# Charge-state dependent vibrational relaxation in a single-molecule junction

Xinya Bian  Department of Materials, University of Oxford, Oxford, OX1 3PH, UK
Zhixin Chen  Department of Materials, University of Oxford, Oxford, OX1 3PH,
UK  Jakub K. Sowa  Department of Chemistry, Rice University, Houston, TX, USA
Charalambos Evangeli  Department of Materials, University of Oxford, Oxford,
OX1 3PH, UK  Bart Limburg  Department of Chemistry, University of Oxford,
Oxford, OX1 3TA, UK  Jacob L. Swett  Department of Materials, University of
Oxford, Oxford, OX1 3PH, UK  Jonathan Baugh  Institute for Quantum Computing,
University of Waterloo, Waterloo, ON N2L 3G1, Canada  G. Andrew D. Briggs
Department of Materials, University of Oxford, Oxford, OX1 3PH, UK  Harry L.
Anderson  Department of Chemistry, University of Oxford, Oxford, OX1 3TA, UK
Jan A. Mol  School of Physical and Chemical Sciences, Queen Mary University,
London, E1 4NS, UK  James O. Thomas<EMAIL_ADDRESS>Department of Materials, University of Oxford, Oxford, OX1 3PH, UK

###### Abstract

The interplay between nuclear and electronic degrees of freedom strongly
influences molecular charge transport. Herein, we report on transport through
a porphyrin dimer molecule, weakly coupled to graphene electrodes, that
displays sequential tunneling within the Coulomb-blockade regime. The
sequential transport is initiated by current-induced phonon absorption and
proceeds by rapid sequential transport via a non-equilibrium vibrational
distribution. We demonstrate this is possible only when the vibrational
dissipation is slow relative to sequential tunneling rates, and obtain a lower
bound for the vibrational relaxation time of 8 ns, a value that is dependent
on the molecular charge state.

Charge transport through nanostructures, and molecules in particular, is
strongly influenced by coupling between electronic and mechanical degrees of
freedom[1, 2, 3, 4, 5, 6]. Geometric differences between molecular charge
states mean that electron transfer steps are accompanied by vibrational
transitions, an effect encompassed by the Franck-Condon principle[7].
Typically vibrational transitions are identified as steps in current that are
equally spaced in voltage in current-voltage (IV) measurements[5, 6, 3, 8, 9].
Transport behavior that results from coupling to equilibrated vibrations has
been extensively studied and well understood. For example, electron-phonon
coupling strength and vibrational frequency are precisely extracted from the
progression of the current step heights, and low-bias suppression in the
strongly coupled condition due to negligible vibrational wavefunction overlap,
known as Franck-Condon blockade, has been observed[5, 6, 8]. Conversely, the
effects of non-equilibrium vibrational distributions are less well-
established. Interesting phenomena such as current-induced molecular heating
and dissociation, giant Fano factors, and avalanche tunneling dynamics have
been predicted theoretically[7, 10, 11, 12, 13, 14] and subsequently observed,
primarily in carbon nanotubes devices[15, 16, 8, 3, 6].

Vibrational effects in single-molecule junctions are particularly prominent in
the weak molecule-electrode coupling limit, as strong electron-electron
interactions lead to Coulomb-blockade (CB) behavior and charge states are well
defined. Transport through the molecule occurs either via resonant sequential
tunneling outside the CB regime or via off-resonant cotunneling inside the CB
regime. Like sequential tunneling processes, cotunneling processes can also
excite vibrational states above a bias threshold through inelastic
cotunneling[17]. If cotunneling-excited vibrational states can promote
sequential tunneling processes, there is a loss of definition of the boundary
between resonant and off-resonant tunneling regimes[12, 18]. Depending on the
interplay between vibrational relaxation, cotunneling, and sequential
tunneling rates, cotunneling-assisted sequential tunneling processes can lead
to distinctive features in the transport spectrum, known as absorption
sidebands, extending into the CB regime[7].

In this Letter we report the first experimental observation of such sidebands
in a single-molecule device and confirm that they originate from a current-
driven, non-equilibrium vibrational distribution. We calculate a lower
threshold for the molecular vibrational relaxation time of 8 ns, and
demonstrate that there is a connection between dissipation rate and molecular
charge state, and by extension, its geometry.

The device architecture is displayed in Fig. 1, and has been described
previously[19]. A high-$\kappa$, $10\text{\,}\mathrm{nm}$-thick, HfO2 gate
dielectric gives a large electrostatic coupling between the molecular states
and the gate potential, $V_{\rm{g}}$ (in this case $\alpha_{g}=0.5$ eV/V). The
molecule under study, a fused zinc-porphyrin dimer (FP2, Fig. 1c), has a small
charging energy which, combined with the large gate coupling, allows multiple
molecular charge states to be studied. FP2 is synthesized by Sonogashira
coupling of a dibromo- edge-fused zinc porphyrin dimer[20] with an
ethynylpyrene derivative[21]111See Supplemental Material for synthetic
details, additional transport data, and fitting results.. Pyrene groups anchor
the molecule to graphene source and drain electrodes through a $\pi$-$\pi$
stacking interaction and provide weak molecule-electrode coupling, $\Gamma$.
Electroburnt graphene nanoelectrodes can display transport features prior to
molecular deposition and therefore devices were measured before and after
deposition to ensure we only study transport features related to
FP2.††footnotemark: . From 98 devices measured at 5 K, the majority displayed
resonant tunneling features prior to molecule deposition, in line with
previous work[21]. Five displayed clear resonant transport only after
deposition and had consistent addition energies with the device studied in
detail here. The presented device has the specific combination of molecule-
electrode and gate coupling to allow the experimental study of absorption
sidebands.

Figure 1: (a) Device structure schematic. (b) False-color SEM image of
graphene (Gr) transferred onto source (S) and (D) leads, and a local gate
electrode (LG). Gr is patterned into a bowtie shape to produce nanoelectrodes
via electroburning. (c) Molecular structure of FP2, Ar:
3,5-bis(trihexylsilyl)phenyl. (d) Wide-range differential conductance map
($dI_{\rm{sd}}/dV_{\rm{sd}}$) of the FP2 device, transitions studied in detail
are outlined in green and blue.

A large $V_{\rm{g}}$-range differential conductance map (Fig. 1d) confirms the
molecule is weakly coupled to the graphene electrodes. Off-resonant transport
is suppressed due to CB and multiple Coulomb diamonds, with addition energies
in the range of 150–300 meV, are observed and separated by resonant transport
regions. Due to the electron-rich anchor groups and p-doping of graphene by
the substrate[23], charge states are assigned to successive oxidized states of
FP2, i.e. $N{-}5$ to $N$, $N$ being the number of electrons on the molecule
when it is neutral. The assignment and addition energies are inline with
studies on the same family of molecules[9, 24, 25]. We focus on electron-
vibration coupling features manifested in the charge transitions at
$V_{\rm{g}}=-1.25$ V, assigned to the $N{-}4$/$N{-}3$ transition (green box),
and $V_{\rm{g}}=-0.6$ V - $N{-}3$/$N{-}2$ (blue box).

Fig. 2a displays high-resolution conductance maps of the $N{-}4$/$N{-}3$
transition. The solid slanted lines define the usual resonant transport region
separating the $N{-}4$ and $N{-}3$ diamonds. Lines of increased conductance
running parallel to the diamond edges are caused by additional transport
channels to the ground-state to ground-state transition between $N{-}4$ and
$N{-}3$. The low energy of these excited-state channels, at around 9 meV
(dashed/dotted lines), point to a vibrational origin. Vibrational transitions
of a similar energy have been observed in a study of a porphyrin monomer[9].
Under weak coupling, resonant transport is dominated by first-order sequential
tunneling processes, i.e. $\ket{N,q}$ to $\ket{N^{\prime},q^{\prime}}$ where
$N^{\prime}=N\pm 1$, and each additional channel involves either the hopping
on/off steps required to transport an electron through the junction to be
accompanied by a change in vibrational state, i.e. $\Delta q=q^{\prime}-q\neq
0$. These steps generate out-of-equilbrium vibrational excitations in the
molecule, but if the rate of vibrational relaxation is much faster than
sequential electron transfer rates then under steady-state conditions the
system can be treated as in equilibrium. Under equilibrium conditions and at
low temperature, only the vibrational ground state is appreciably populated,
CB is maintained, and sequential tunneling current should be strictly bound
within the solid lines defining the ground-state to ground-state resonant
tunneling transition. Here, however, more complex behavior is observed, and
the additional transport channels at energies of $\hbar\omega_{q}=9$ meV (the
dashed/dotted lines slanted lines) do not terminate at the boundaries of CB
regions but instead extend into the $N{-}4$ diamond, forming sidebands to the
Coulomb peak. The $N{-}4$ sidebands run parallel to diamond edges, indicating
current within them results from sequential tunneling processes. The first
sideband corresponds to a transition with $\Delta q=-1$ and thus sequential
tunneling must involve at least the first vibrationally excited state of
$N{-}4$. For the second sideband, $\Delta q=-2$ and sequential tunneling must
originate from at least the second vibrationally excited state of $N{-}4$. The
sidebands are present at 5 K, despite the absence of thermal population in
$\ket{N{-}4,1}$.

