{S}$ as the time parameter conjugate to their quantum energy measurements, as will be motivated in the Appendix.

Near-horizon Painlevé-Gullstrand

Next, we examine the near-horizon form of the Painlevé-Gullstrand metric. We will define $\Delta\equiv\frac{r-R_{S}}{R_{S}}$, and examine local analytic coordinate transformation to a new time coordinate $\tilde{ct}$:

[TABLE]

The metric (2.2) transforms into a diagonal form given by

[TABLE]

It is clear that the transformation is not analytic at the horizon $\Delta\rightarrow 0$. This is expected since no analytic transformation can take non-singular coordinates into singular coordinates. This local transformation can be utilized by even distant observers, but asymptotic Schwarzschild observers require a singular transformation on (2.2) in order to identify $ct_{S}$ as $\tilde{ct}$.

Probes of inertial and accelerating ground states

As will be motivated in the Appendix, inertial probing of accelerating ground states, as well as probing of inertial ground states by accelerating probes, involve representing the quantum operators of one system in terms of the other’s using canonical transformations. For outgoing massless radiations, the formulations requires a calculation of the inertial conformal coordinate in terms of the accelerating conformal coordinate, $\mathcal{U}(u)$.

For a Rindler observer undergoing constant acceleration in Minkowski space-time, the needed relations $(\mathcal{U}=\xi^{0}-\xi^{1},\mathcal{V}=\xi^{0}+\xi^{1})$ and $(\tilde{u}=ct-x,\tilde{v}=ct+x)$ can be calculated from (4.26) with $\rho_{a}=\frac{c^{2}}{a}e^{\frac{a}{c^{2}}x}$, yielding

[TABLE]

The form $\mathcal{X}=-e^{-x}$ is unique in that rescaling the inertial coordinate $\mathcal{X}\rightarrow\lambda\mathcal{X}$ simply results in a shift of the accelerating coordinate $\lambda\mathcal{X}=-e^{-(x-log\lambda)}$. This can be interpreted for uniform acceleration as implying that the rescaling of inertial light-like trajectories is equivalent to a Lorentz (Doppler) transformation due to shifting the onset of acceleration.

For near horizon, as well as asymptotic Schwarzschild observers in 1+1 dimensions, we expect relations analogous to (4.31) with a finite effective acceleration to be valid. For Painlevé-Gullstrand observers, the result obtained using the principle of equivalence (3.21) is likewise consistent with a global form $\Delta\mathcal{U}(\Delta u)=-\frac{c^{2}}{a_{proper}}\>e^{-\frac{a_{proper}}{c^{2}}\>\Delta u}$, only with effective accelerations that becomes vanishingly smaller for more distant observers.

5 Discussion and Conclusions

In this paper, the principle of equivalence was used to examine macro- and micro-physical measurements using the Painlevé-Gullstrand metric. To do this, a global set of freely falling coordinates utilizing the proper time of these fallers as the non-singular global temporal coordinate was developed. Since these proper times represent observable clocks, one is expected to be able to covariantly examine quantum dynamics using conjugate energies with minimal interpretation. From these global coordinates, locally-flat freely-falling coordinates that incorporate curvature (and generally, local energy density) scales were obtained.

Un-ambiguous algebraic relationships were established between the coordinates (including conformal coordinates) of stationary accelerating observers with fixed coordinate $r=r_{p}$ and those of the freely-falling observers that are momentarily at rest with the stationary observers, allowing the mutual probing of local phenomena. In particular, the relationships demonstrate that the conformal coordinates used by the two sets of observers differ only through the proper acceleration of the $(ct_{p},r_{p})$ observers, without Doppler shift.

Phenomena associated with the local proper accelerations of stationary observers were found to be consistent with those experienced by Rindler observers away from the horizon in regions of sufficient flatness. This means that globally, any local radiations detected by observers with static curvilinear coordinates in this geometry are related only to local proper accelerations. We expect increasingly distant stationary observers to measure vanishingly smaller phenomena associated with their proper accelerations, decreasing to zero for very distant observers in contrast to what is expected for asymptotic Schwarzschild observers. The results of measurements of accelerating probes on inertial ground states are found to be consistent with those done by observers in Rindler space.