Figure 2: (a) Differential conductance maps of the $N{-}4$/$N{-}3$ transition
measured at different temperatures, showing quasi-periodic resonant transport
features. The onset of the second sideband is indicated by the orange arrow.
(b) Conductance measured along the first (grey marker/arrow on (a)) and second
(pink marker/arrow on (a)) sidebands shows suppression below
$eV_{\rm{sd}}<n\hbar\omega_{q}$. (c) Temperature-dependence of the Coulomb
peak (CP) and the first sideband (SB), (red/blue circular markers on (a)). The
ground state shows a simple $1/T$ temperature dependence (black line), the
first sideband is suppressed, increasing due to thermal population of $q=1$,
as fitted by 4 (black line). (d) Temperature-dependent conductance through the
first and second sideband (square marker positions on (a)) at
$eV_{\rm{sd}}=\hbar\omega_{q}$, the conductance of the first sideband is high
and decays as $1/T$ (black line) whilst the second grows with temperature.
Figure 3: (a) The onset of first sideband (red dot in panel (d)). Blue $N{-}4$
parabolas correspond to an electron at the Fermi level of left
$\ket{\mu_{L},N{-}4,0}$ or right $\ket{0,N{-}4,\mu_{R}}$ electrode. The green
parabola corresponds to the electron being on the molecule: $\ket{0,N{-}3,0}$.
$N{-}4$ parabola have the same $x$-position, but are offset from the $N{-}3$
by $\pm l_{osc}\sqrt{2}\lambda_{q}$ to visualise electron flow from right to
left lead. Inelastic cotunneling promotes the molecule to $q=1$. If relaxation
($1/\tau_{N{-}4}$) is smaller than $W^{N{-}4,N{-}3}_{1,0}$, sequential
tunneling proceeds via excited states of FP2, leading to sidebands. (b)
Parabolas at the potential onset of the first $N{-}3$ sideband (grey dot) are
not observed as $1/\tau_{N{-}3}$ out-competes sequential tunneling. (c) Fits
to zero-bias gate traces yield $\lambda_{q}$ and $\bar{\Gamma}$ values, here
$\lambda_{q}=2.7$, $\bar{\Gamma}=$$40\text{\,}\mathrm{\SIUnitSymbolMicro eV}$.
Traces are offset for clarity. (d) Modelled conductance map at 5 K, using
$\hbar\omega_{q}=9$ meV, $\lambda_{q}=2.7$,
$\bar{\Gamma}=$$40\text{\,}\mathrm{\SIUnitSymbolMicro eV}$,
$1/\tau_{N-4}=100\times W^{N-4,N-3}_{1,0}$, $1/\tau_{N-3}=1/100\times
W^{N-3,N-4}_{1,0}$; sequential tunneling rates are calculated at the red and
grey points respectively.

As shown in Fig. 2b, the sidebands do not cross the zero-bias axis. At 5 K the
conductance of the first sideband (Fig. 2b, grey) is suppressed below
$|eV_{\rm{sd}}|=\hbar\omega_{q}$, and the second sideband conductance (pink)
is suppressed below $2\hbar\omega$. This suggests that the features within the
$N{-}4$ Coulomb diamond are due to cotunneling-assisted sequential tunneling
processes[18, 12]. The emergence of the first sideband is shown in Fig. 3a. At
low voltages ($|eV_{\rm{sd}}|<\hbar\omega_{q}$) cotunneling must be elastic
($q^{\prime}=q$) and will contribute a small portion of tunneling current
(cotunneling is second-order in $\Gamma$). Above
$|eV_{\rm{sd}}|=\hbar\omega_{q}$ inelastic cotunneling events, which leave the
molecule in an excited vibrational state, are energetically allowed (Fig. 3a).
Under weak molecule-electrode coupling, inelastic cotunneling processes carry
little net current, and we do not observe gate-independent co-tunneling lines
within the Coulomb diamonds[26]. If the vibrational excitation that results
from inelastic cotunneling is slow to relax compared to the sequential
tunneling rates, $W^{N,N^{\prime}}_{q,q^{\prime}}$ (where $N^{\prime}=N\pm
1$), specifically $W^{N{-}4,N{-}3}_{1,0}$, then a sequential tunneling pathway
to the ground state of $N{-}3$ is opened up (Fig. 3a). The $\ket{N{-}3,q={0}}$
state can subsequently undergo sequential tunneling to up to the second
vibrationally excited state of $N{-}4$. If electron-phonon coupling,
parameterized by the coupling constant: $\lambda_{q}$, is weak
($\lambda_{q}<1$) then the transition to the $\ket{N{-}4,q{=}0}$ is most
likely, and the molecule returns to the vibrational ground state until another
inelastic cotunneling begins the cycle again. If $\lambda_{q}>1$ then
transitions to $\ket{N{-}4,q{=}1,2}$ out-compete the return to the $N{-}4$
ground state, and a single inelastic cotunneling event leads to sustained
sequential electron tunneling through the molecule, even in the CB region, via
vibrationally excited states of $N{-}4$. Therefore the prominence of the
sidebands is enhanced when electron-phonon coupling is strong.

We look to understand these mechanisms quantitatively. Conductance is a sum of
the cotunneling and sequential tunneling contributions, i.e., $G_{sd}\approx
G_{seq}+G_{cot}$. At low temperature and zero-bias these are given
analytically by[27]:

$G_{seq}=-\frac{2e^{2}}{\hbar}\frac{\Gamma_{L}\Gamma_{R}}{\Gamma_{L}+\Gamma_{R}}\frac{f^{\prime}(\epsilon_{d})}{1+f(\epsilon_{d})}\rm{e}^{-\lambda^{2}}$
(1)

and

$G_{cot}=\frac{2e^{2}}{h}\frac{\Gamma_{L}\Gamma_{R}}{1+f(\epsilon_{d})}\left[\frac{1-f(\epsilon_{d})}{(\epsilon_{d}+\lambda^{2}\hbar\omega)^{2}}+\frac{2f(\epsilon_{d})}{(\epsilon_{d}-\lambda^{2}\hbar\omega)^{2}}\right]$
(2)

where $\Gamma_{L}$ and $\Gamma_{R}$ are the couplings of the molecule to left
and right electrodes, $f(\epsilon_{d})$ is the Fermi distribution at the
energy of the molecular transition ($\epsilon_{d}$), and
$f^{\prime}(\epsilon_{d})$ is its derivative. We fit zero-bias conductance
traces (at 10 K and below) to $\bar{\Gamma}$ and $\lambda_{q}$, shown in Fig.
3c ($\bar{\Gamma}=\Gamma_{L}\Gamma_{R}/(\Gamma_{L}+\Gamma_{R})$). The two
parameters cannot be determined independently (they are positively
covariant††footnotemark: ), although it is possible to determine the range in
which they lie. Sequential tunneling is exponentially suppressed in
$\lambda_{q}$, however cotunneling can still occur via highly excited virtual
states of $N{-}3$, therefore at large $\lambda_{q}$ (> 2.8), and with the
corresponding increase in $\bar{\Gamma}$, there are significant contributions
from elastic cotunneling that are not observed experimentally. This sets an
upper bound for $\lambda_{q}$. Simulations (vide infra) of the conductance map
show at $\lambda_{q}<2$ (corresponding to
$\bar{\Gamma}<$$1\text{\,}\mathrm{\SIUnitSymbolMicro eV}$) the sidebands
become too low in conductance to observe. The lack of sidebands at low values
of $\lambda_{q}$ and $\bar{\Gamma}$ results from reduced inelastic cotunneling
rates that initiate the sidebands, along with the enhanced rates of sequential
transfer from excited vibrational states of $\ket{N{-}3,q}$ to the ground
state, $\ket{N{-}4,q={0}}$ due to larger Franck-Condon factors. Therefore, we
find pairs of parameters within ranges of $2-3$ for $\lambda_{q}$ and
1–$100\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ for $\bar{\Gamma}$. Theoretical
work predicted that slow vibrational relaxation coupled with values
$\lambda_{q}$ in the range of 2–3 would result in negative differential
conductance (NDC) after the conductance peaks of the vibrational sidebands,
due to the depopulation of non-equilibrium vibrational distribution to the
ground state by sequential transfer as the sidebands enter the bias
window[11]. Our experimental observation of NDC (Fig. 2a) and calculated range
of $\lambda_{q}$ are consistent with this prediction. The criteria for the
emergence of the first $N{-}4$ sideband is: $1/\tau_{N{-}4}\leq
W^{N{-}4,N{-}3}_{1,0}$. Therefore a calculation of the sequential rate yields
a lower bound for the vibrational relaxation time, $\tau_{N{-}4}$, within the
junction. Sequential tunneling rates are given by[27]:

$W^{N-4,N-3}_{q,q^{\prime};a}=2\Gamma_{a}|M_{q,q^{\prime}}|^{2}f_{a}(\epsilon+[q^{\prime}-q]\hbar\omega)$
(3)

where $M_{q,q^{\prime}}$ are the Franck-Condon matrix elements. Across the
range of $\lambda_{q},\bar{\Gamma}$, the calculated values of
$W^{N{-}4,N{-}3}_{1,0;L}$ are in the narrow range of 56 – 120
MHz,††footnotemark:  giving a lower bound for the vibrational relaxation time
of $\tau_{N{-}4}>8$ ns. The absence of sidebands within the $N{-}3$ charge
state is an intriguing part of the system dynamics. If
$1/\tau_{N{-}3}>W^{N{-}3,N{-}4}_{1,0}$ the sidebands are not present (Fig. 3b)
pointing to a charge-state dependence of the vibrational relaxation rates.
Using a minor extension to the model developed by Koch et al. to include
charge-state dependent relaxation times, $\tau$, we calculate the conductance
map (Fig. 3d) using these inequalities[27], which shows good correspondence
with the experimental data, reproducing the key features of the transport
spectrum.

The temperature dependence of the conductance supports the assignment of the
sidebands as cotunneling-assisted sidebands. At zero bias the Coulomb peak and
the first sideband follow equilibrium behavior (Fig. 2c). The conductance of
the Coulomb peak ($V_{\rm{CP}}=-1.08$ V) decays as
$G_{max}\propto\frac{1}{k_{B}T}$, due to thermal broadening of the electrode
Fermi-Dirac distributions, and the conductance at the point where the first
sideband would cross $V_{sd}=0$ (from which electron transfer is driven from
$q=1$, red circle, Fig. 2a) can be fitted to the product of thermal broadening
and the Bose distribution[8]:

$G_{max}\propto\frac{1}{k_{B}T}\times\frac{1}{exp(\hbar\omega_{q}/k_{B}T)-1}$
(4)

with $\hbar\omega_{q}=8\pm 1$ meV. This demonstrates the zero-bias conductance
increase is driven thermally by an increasing equilibrium population in $q=1$.
The picture is different at higher bias however (Fig. 2d). At
$eV_{\rm{sd}}=\hbar\omega_{q}$ the conductance of the first sideband is
already high at 5 K and decreases with increasing temperature, indicating
there is a substantial non-equilibrium population at
$eV_{\rm{sd}}=\hbar\omega_{q}$, that must be driven by the tunneling dynamics
of the system, i.e. inelastic co-tunneling and slow relaxation. For the second
sideband (orange line in Fig. 2) sequential tunneling begins from
$\ket{N{-}4,q{=}2}$. At $|eV_{\rm{sd}}|=\hbar\omega_{q}$ (red square), this
state can only arise as the result of two inelastic excitations of the
vibrational ground state or due to an inelastic excitation of a thermally-
populated $\ket{N-4,q{=}1}$ state. The latter becomes appreciable only at
higher temperature. Since the emergence of the second sideband at positive
bias is observed only above 10 K, this suggests the second mechanism
dominates. A fit to equation 4 gives $\hbar\omega_{q}=7.4\pm 1.5$ meV. The
sidebands are stronger at negative bias, likely due to asymmetries resulting
from both molecule-electrode coupling, common in single-molecule devices[21],
and an asymmetric voltage drop across the junction ($\alpha_{\rm{S}}=0.24$,
extracted from the Coulomb diamonds slopes).

The neighboring transition, $N{-}3$/$N{-}2$ (blue box on Fig. 1d), as with
$N{-}4$/$N{-}3$ , displays signatures of strong electron-phonon coupling,
i.e., Franck-Condon blockade involving a mode of 10 meV††footnotemark: .
Vibrational modes that couple strongly to the electron transfer process are
those that displace atoms along the same vectors that define the nuclear
rearrangement. Density functional theory calculations aid our understanding of
strong electron-vibrational coupling in the oxidations of FP2[28]. Frequency
calculations (B3LYP/6-31G(d) functional/basis set) on FP2 optimized in the $N$
to $N{-}4$ states show that modes around 9 meV correspond to out-of-plane
motions of the molecule, i.e., saddling of the porphyrin unit, or torsional
motions between the pyrene/porphyrin $\pi$-systems. Fig. 4 shows the angle
between the $\pi$-systems indeed vary with charge state, providing an
explanation for why a mode of this nature would couple strongly to charge
transfer in the oxidations of FP2, and similar motions were observed to couple
to electron transfer in porphyrin monomers with pyrene anchoring groups[9].

Figure 4: (a) An example of vibrational mode around the energy (9 meV), the
energy responsible for FC blockade in $N{-}3$/$N{-}2$ and the sidebands in
$N{-}4$/$N{-}3$. (b) DFT-calculated dihedral angle, $\theta$ (shown in (a)),
as a function of charge state. For $N{-}2$, the closely spaced singlet and
triplet (the ground state) are labelled by (s) and (t) respectively.

The origins of slow vibrational relaxation are more difficult to pin down;
dissipation is affected by energy-dependent coupling to the phononic
background of the substrate and electrodes, as well as intramolecular
vibrational redistribution[29, 30]. A room-temperature study of molecular
vibrational lifetimes in anthracene derivatives in molecular junctions gave
values in the 10–100 picosecond range, similar to those measured in solution,
where picosecond relaxation is typical[31]. In general vibration relaxation
will slow as temperature decreases[30]. For example, our measured relaxation
time of $\tau_{N{-}4}>8$ ns, giving a $Q$-factor ($\tau E/h$) of $>20,000$, is
comparable to that found for the radial breathing mode of a carbon nanotube at
5 K[32]. Particularly interesting is the link between the molecular charge
state, and consequently, its geometry, and the vibrational relaxation rate. It
is noteworthy that FP2 in the $N{-}4$ oxidation state has a coplanar
porphyrin-pyrene system (Fig. 4). Therefore the potential resulting from
rotation around the torsional angle, $\theta$, is approximately harmonic (an
assumption on which the above analysis is based), whereas for $N$ to $N{-}3$
with non-zero values of $\theta$ the potential is quartic with two local
minima, corresponding to syn and anti conformations of the pyrene-porphyrin-
pyrene system. Quartic potentials increase the energy density of low-energy
modes and, combined with the asymmetry induced by the substrate, could
expedite intramolecular vibrational relaxation for the $N$ to $N{-}3$
states[33, 34].

In summary, we demonstrate that low-energy, torsional motions of a pyrene-
porphyrin dimer-pyrene couple strongly to electron transfer and can lead to
cotunneling-assisted absorption sidebands. We observe a strong dependence of
the vibrational relaxation rate on the charge state of the molecule, and
moving beyond the harmonic approximation may be required to fully unravel the
tunneling dynamics at play. This study of non-equilibrium vibrational dynamics
and dissipation sheds light on heat and energy flow in molecular devices.

###### Acknowledgements.

This work was supported by the EPSRC (grants EP/N017188/1 and EP/R029229/1).
JAM acknowledges funding from the Royal Academy of Engineering and a UKRI
Future Leaders Fellowship, Grant No. MR/S032541/1. The authors would like to
acknowledge the use of the University of Oxford Advanced Research Computing
(ARC) facility in carrying out this work.
http://dx.doi.org/10.5281/zenodo.22558

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##  I Supplementary Information

###  I.1 Synthetic details

FP2. The dibromo- precursor Br2FP2 (6.88 mg, 1.99 µmol) and
1-ethynyl-3,6,8-tridodecoxypyrene (7.79 mg, 10 µmol) were dissolved in toluene
(4 mL) and DIPA (1 mL). The solution was degassed by freeze-pump-thaw three
times before [Pd(PPh3)4] (0.23 mg, 0.2 µmol) and CuI (0.04 mg, 0.2 µmol) were
added under a flow of argon. After another freeze-pump-thaw cycle, the mixture
was heated to 50 °C for 20 hours. The solvents were removed and the product
purified by repeated chromatography (SiO2, PE/DCM 5:1 followed by size-
exclusion, CHCl3), to give FP2 as a green/brown solid
($6.38\text{\,}\mathrm{mg}$, yield = 64%). For deposition onto graphene
electrodes $2\text{\,}\mathrm{\SIUnitSymbolMicro l}$ of a
$2\text{\,}\mathrm{\SIUnitSymbolMicro M}$ toluene solution of FP2 was dropcast
onto the devices. 1H NMR (400 MHz, CDCl3) $\delta$ 8.67 (d, J = 4.6 Hz, 4H),
8.50 (d, J = 9.3 Hz, 2H), 8.31 (d, J = 9.3 Hz, 2H), 8.26 (d, J = 9.4 Hz, 2H),
8.18 (d, J = 9.3 Hz, 2H), 7.78 (s, 8H), 7.76 (s, 4H), 7.75 (s, 2H), 7.49 (d, J
= 4.5 Hz, 4H), 7.06 – 7.05 (m, 6H), 4.32 – 4.19 (m, 12H), 2.02 – 1.86 (m,
12H), 1.63 – 1.50 (m, 12H), 1.43 – 1.01 (m, 288H), 0.86 – 0.64 (m, 138H). m/z
(MALDI-TOF, dithranol): 4843.6958 ([M]+ calcd. 4860.95).

###  I.2 Additional transport data

Figure S1: Current stability diagrams, measured at 4 K, before and after FP2
deposition.

###  I.3 Determination of ranges of molecule-electrode and electron-phonon
coupling

Figure S2: (a) $R^{2}$ (blue markers) of fit for zero-bias gate traces from 5
K to 10 K, with different values of $\lambda_{q}$, the corresponding molecule-
electrode coupling is given in red. (b) For each set of $\lambda_{q}$ and
$\bar{\Gamma}$ the calculated sequential tunneling rate $W^{N-4,N-3}_{1,0;L}$
and cotunneling rates $W^{N-4,N-4}_{0,1}$ at the onset of the first sideband
in $N{-}4$.

###  I.4 Franck-Condon blockade in $N{-}3$/$N{-}2$

Figure S3: (a) Conductance maps at different temperatures of the
$N{-}3$/$N{-}2$ transition (blue box in Fig. 1), showing conductance
suppression at low bias and low temperature. (b) Zero-bias gate traces
displaying the lifting of the blockade with increasing temperature, consistent
with FC blockade, and a fit the temperature-dependence of the Coulomb peak
(equation 4) gives the mode energy of $\lambda_{q}$.


# Thermality and Heat Content of horizons from infinitesimal coordinate
transformations

Bibhas Ranjan Majhi and T. Padmanabhan
IUCAA, Pune University Campus,
Ganeshkhind, Pune- 411 007.
email<EMAIL_ADDRESS>email<EMAIL_ADDRESS>
###### Abstract

Thermal properties of a static horizon, (like the entropy $S$, heat content
$TS$ etc.) can be obtained either from the surface term of the Einstein-
Hilbert action or by evaluating the Noether charge, corresponding to the
diffeomorphisms generated by the timelike Killing vector field. We show that,
for a wide class of geometries, the same results can be obtained using the
vector field which produces an infinitesimal coordinate transformation between
two physically relevant reference frames, viz. the freely falling frame near
the horizon and the static, accelerated, frame. In particular, the
infinitesimal coordinate transformation from inertial coordinates to uniformly
accelerated frame can be used to obtain the heat content and entropy of the
Rindler horizon. This result offers insight into understanding the observer
dependent degrees of freedom which contribute to the entropy of null surfaces.