Appendix

For completeness, the spectral distribution of radiations from the ground states of others as measured by inertial and accelerating observers will be briefly sketched. A more complete exposition is available in the references[6, 7].

A massless scalar field $\hat{\phi}(ct,x)$ can be expressed using operators $\hat{b}(k)$ quantized in the laboratory frame of reference $(ct,x)$ in the form

[TABLE]

The operators $\hat{b}(k)$ satisfy canonical commutation relations as a function of wave-number $k$, and are assumed to annihilate the ground state of radiations in the frame $(ct,x)$. Assuming that conformal coordinates can be directly expressed using $u=ct-x$ and $v=ct+x$, the field can be written (with $\hat{b}\left(\frac{\omega}{c}\right)\equiv\hat{b}_{\omega}$) as

[TABLE]

where the dispersion relation $\omega=c|k|$ for positive frequencies has been utilized.

In a similar way, the scalar field can be expressed using operators quantized in the freely falling or inertial frame $(Y^{0},Y^{1})$ as:

[TABLE]

where operators $\hat{B}(K)$ expressed in terms of inertially defined frequencies replace the operators $\hat{b}(k)$ in equation (A.2). A canonical Bogoliubov transformation can be found to connect the operators $\left(\hat{b}(k),\hat{b}^{\dagger}(k)\right)$ to $\left(\hat{B}(K),\hat{B}^{\dagger}(K)\right)$, preserving the commutation relationships. Since outgoing/ingoing light-like trajectories must be represented by constant $(u,\mathcal{U})/(v,\mathcal{V})$ for either observer, one can conclude that $\mathcal{U}=\mathcal{U}(u)$ and $\mathcal{V}=\mathcal{V}(v)$, i.e. the transformations do not mix ingoing with outgoing fields in (A.3). Thus $\hat{\Phi}_{+}(\mathcal{U}(u))=\hat{\phi}_{+}(u)$ and $\hat{\Phi}_{-}(\mathcal{V}(v))=\hat{\phi}_{-}(v)$.

Explicitly, for outgoing massless radiations

[TABLE]

Recall that, as was done in (A.2), Fourier transforms of fields $\hat{\Psi}(u)$ over all wave-numbers $k$ can be alternatively expressed in terms of integrals over positive frequencies $\omega$ only, generally requiring that the Fourier transformed fields $\tilde{\hat{\Psi}}(k)$ satisfy

[TABLE]

Thus, we can make the following identification for the Fourier transform:

[TABLE]

Again utilizing (A.5), the Fourier transform of $\hat{\Phi}_{+}(\mathcal{U}(u))$ results in the expression

[TABLE]

where

[TABLE]

From (A.6), we obtain the following relations between the canonical operators:

[TABLE]

This general equation relates canonical massless scalar operators that act upon accelerating and inertial ground states used by the respective observers.

In particular, we demonstrate the spectral characteristics of outgoing massless radiations from an inertial Minkowski ground state as measured by a Rindler probe with proper acceleration $a$. Substituting (4.31) to relate inertial and accelerating conformal coordinates, the auxiliary function (A.8) takes the form

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Calculating the Minkowski ground state expectation value of the accelerating number operator of mode frequency $\omega$ gives the number of excitations of this mode in the Minkowski ground state $\left|\,0_{M}\right\rangle$:

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The auxiliary function (A.10) for Rindler observers can be expressed in terms of a gamma function:

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Furthermore, the canonical commutation relation $\left[\hat{b}_{\omega},\hat{b}^{\dagger}_{\omega^{\prime}}\right]$ gives

[TABLE]

Substituting (A.12) into (Appendix), and setting $\omega=\omega^{\prime}$, we get:

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Using the common $L\rightarrow\infty$ identification $\frac{L}{2\pi}\delta_{k_{r}^{\prime},k_{s}}\doteq\delta(k^{\prime}-k)$ in the result from(A.14), the number density in 1+1 dimensions is effectively identified as $\langle n\rangle_{\omega}\doteq\frac{1}{2\pi\delta(0)}\langle N\rangle_{\omega}$. yielding

[TABLE]

This is precisely the Planck form of black body radiation, if the effective temperature is given by

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