##  1 Introduction

Classical general relativity is built on the principle of general covariance
which denies special status to any particular class of coordinate systems or
associated observers. Phenomena which are specifically coordinate dependent
are usually treated with suspicion by classical relativists. The situation,
however, is different when one brings in quantum effects through the study of
quantum field theory in non-trivial background spacetimes. The pioneering work
by Davies and Unruh [1] showed that an accelerating observer will attribute a
non-zero temperature to the quantum state, interpreted as a zero-temperature
vacuum state by the inertial observers. Similarly the thermal phenomenon
associated with a black hole will also depend on the observer; while a
stationary observer outside the event horizon will attribute an entropy and
temperature to the event horizon, another observer freely falling through the
event horizon will perceive the physics quite differently. It appears
inevitable that these results require one to attribute an observer dependence
to all thermal phenomena in nature. (For more details on this point of view,
see e.g.,[2, 3]).

The two physical contexts mentioned above — the black hole horizon and the
Rindler horizon — are, in fact, related to each other in a fairly simple
manner. The metric near the event horizon of a massive ($M\gg M_{Planck}$)
black hole can be approximated by the metric of the accelerated (Rindler)
frame in flat spacetime. The thermal phenomena perceived by observers close to
the event horizon can then be mapped to those perceived by Rindler observers
in flat spacetime and one obtains the correct expression for the temperature
by this mapping. In this correspondence, the freely falling observers through
the event horizon will be analogous to inertial observers in flat spacetime
while the stationary observers outside the event horizon will be similar to
uniformly accelerated observers in flat spacetime. Given the fact that
stationary observers in black hole spacetime attribute an entropy density
(entropy per unit area) of 1/4 in Planck units to the event horizon, it seems
inevitable that accelerated observers in flat spacetime will also attribute an
entropy density of 1/4 to the Rindler horizon. The quasi-local nature of
physics implies that any null surface which acts as a local Rindler horizon to
a class of observers should have an observer dependent entropy density of 1/4
vis-a-vis this class of observers.

While the above results do not seem to lead to any clear-cut paradoxes (like
violation of conservation laws or perpetual motion) they do appear somewhat
perplexing and counter-intuitive. In non-gravitational physics, we are
accustomed to thinking of entropy as an observer-independent quantity being
related to certain unobserved degrees of freedom. It is, therefore tempting to
think of the entropy of a black hole as being absolute and is observer
independent and related to some specific degrees of freedom. This, however,
cannot be true in view of the argument given above, if the quasi-local nature
of physics has to be maintained. The fact that an observer freely falling
through the black hole event horizon has access to more spacetime region than
the observer eternally confined to the outside of the black hole shows how
such an observer dependence can indeed arise in this context. While this is
intuitively understandable, it would be nice to see how this effect arises in
the standard procedures we adopt for the computation of the entropy of the
horizons. We will illustrate one class of such examples in this paper.

A possible mechanism behind the origin of observer dependent gravitational
degrees of freedom was suggested in Ref.[4] based on the approach pioneered by
Carlip [[5]; also see [6]]. The standard description of gravity is invariant
under the set $\mathcal{C}$ of all possible diffeomorphisms. These
diffeomorphisms allow us, in principle, to remove all the gauge degrees of
freedom retaining only the diffeomorphism invariant physical degrees of
freedom. When we limit our attention to a set of observers who perceive a null
surface as a horizon, the physical context gets restricted to a situation in
which only those diffeomorphisms in the set $\mathcal{C}^{\prime}$, which
retain the horizon structure, are allowed. While it is a difficult (and
unsolved) problem to precisely quantify what this restricted class
$\mathcal{C}^{\prime}$ is, it seems reasonable to assume that
$\mathcal{C}^{\prime}$ is a proper subset of $\mathcal{C}$. This, in turn,
implies that using the diffeomorphisms in $\mathcal{C}^{\prime}$, we cannot
remove all the redundant gravitational degrees of freedom which we could have
originally removed using the full set $\mathcal{C}$. In other words, certain
degrees of freedom which would have been treated as pure gauge (when the
theory was invariant under $\mathcal{C}$) now gets ‘upgraded’ to physical
degrees of freedom (when the theory is invariant under
$\mathcal{C}^{\prime}$). The entropy attributed to the null surface, by the
observers who perceive it as a horizon, could arise from these degrees of
freedom. (See [7] for related ideas in a different context.)

The above description is purely intuitive and mathematically ill-defined. To
make progress towards a somewhat concrete realization of the above ideas and
test their conceptual veracity, we will investigate the following issue: In a
flat spacetime, the inertial observers do not attribute any thermal properties
to any null surface. Consider now an infinitesimal coordinate transformation
$x^{a}\to x^{a}+q^{a}(x)$ from the inertial coordinate system to the Rindler
coordinate system. We then ask: Is it possible to obtain the thermal
properties, associated with a null surface in flat spacetime, in terms of the
vector field $q^{a}$? We will show that this is indeed possible and — in this
sense — one can consider the infinitesimal transformations described by the
vector field $q^{a}$ as having upgraded some of the gauge degrees of freedom
to physical degrees of freedom. We stress that $q^{a}(x)$ is not the usual
Killing vector field associated with the Rindler time translation; in fact, we
are not aware of any previous discussion in the literature of this particular
vector field. The same ideas work in a much wider class of spacetimes with
horizons, when we consider the infinitesimal coordinate transformations
between the freely falling frame near the horizon and the frame of the static
observers (like, for e.g., in terms of the infinitesimal coordinate
transformation from Kruskal to Schwarzschild coordinates) and even in a more
general context.

The paper is organized as follows: In Section 2 we review and rephrase some of
the standard known results to motivate and show how the surface Hamiltonian,
$H_{\textrm{sur}}\equiv TS$ (which represents the heat containt of the
horizon), is related to the surface action and the Noether charge. Next, we
introduce the infinitesimal coordinate transformation from inertial to Rindler
frame and the vector field corresponding to this transformation. In Section 4,
both the surface term of the action and the Noether charge are evaluated in
terms of this vector field and shown to lead to identical expression. We
explain how this arises and provide a discussion of related issues. Section 5
generalizes our findings for a general static, spherically symmetric metric
and describes how a more general spacetime can be handled by this approach. In
the final section we discuss our results.

##  2 Heat content and thermality of horizons

We begin by briefly reviewing some of the standard results rephrasing them in
a useful manner. A convenient physical quantity which captures the thermality
of a any system is the difference between the energy $E$ and the free energy
$F=E-TS$, viz. $TS$. In the case of a spacetime horizon, as we will see, this
heat content can be related [3, 8] to the surface Hamiltonian

$H_{\rm sur}\equiv TS=\frac{\kappa}{2\pi}\ S=\frac{\kappa A_{\perp}}{8\pi}$
(1)

where $S=(A_{\perp}/4)$ is the entropy of the horizon with $A_{\perp}$ being
its area, $T=(\kappa/2\pi)$ is the temperature with $\kappa$ being the
relevant acceleration (Rindler acceleration or the surface gravity). In the
case of non-compact horizons we will interpret $(H_{\rm sur}/A_{\perp})$ as
$Ts$ where $s$ is the entropy density, viz. entropy per unit area. The
acceleration $\kappa$ can be defined through the relation
$\xi^{b}\nabla_{b}\xi^{a}=\kappa\xi^{a}$ in the case of Killing horizons
associated with a time-like Killing vector field $\xi^{a}$, which becomes null
on the horizon surface. Defined in this manner, the numerical value of
$\kappa$ changes if we rescale $\xi^{a}$. This ambiguity is avoided by either
defining the normalization of $\xi^{a}$ through some physical consideration or
by dealing with quantities which are invariant under the rescaling.

The heat content $H_{\rm sur}$ can be computed in two equivalent ways, both of
which capture its physical meaning [3, 8]. The first method is to define
$H_{\rm sur}$ as the Hamiltonian associated with the surface term of the
Einstein-Hilbert action through $H_{\rm sur}=-(\partial\mathcal{A}_{\rm
sur}/\partial t)$. As is well-known, the Einstein-Hilbert Lagrangian can be
written as a sum of a bulk term ($L_{bulk}$, which is quadratic in $\Gamma$)
and a surface term $L_{sur}$ with

$L_{\rm sur}=\frac{1}{16\pi}\partial_{c}(\sqrt{-g}V^{c});\qquad
V^{c}\equiv-\frac{1}{g}\partial_{b}(gg^{bc})$ (2)

(see e.g., eq (6.15) of [9]). The surface action $\mathcal{A}_{\rm sur}$ is
defined as the integral over $L_{sur}$. If the near horizon metric is
approximated as a Rindler metric (with $-g_{00}=1/g_{xx}=N^{2}=2\kappa x$ to
evaluate these expressions on $N=$const surface, we will get:

$\mathcal{A}_{\rm sur}=\frac{1}{16\pi}\int_{x}dtd^{2}x_{\perp}V^{x}=\pm
t\left(\frac{\kappa A_{\perp}}{8\pi}\right)$ (3)

where $A_{\perp}$ is the transverse area. (The sign depends on the convention
chosen for the outward normal or whether the contribution of the integral is
taken at the inner or outer boundaries; see e.g., the discussion in [15]. We
will choose the negative sign in Eq. (3).) More generally, a static, near-
horizon, geometry can be described by the metric [16, 17]

$ds^{2}=-N^{2}dt^{2}+dl^{2}+\sigma_{AB}dx^{A}dx^{B};$ (4)

where $N$ and $\sigma_{AB}$ have the near-horizon behaviour of the form

$N=\kappa l+O(l^{3});\quad\sigma_{AB}=\mu_{AB}(x^{A})+O(l^{2})$ (5)

with $l=0$ surface being the horizon. The integrals in Eq. (3) again leads to
the same result. Near the static horizon, the integrand is independent of $t$
and transverse coordinates and hence $\mathcal{A}_{\rm sur}$ must be
proportional to $tA_{\perp}$. The proportionality constant turns out to be
$\kappa/8\pi$.

With future applications in mind, we stress that the $A_{sur}$ used in Eq. (3)
is not the full surface action but only the contribution of it on the horizon.
This is relevant when $L_{sur}\propto\partial_{c}(\sqrt{-g}V^{c})$ vanishes
because $\sqrt{h}n_{c}V^{c}=$ constant. If we define the action as the
integral of $L_{sur}$ over a spacetime domain $\mathcal{V}$, the action can be
expressed as the surface integral over the boundary $\partial\mathcal{V}$ of
this domain and this will also vanish. But the contribution of the surface
integral from part of the boundary surface (like the horizon, say) will be
given by the integral of $\sqrt{h}n_{c}V^{c}$ over that part of the surface
and need not vanish. It is this contribution, computed over the horizon, which
we have defined as $A_{sur}$ and this can be nonzero even when
$L_{sur}\propto\partial_{c}(\sqrt{-g}V^{c})=0$ provided $V^{c}$ itself is not
zero. We will have occasion to consider such a case later on.

Given $A_{sur}$, the surface Hamiltonian is defined in a standard manner as:

$H_{sur}\equiv-\frac{\partial\mathcal{A}_{sur}}{\partial
t}=\frac{1}{16\pi}\int_{x}d^{2}x_{\perp}V^{x}=\left(\frac{\kappa
A_{\perp}}{8\pi}\right)=TS$ (6)

with suitable choice of sign. The $H_{sur}$ is the difference between the
energy $E$ and the free energy $F=E-TS$ of a finite temperature system and
measures the heat content of the horizon.

The second method [3, 8] to compute the heat content is by using the Noether
potential. In any gravitational theory which is invariant under the
diffeomorphism $x^{a}\to x^{a}+q^{a}(x)$, one can define a conserved Noether
current $J^{a}$ (which depends on $q^{a}$) related to an anti-symmetric
Noether potential $J^{ab}$ through $J^{a}=\nabla_{b}J^{ab}$. In Einstein’s
theory one can choose the Noether potential to be:

$J_{ab}=\frac{1}{16\pi}\left(\nabla_{a}q_{b}-\nabla_{b}q_{a}\right)=\frac{1}{16\pi}\left(\partial_{a}q_{b}-\partial_{b}q_{a}\right)$
(7)

The surface Hamiltonian can then be defined as the surface integral over a co-
dimension-2 cross-section of the null surface:

$H_{\rm sur}=TS\equiv\int d^{D-2}\Sigma_{ab}\;J^{ab}$ (8)

where the Noether potential is computed for $q^{a}=\xi^{a}$ with $\xi^{a}$
being the local Killing vector corresponding to time translation symmetry of
the local Rindler frame near the horizon. (The connection between Noether
charge and energy has been previously noted in, e.g., Refs. [10]) Note that
both $J_{ab}$ and $T$ rescale in the same manner, if $\xi^{a}$ is rescaled,
leaving the entropy invariant.

While the two ways of computing $H_{\rm sur}$ described above — from the
surface term of the Einstein-Hilbert action in Eq. (6) or from the Noether
potential related to the time-like Killing vector in Eq. (8) — lead to the
same expression for $H_{\rm sur}$, there is a subtle difference between the
two results. The surface term of Einstein-Hilbert action defined through
$L_{sur}$ is not generally covariant. In flat spacetime, $A_{sur}$ will vanish
in inertial coordinates while it can be non-zero in the non-inertial
coordinates. To obtain a non-zero value for $H_{\rm sur}$ from
$\mathcal{A}_{\rm sur}$ using Eq. (6) we need to explicitly work with the
noninertial, Rindler, coordinates in flat spacetime. More generally, in a
curved spacetime, the expression for $\mathcal{A}_{\rm sur}$ can depend on the
coordinate system which is used. The expression for $H_{\rm sur}$ in terms of
the Noether potential in Eq. (8), on the other hand, is manifestly covariant
and will lead to the same result in all coordinate systems. While this may be
surprising at first sight, it should be noted that — while the value of the
integral in Eq. (8) will be the same in all coordinate systems — the physical
interpretation of $\xi^{a}$ used to define the Noether potential is different
in different coordinate systems. In flat spacetime, $\xi^{a}$ will be thought
of as generator of time translation in Rindler frame but it is (proportional
to) the Lorentz boost generator in the inertial frame.

It is worth mentioning at this stage that there is another (York-Gibbons-
Hawking, [11]) surface term which is often discussed in the gravitational
context and is given by

$\mathcal{A}_{\rm
YGH}\equiv\pm\frac{1}{16\pi}\int_{\mathcal{S}}d^{3}x\,\sqrt{h}\,(2K)$ (9)

where $K$ is the extrinsic curvature of the surface $\mathcal{S}$ and $h_{ab}$
is the metric induced on this surface. (The overall sign depends on whether
the surface is spacelike or timelike.) This term is usually added to the
Einstein-Hilbert action to cancel the variations of the metric normal to the
boundary. Contrary to the popular myths: (i) The action $\mathcal{A}_{\rm
YGH}$ does not cancel the surface term $\mathcal{A}_{\rm sur}$ in general.
(ii) Even their variations do not cancel each other for arbitrary variations
of the metric but they do cancel each other if the metric is held fixed on the
boundary [see e.g., pages 249-250 of Ref. [9]]. This property is shared by an
infinite set of surface functionals [12] and the choice of $\mathcal{A}_{\rm
YGH}$ is merely a convenient one to make the variational principle in gravity
well defined. In this sense, $\mathcal{A}_{\rm sur}$ has a better standing as
a well defined surface term arising naturally from the Einstein-Hilbert action
itself than $\mathcal{A}_{\rm YGH}$. Nevertheless, given the popularity
enjoyed by $\mathcal{A}_{\rm YGH}$ in the literature, we will briefly mention
the corresponding results for this action.

It is easy to show that (see, for e.g., Ex. (6.3) of Ref. [9]), in general,
the integrands of the surface integral in $\mathcal{A}_{\rm sur}$ and
$\mathcal{A}_{\rm YGH}$ are related as follows:

$V^{a}n_{a}=2K+2h^{ab}\partial_{b}n_{a}-n^{m}h^{ns}\partial_{n}g_{sm}$ (10)

The second and third terms on the right hand side will not vanish for an
arbitrary surface and coordinates because of which $n_{c}V^{c}$ is obviously
not a scalar even though $2K$ is. However, if the coordinate system is chosen
such that the surface corresponds to, say, $x^{1}=$ constant with the metric
being block diagonal with respect to the $x^{1}$ coordinate (i.e., $g_{1k}=0$
for $k\neq 1$) then it is easy to show that $V^{a}n_{a}=2K$ (see, for e.g.,
the Appendix of Ref. [13]). Because most of the metrics we deal with (and
certainly the Rindler metric which is valid close to the horizon) belongs to
this class, we can use $2K$ and $V^{c}n_{c}$ interchangeably for our purpose.
This implies that, in the Rindler frame, we should get the same answer for
$\mathcal{A}_{\rm YGH}$ and $\mathcal{A}_{\rm sur}$ when evaluated on a
stretched horizon corresponding to $x=\epsilon$ with the limit $\epsilon\to 0$
taken in the end. This is indeed true since in the Rindler frame
$K\sqrt{h}=\kappa$=constant, independent of $\epsilon$, leading to the result:

$\mathcal{A}_{\rm YGH}=-t\left(\frac{\kappa A_{\perp}}{8\pi}\right)$ (11)

So, in the Rindler frame, we can also write the result in Eq. (6) as

$H_{\rm sur}=-\xi^{a}\nabla_{a}\mathcal{A}_{\rm
YGH}=\xi^{a}\nabla_{a}\left[\frac{1}{8\pi}\int_{\mathcal{S}}d^{3}x\,\sqrt{h}\,K\right]$
(12)

This provides a generally covariant expression for the heat content as long as
the expression on the right hand side is well defined. If we repeat the
calculation in the inertial coordinates, it turns out that we obtain the same
result provided: (a) We perform the differentiation in Eq. (12) before taking
the $\epsilon\to 0$ limit. (b) We choose $\xi^{a}$ with components
$\xi^{a}=\kappa(X,T,\mathbf{0})$ in the inertial frame. This is the standard
Killing vector for generation of the Lorentz boosts scaled up by $\kappa$. The
inertial frame has no intrinsic $\kappa$ in it and we can obtain the same
result for $H_{\rm sur}$ in both frames only through this scaling. We will
come back to this issue a little later.

Given the fact that $\mathcal{A}_{\rm sur}=0$ in the inertial coordinates
(because $\Gamma$s vanish), while $\mathcal{A}_{\rm sur}\neq 0$ and is given
by Eq. (3) in the Rindler coordinates, one might wonder what we get for
$\mathcal{A}_{\rm YGH}$ if evaluated in inertial and Rindler frames. As we
said before, we get $\mathcal{A}_{\rm YGH}=\mathcal{A}_{\rm sur}$ if we
evaluate both in the Rindler frame. If we compute $\mathcal{A}_{\rm YGH}$ in
the inertial coordinates taking the surface $\mathcal{S}$ to be given by
$X^{2}-T^{2}=\epsilon^{2}$ then we get the result

$\mathcal{A}_{\rm YGH}=-\frac{A_{\perp}}{8\pi}\,{\rm
tanh}^{-1}\left(\frac{T}{X}\right)\Bigg{|}_{\mathcal{S}}$ (13)

which, of course, matches with the result in Eq. (11) for non-zero $\epsilon$
because ${\rm tanh}^{-1}(T/X)=\kappa t$ ensuring general covariance. But a
naive limit of $\epsilon\to 0$ corresponding to $(T/X)\to\pm 1$ leads to a
divergence in Eq. (13) so that this expression is not well defined when we
take the null surface limit. (One can do slightly better by working in
Euclidean coordinates when one would have got $\tan^{-1}(T_{E}/X)$ instead of
the tanh${}^{-1}(T/X)$. The null surface limit now corresponds to $T_{E}\to
0,X\to 0$ and the results depend on the order in which these limits are taken.
A choice of taking $T_{E}\to 0$ first with $X$ finite will give
$\mathcal{A}_{\rm YGH}=0$ which will match with $\mathcal{A}_{\rm sur}=0$ in
inertial coordinates but not with Eq. (11).) Thus we find that
$\mathcal{A}_{\rm YGH}$ is not well defined in the null surface limit when
evaluated in the inertial coordinates even though the heat content, obtained
after the differentiation is carried out in Eq. (12), remains well defined in
both coordinates. Because of these issues and the comments made earlier about
the ad-hoc nature of $\mathcal{A}_{\rm YGH}$, we will not use it further in
our discussion.

We will next consider the corresponding aspects in the case of $H_{\rm sur}$
evaluated from Eq. (8) by explicitly computing it from Eq. (8) in flat
spacetime in both Rindler and inertial coordinates. In the local Rindler frame
with $-g_{00}=1/g_{xx}=N^{2}=2\kappa x$ if we take

$\xi^{a}=\delta^{a}_{0},\quad
d^{2}\Sigma^{ab}=\frac{1}{2}(n^{a}m^{b}-n^{b}m^{a})\sqrt{\sigma}\,d^{2}x_{\perp},$
(14)

where $n^{a}=(2\kappa x)^{1/2}\,\delta^{a}_{1}$, $m^{a}=-(2\kappa
x)^{-1/2}\delta^{a}_{0}$, then a straightforward calculation gives

$H_{\rm sur}=\int_{\mathcal{H}}d^{2}\Sigma_{ab}\,J^{ab}=\frac{1}{16\pi}\int
d^{2}x_{\perp}\,\sqrt{\sigma}\,(2\kappa)=\frac{\kappa A_{\perp}}{8\pi}$ (15)

The above calculation was done in the Rindler coordinates which has the
parameter $\kappa$ in the metric. But since the integral defining $H_{\rm
sur}$ is generally covariant we could have evaluated the same in the inertial
coordinates as well. In the inertial coordinates ($T,X,\mathbf{X}_{\perp}$),
the components of $\xi^{a}$ will be $\kappa(X,T,\mathbf{0})$ which represents
the Lorentz boost vector with an arbitrary scaling factor $\kappa$. This
vector becomes null on $X=T$ and hence $\xi^{b}\nabla_{b}\,\xi^{a}$ must be
proportional to $\xi^{a}$ on this surface; that is,
$\xi^{b}\nabla_{b}\,\xi^{a}=\lambda\xi^{a}$ on $X=T$. An elementary
calculation shows that $\lambda=\kappa$ which can be thought of as the non-
affinity parameter associated with the generators of the null surface $X=T$.
(Obviously there is no thermal interpretation for $\kappa$ in the inertial
coordinates.) The relevant non-zero component of the Noether potential in the
inertial frame is $J_{TX}=2\kappa$ while the measure for transverse
integration is just $d\Sigma^{TX}=\frac{1}{2}\sqrt{\sigma}d^{2}x_{\perp}$. The
integral in Eq. (8) now becomes

$H_{\rm
sur}=\frac{1}{16\pi}\,\int_{\mathcal{H}}d^{2}x_{\perp}\,\sqrt{\sigma}\,(2\kappa)=\frac{\kappa
A_{\perp}}{8\pi}$ (16)

Thus we get the same result, as we should, when we use the same vector
$\xi^{a}$ in either coordinate system. If we had known that non-affinity
parameter $\kappa$ has the interpretation as horizon temperature in the
Rindler frame, we could have done the entire computation in the inertial
coordinate itself. (In Section 4 we will describe a similar procedure but with
a different vector field.)

There is a related issue, regarding the dimensions of $\xi^{a}$, which is
worth noting. If we think of the coordinates $x^{a}$ having the dimensions of
length, then it is natural to assume that the deformation vector $q^{a}$
appearing in the diffeomorphism $x^{a}\to x^{a}+q^{a}$ also has the dimensions
of length. But it is conventional to take the time-like Killing vector
$\xi^{a}$ in static spacetimes to be dimensionless with
$\xi^{a}=\delta^{a}_{0}$ as we did in the above calculation. In that case, one
cannot take $q^{a}=\xi^{a}$ unless we change the dimensions of either one of
them. In the inertial coordinates it is more natural to work with
$\chi^{a}=\xi^{a}/\kappa=(X,T,\mathbf{0})$ (which is the usual Killing vector
corresponding to Lorentz boosts with the dimension of length). This is
equivalent to setting $\kappa=1$ and the non-affinity parameter for the null
congruence generating the $X=T$ surface becomes unity; that is,
$\xi^{b}\nabla_{b}\,\xi^{a}=\xi^{a}$ on $X=T$. This rescales $H_{\rm sur}$ but
note that the corresponding Rindler temperature is now just $(1/2\pi)$. The
simultaneous rescaling of $H_{\rm sur}$ and $\kappa$ keeps the entropy $S=2\pi
H_{\rm sur}/\kappa=A_{\perp}/4$ invariant.

Based on these results, we can give two prescriptions for computing the
entropy. The first one is from the surface term in the action:

$S=A_{\rm sur}\Big{|}_{t=2\pi/\kappa}$ (17)

which of course has a natural interpretation in the Euclidean Rindler sector
since the periodicity of Euclidean time is $2\pi/\kappa$; but as a
prescription it can be used even with Lorentzian Rindler time. The second
prescription is essentially that of Wald entropy [14]:

$S=\frac{2\pi}{\kappa}\,\int d^{D-2}\Sigma_{ab}\;J^{ab}$ (18)

In terms of $H_{\rm sur}$ the two definitions given above are equivalent to

$S=\int_{0}^{2\pi/\kappa}dt\,H_{\rm sur}=\frac{2\pi}{\kappa}\,H_{\rm sur}$
(19)

This shows that, in either way, the central quantity which captures the
thermality of the null surface is its heat content, $H_{\rm
sur}=(\kappa/2\pi)S$.

From the heat content $H_{sur}$ of the horizon we can obtain the heat energy
per unit area of the horizon, $H_{sur}/A_{\perp}=\kappa/8\pi=P$, which also
appears as the pressure term in the Navier-Stokes equation obtained by
projecting Einstein’s equation on to the null surface [18, 19] and leads to
the equation of state $PA_{\perp}=TS$. This heat energy density on the
stretched horizon at $x^{1}=const$ with $n_{c}=\delta_{c}^{1}$ is

$\mathcal{H}=\frac{NK}{8\pi}=\frac{1}{16\pi}\sqrt{-g}V^{c}n_{c}=-\frac{1}{16\pi}\,n_{c}(f^{ab}N^{c}_{ab})$
(20)

where $f^{ab}=\sqrt{-g}\,g^{ab}$ and $N^{c}_{ab}$ is the corresponding
conjugate momentum defined by

$N^{i}_{jk}\equiv\frac{\partial(\sqrt{-g}L_{bulk})}{\partial(\partial_{i}f^{jk})}=-[\Gamma^{i}_{jk}-\frac{1}{2}(\delta^{i}_{j}\Gamma^{a}_{ka}+\delta^{i}_{k}\Gamma^{a}_{ja})].$
(21)

Closing the loop, one can express the surface term in the Hilbert action in
terms of the heat energy density of the horizon as:

$A_{sur}=\frac{1}{16\pi}\int dt\int d^{2}x\ n_{c}(f^{ab}N^{c}_{ab})=-\int
dt\int d^{2}x\mathcal{H}$ (22)

These results show that the heat content is directly related to gravitational
degrees of freedom $f^{ab}$ and their conjugate momenta $N^{c}_{ab}$. When
expressed in terms of $N^{c}_{ab}$ the expression is not generally covariant;
the $N^{a}_{bc}$ vanishes in the inertial coordinates but not in the Rindler
coordinates.

We conclude this section with a brief mention of the physical importance of
$H_{sur}$, which also arises in the study of black hole horizons [15] and is
closely related to the phase of the semiclassical wave function of the black
hole. The phase of the semiclassical wave function describing a geometry with
a horizon will pick up a term from the surface term in the action given by:

$\Psi\propto\exp(iA_{\rm sur})=\exp\left(-i\int dtH_{\rm sur}\right)$ (23)

When a semiclassical black hole interacts with matter fields, the probability
for its area to change by $\Delta A_{\perp}$ is determined by the integral of
the form

$\mathcal{P}(\Delta A_{\perp})=\int_{-\infty}^{\infty}dtF_{m}(t)\exp[-it\Delta
H_{sur}]=\int_{-\infty}^{\infty}dtF_{m}(t)\exp[-it\frac{\kappa}{8\pi}\Delta
A_{\perp}]$ (24)

where $F_{m}(t)$ is a suitable matter variable. The exponential redshift near
the horizon will result in the time evolution of $F_{m}(t)$ having the
asymptotic form $\exp[-iC\exp(-\kappa t)]$ where $C$ is a constant. An
elementary calculation now shows that the relative probability for the horizon
to change its area by $\Delta A_{\perp}$ is given by $\exp[\Delta
A_{\perp}/4]$.

##  3 Infinitesimal coordinate transformation from the inertial to the Rindler
frame

The calculation of $H_{\rm sur}=TS$ in the Rindler frame, performed in the
last section, used the full non-linear transformation between the inertial and
Rindler coordinates in flat spacetime (or between the freely falling and
static coordinate systems near a horizon in a curved spacetime). The
calculation based on the Noether potential also used the Killing vector field
associated with the (local) Rindler frame. These results bring to focus the
issue we raised in Sec. 1 viz., how certain degrees of freedom contributing to
the heat content of a null surface arises when we switch from one coordinate
system to another. Because degrees of freedom are closely associated with the
symmetries of the theories — which in this case are infinitesimal coordinate
transformations — it would be nice to see whether the results in Sec. 2 can be
obtained using only infinitesimal coordinate transformations. As a preliminary
to this investigation in Sec.4, we will now set up the necessary
transformation.

Let us begin with a flat spacetime described in the inertial coordinates
($T,X,\mathbf{X}_{\perp}$) with the metric

$ds^{2}=-dT^{2}+dX^{2}+dL_{\perp}^{2}~{},$ (25)

where $dL_{\perp}^{2}$ denotes the flat transverse metric which will not play
any significant role in the discussion. The usual coordinate transformation
from the inertial coordinates ($T,X,\mathbf{X}_{\perp}$) to the Rindler
coordinates $(t,x,\mathbf{X}_{\perp})$ can be taken to be

$\displaystyle T=\kappa^{-1}(1+2\kappa x)^{1/2}\sinh(\kappa t);\quad
X=\kappa^{-1}(1+2\kappa x)^{1/2}\cosh(\kappa t)-\kappa^{-1}.$ (26)

leading to the metric:

$\displaystyle ds^{2}=-(1+2\kappa x)dt^{2}+\frac{dx^{2}}{1+2\kappa
x}+dL_{\perp}^{2}~{}.$ (27)

These transformations are structured such that they reduce to identity when
the acceleration $\kappa=0$ making the metric in Eq. (27) reduce to the metric
in Eq. (25).

We are interested in the infinitesimal form of these coordinate
transformations which we will define as the ones obtained by retaining only
terms up to linear order in $\kappa$ in Eq. (26). It is easy to verify that,
to this order, the transformations in Eq. (26) become:

$T=t+\kappa xt+{\cal{O}}(\kappa^{2});\,\,\,\ X=x+\frac{1}{2}\kappa
t^{2}-\frac{1}{2}\kappa x^{2}+{\cal{O}}(\kappa^{2})~{}.$ (28)

The inverse transformations are:

$t=T-\kappa XT+{\cal{O}}(\kappa^{2});\,\,\,\ x=X-\frac{1}{2}\kappa
T^{2}+\frac{1}{2}\kappa X^{2}+{\cal{O}}(\kappa^{2})~{}.$ (29)

(Since $\kappa$ is a dimensionfull parameter, the smallness or otherwise of
$\kappa$ is somewhat ill-defined. To handle this, we can replace $\kappa$ by
$\epsilon\kappa$ in all expressions where $\epsilon$ is a dimensionless
infinitesimal parameter and do the Taylor series expansion in $\epsilon$. We
will not bother to do this since it clutters up the notation and ultimately
leads to the same results.) In general, an infinitesimal transformation
between two coordinate systems $x^{a}\to\bar{x}^{a}=x^{a}+q^{a}(x)$ is
implemented by a vector field $q^{a}(x)$ in the spacetime. (Once again,
$q^{a}$ should be thought of as $\epsilon q^{a}$ with all expansions treated
as Taylor series in $\epsilon$.) In the case of infinitesimal transformation
from the inertial to Rindler coordinate system the vector field $q^{a}\equiv
x^{a}-X^{a}$ has the components in the inertial frame given by:

$\displaystyle q^{T}=t-T=-\kappa XT;$ $\displaystyle
q^{X}=x-X=-\frac{1}{2}\kappa T^{2}+\frac{1}{2}\kappa X^{2}~{}.$ (30)

It can be easily verified that $q^{a}$ is not a Killing vector; its integral
curves are given by

$\displaystyle X^{2}T-\frac{T^{3}}{3}=C~{},$ (31)

where $C$ is a constant (In fact, as far as we know, this vector field has not
been studied in the literature).

To avoid possible misunderstanding, we emphasize again how the vector field
$q^{a}$ is related to the transformation from inertial to Rindler coordinate
system. This is important because, this transformation in Eq. (26) does not
reduce to identity at infinity; similarly, the Rindler metric in Eq. (27) is
not asymptotically flat. So one cannot obtain an infinitesimal version of the
transformations at asymptotic infinity. Instead, we imagine a family of
Rindler transformations, parametrized by a dimensionless parameter $\epsilon$
with an acceleration $\epsilon\kappa$. When we vary the parameter $\epsilon$
from zero to unity, we move from identity transformation (at $\epsilon=0$) to
the final Rindler transformation (at $\epsilon=1$). Now we can consider
transformations in this family, close to the identity by doing a Taylor series
expansion of Eq. (26) in the parameter $\epsilon$. To the linear order in
$\epsilon$ we will get $x^{a}=X^{a}+\epsilon q^{a}$ which defines the vector
field $q^{a}$ we work with. Once this exercise is carried out we can set
$\epsilon$ to unity. As mentioned earlier, it is obvious that — since
$\epsilon$ occurs only in the combination $\epsilon\kappa$ — the same $q^{a}$
is obtained by simply expanding the transformation in Eq. (26) in a Taylor
series in $\kappa$. This procedure, in fact, is quite general. If we consider
any finite coordinate transformation $\bar{x}^{a}=f^{a}(x^{i},\lambda)$ where
$\lambda$ is a (in general, dimensionfull) parameter defined such that we get
identity transformation when $\lambda=0$, then by considering the family of
transformations $\bar{x}^{a}=f^{a}(x^{i},\epsilon\lambda)$ we can smoothly
interpolate between identity transformation and finite transformation when
$\epsilon$ varies between 0 and 1. Expanding
$\bar{x}^{a}=f^{a}(x^{i},\epsilon\lambda)$ in a Taylor series in $\epsilon$ we
will get $\bar{x}^{a}=x^{a}+\epsilon q^{a}$ which identifies the vector field
$q^{a}$.

##  4 Thermality from the infinitesimal coordinate transformation

We will now relate the infinitesimal coordinate transformations between
inertial and Rindler coordinates to the thermality of the null surface $X=T$.
This is probably the most instructive example because it shows such an effect
can arise even in the flat spacetime. The idea generalizes in a straight
forward manner to all static horizons and we will comment on these
generalizations later on.

How does the surface $X=T$ acquire an interpretation in terms of a heat
content $H_{\rm sur}$ when we made a coordinate transformation from inertial
to Rindler frame? Of the two methods described in Sec.2 for computing $H_{\rm
sur}$, the one based on surface term in the action is conceptually simpler to
understand, which we will discuss first.

We know that neither the bulk nor the surface term of the Einstein-Hilbert
action is generally covariant individually though, of course, $R=0$ in all
coordinate systems. In the inertial coordinates $\Gamma$s vanish making
$A_{\rm sur}=0=A_{\rm bulk}$ and hence $H_{\rm sur}=0$; so we cannot attribute
any heat content to the null surface $T=X$. But since $A_{\rm sur}$ is not
generally covariant, it acquires non-zero value under infinitesimal coordinate
transformations $x^{a}\to x^{a}+q^{a}(x)$. The connection generated to the
lowest order by this transformation is given by

$\Gamma^{a}_{bc}=-\partial_{b}\partial_{c}q^{a}.$ (32)

For the infinitesimal transformation between inertial and Rindler coordinates
in Eq. (30), this gives the non-zero components (in the Rindler frame) to be
$\Gamma^{t}_{tx}=\Gamma^{x}_{tt}=\Gamma^{x}_{xx}=\kappa$. Using this we find
that $V^{c}$ now has the component $V^{x}=-2\kappa$. Therefore, the non-zero
surface term in the action which is generated is now given by

$\mathcal{A}_{\rm sur}=-t\left(\frac{\kappa A_{\perp}}{8\pi}\right)$ (33)

leading to the standard results

$H_{\rm sur}=-\frac{\partial A_{\rm sur}}{\partial t}=\left(\frac{\kappa
A_{\perp}}{8\pi}\right)$ (34)

and, using Eq. (17) we get:

$S=A_{\rm sur}\Big{|}_{t=2\pi/\kappa}=\frac{A_{\perp}}{4}$ (35)

In this approach, it is not surprising that we get $H_{\rm sur}\neq 0$ because
$A_{\rm sur}$ is not generally covariant. The infinitesimal coordinate
transformation from inertial to Rindler coordinates generates a non-zero
$V^{c}$ and thus a nonzero contribution to $A_{\rm sur}$ on the horizon
leading to a non-zero $H_{\rm sur}$. (Aside: Note that this corresponds to the
situation mentioned earlier, in the para after Eq. (5). When we make an
infinitesimal coordinate transformation $x^{a}\to x^{a}+\epsilon q^{a}$, the
$\Gamma$ that is produced are of $\mathcal{O}(\epsilon)$. But from the
structure of the scalar curvature
$R\sim\Gamma^{2}+\partial\Gamma=\mathcal{O}(\epsilon^{2})+\mathcal{O}(\epsilon)$
we see that the two terms are of different order. When $R=0$, this will
require the two terms to vanish separately, which — in turn — implies that
$L_{sur}\propto\partial_{c}(\sqrt{-g}V^{c})=0$. The integral of $L_{sur}$,
which is a surface term, will also vanish when the contribution from the
entire surface is taken into account. However, the contribution of this
surface term from just on one surface need not vanish as long as
$\sqrt{-g}V^{c}$ itself is not zero — but a constant, which is the current
situation.)

What is surprising is that the infinitesimal coordinate transformations
captures the exact result! The transformations in Eq. (28) are obtained by
linearising the full transformations in Eq. (26) in $\kappa$. The $A_{\rm
sur}$ was computed correct only to linear order in $\kappa$ because we used
the linearized expressions in Eq. (32) in its calculation. In spite of this,
the result in Eq. (33) matches with the exact result in Eq. (3).

A partial, mathematical reason for this result is the following. Since the
exact form of $A_{\rm sur}$ has to be proportional to
$t(A_{\perp}/L_{P}^{2})$, dimensional considerations show that the
proportionality constant must have the dimensions of inverse length. The only
such constant available to us is $\kappa$ and hence it follows that the
expression should have the form $C(tA_{\perp}\kappa/L_{P}^{2})$ where $C$ is a
dimensionless numerical factor. Therefore, $A_{\rm sur}$ must be linear in
$\kappa$ and an infinitesimal transformation accurate to linear order in
$\kappa$ can reproduce the result except for a numerical factor.

This argument, however, does not explain why we obtain the correct numerical
factor which is not guaranteed by the dimensional analysis. A more physical
reasoning will be to attribute this result to the fact that infinitesimal
coordinate transformations are all that is required to study the nature of
gauge degrees of freedom. As we conjectured in Sec.1, the observer dependent
thermality of null surfaces has to do with upgrading of some gauge degrees of
freedom to true degrees of freedom. Infinitesimal coordinate transformations
should be capable of capturing this effect, which is what has happened.

We will now turn into the second procedure described in Sec.2 for computing
$H_{\rm sur}$ using Noether potential. A natural vector field to use in this
context is, of course, $q^{a}\equiv x^{a}-X^{a}$, which defines the
infinitesimal coordinate transformation between the inertial and Rindler
coordinates. We will now compute the right hand side of Eq. (8) using $J^{ab}$
defined for $q^{a}=x^{a}-X^{a}$ and working entirely in inertial coordinates.
The only relevant non-trivial component of $J^{ab}$ is
$J^{TX}=(\kappa/8\pi)T\approx(\kappa/8\pi)t$ to order $\mathcal{O}(\kappa)$.
Using $d\Sigma_{TX}=-\frac{1}{2}\sqrt{\sigma}\,d^{2}X_{\perp}$ we find that
the Noether integral now gives 111Incidentally, the null horizon surface for
the exact transformations in Eq (26) is at $T=X+1/\kappa$. If we evaluate the
charge on this surface, instead of on the surface $X=T$, and express the
result in terms of Rindler time $t$, we will get the same result in Eq (36).,

$Q=\int d^{2}\Sigma_{ab}\;J^{ab}=-T\,\frac{\kappa
A_{\perp}}{8\pi}\approx-t\,\frac{\kappa A_{\perp}}{8\pi}$ (36)

This is not the $H_{\rm sur}$ computed earlier; and indeed it is not expected
to be since the vector field we are using now is $q^{a}=x^{a}-X^{a}$ and not
$\xi^{a}$ and the result of the integration, of course, will depend on the
vector field. What is curious is that the expression matches with the
expression for $A_{\rm sur}$. This could be, at first sight, surprising for
two reasons: (i) It is not obvious why the Noether integral for the
infinitesimal coordinate transformation, calculated in the inertial frame,
should give $A_{\rm sur}$, computed in the Rindler frame. (ii) We have now
provided a generally covariant expression for $A_{\rm sur}$ which — when
treated as the surface term of the action — is obviously not generally
covariant.

It is possible to understand both these features by probing a little deeper
into the structure of the Noether charge and Noether potential. In general
relativity, the Noether current for a diffeomorphism generated by a vector
field $q^{a}$ is given by

$J^{c}=\nabla_{l}(\nabla^{c}q^{l}-\nabla^{l}q^{c})=2R^{c}_{m}q^{m}+\mathcal{V}^{c};\quad\mathcal{V}^{c}\equiv
g^{lm}\pounds_{q}N^{c}_{lm}$ (37)

where $N^{c}_{lm}$ is the canonical momentum defined in Eq. (21). In flat
spacetime, $R^{a}_{b}=0$ and we have $J^{c}=\mathcal{V}^{c}$. Note that even
though $N^{c}_{lm}=0$ in inertial coordinates, its Lie derivative along a
vector field:

$\pounds_{q}N^{c}_{lm}=-\partial_{l}\partial_{m}q^{c}+\frac{1}{2}\Big{(}\delta^{c}_{l}\partial_{m}\partial_{i}q^{i}+\delta^{c}_{m}\partial_{l}\partial_{i}q^{i}\Big{)}$
(38)

is nonzero! To the lowest order we can interpret the Lie derivative in
$\pounds_{q}N^{c}_{lm}$ as giving the variation $\delta N^{c}_{lm}$ under the
diffeomorphism generated by $q^{a}$. But since $N^{c}_{lm}=0$ in inertial
coordinates, we find that $\delta N^{c}_{lm}=\pounds_{q}N^{c}_{lm}$ is equal
to $N^{c}_{lm}$ in the the non-inertial coordinates. Thus, in this case,
$\mathcal{V}^{c}=g^{lm}\pounds_{q}N^{c}_{lm}=g^{lm}N^{c}_{lm}$ where
$N^{c}_{lm}$ now denotes the canonical momenta in the noninertial frame
generated by the transformation. It then follows that, in our case,

$J^{c}=\mathcal{V}^{c}=g^{lm}N^{c}_{lm}$ (39)

The integral in Eq. (36) over Noether potential $J^{ab}$ can be expressed as
an integral over Noether current $J^{c}$ using Stokes theorem:

$Q=\int d^{2}\Sigma_{ab}\;J^{ab}=\int d^{3}\Sigma_{c}\;J^{c}\to\int
d^{3}\Sigma_{c}\,\mathcal{V}^{c}=A_{\rm sur}$ (40)

In arriving at the third equality we have restricted ourselves to the
contribution from the horizon surface, as mentioned earlier; in obtaining the
last equality, we have used Eq. (22) giving $A_{\rm sur}$ in terms of
$N^{c}_{lm}$. This explains why in this case the Noether integral leads to
$A_{\rm sur}$.

As an aside, we note the following: The dimensions of $A_{\rm sur}$ and
$H_{\rm sur}$ are different and it may be surprising at first sight that the
integral over $d\Sigma_{ab}J^{ab}$ leads to $H_{\rm sur}$ in Eq. (8) and
$A_{\rm sur}$ in Eq. (36). This arises because $\xi^{a}$ used to define
$J^{ab}$ appearing in Eq. (8) is dimensionless while $q^{a}$ used to define
$J^{ab}$ appearing in Eq. (36) has the dimensions of length. Any deformation
vector field, appearing in $x^{a}\to x^{a}+q^{a}$ with length dimensions,
cannot lead to a result with dimensions of $H_{\rm sur}$ but will lead to a
result with dimensions of $A_{\rm sur}$ in the Noether integral. So the
dimensions are consistent.

We can now define the heat content of the null surface through the relation:

$H_{\rm sur}=-\frac{\partial}{\partial t}\,\int
d^{2}\Sigma_{ab}\;J^{ab}=-\xi^{a}\nabla_{a}\int d^{2}\Sigma_{ab}\;J^{ab}$ (41)

which shows that the heat content can be given a generally covariant
definition even with our $q^{a}$. In the Rindler coordinates we can also write

$\int dt\,H_{\rm sur}=-\int d^{2}\Sigma_{ab}\;J^{ab};$ (42)

Since the right hand side of the first relation is generally covariant, we
expect the integral of $H_{\rm sur}dt$ also to be generally covariant. In the
static situation we are considering, we again get the standard result that
$H_{\rm sur}=(\kappa A_{\perp}/8\pi)$.

It should be stressed that the entire calculation given above was carried out
in inertial coordinates. The only hint of the Rindler frame was through the
vector field $q^{a}$ which was defined by the infinitesimal coordinate
transformation connecting the inertial and Rindler frames. After defining
$q^{a}$ by this procedure, we computed the Noether potential corresponding to
this infinitesimal diffeomorphism. This Noether integral, evaluated over the
null surface $X=T$ in the inertial frame, then reproduces the surface term of
the action evaluated using the exact metric in the Rindler frame. From this
expression, we can obtain the heat content by Eq. (41) working entirely in the
inertial frame. The fact that the Noether potential for the infinitesimal
coordinate transformation to Rindler frame is directly related to the degrees
of freedom — which ‘come alive’ through non-zero $N^{a}_{bc}$ — is the
physical reason why the procedure works. We believe this is a first step in
demonstrating in a concrete setting the relationship between coordinate
transformations and observer dependent notion of thermality.

Before we conclude the section, we clarify certain aspects of the above
analysis in order to avoid possible misunderstanding. We begin by noting that
the Noether charge

$Q=\int_{\mathcal{S}}d^{2}\Sigma_{ab}J^{ab}[q]$ (43)

is well-defined for any 2D surface $\mathcal{S}$ and any diffeomorphism
$x^{a}\to x^{a}+\epsilon q^{a}$ which introduces the vector field $q^{a}$.
Normally, in the case static horizons, one takes $\mathcal{S}$ to be the
horizon surface (usually treated as a limit of a timelike stretched horizon)
and takes $q^{a}$ to be the timelike Killing vector field; i.e
$q^{a}=\xi^{a}=(1,0,0,0)$. One then finds that $Q$ can be related to the
entropy of the horizon.

But one can certainly explore the integral in Eq. (43) over the horizon
surface $\mathcal{S}$ for any diffeomorphism $x^{a}\to x^{a}+\epsilon q^{a}$.
The diffeomorphism can be chosen independently of the surface $\mathcal{S}$.
In our approach, we make these choices as follows:

  1. 1.

We choose $\mathcal{S}$ to be the null surface $X=T$ which acts as the horizon
for a class of Rindler observers with acceleration $\kappa$. This allows one
to define the temperature of the horizon using the standard, finite,
coordinate transformations in Eq. (26), using, say, periodicity in Euclidean
continuation.

  2. 2.

We obtain $q^{a}$ from the finite Rindler transformation by a well-defined
Taylor series expansion described at the end of Sec.3. This essentially
involves introducing a book-keeping parameter $\epsilon$, defining a family of
transformations and picking $q^{a}$ from the structure of transformations near
the identity transformation.

  3. 3.

The choice of $\mathcal{S}$ and choice of $q^{a}$ in the above two items are
independent of each other. The thermodynamics of $\mathcal{S}$ (like e.g.,
definition of temperature) arises from the full Rindler transformation and
could not be generated just from $q^{a}$. This is obvious from the fact that
several finite transformations might have the same infinitesimal limit.

  4. 4.

So, a priori, if we use $\mathcal{S}$ and $q^{a}$ defined by the above
procedure and evaluate the integral in Eq. (43) we do not expect anything
physically meaningful to emerge. One would have naively thought that the
infinitesimal limit of this transformation does not contain adequate structure
to reproduce the temperature etc. originally defined by the full
transformation.

  5. 5.

The key point of this paper is that, contrary to this naive expectation, the
$Q$ in Eq. (43) evaluated by this procedure captures the thermodynamic
features of $\mathcal{S}$, determined originally via the full Rindler
transformations. That is, one can indeed reconstruct the thermality of the
horizon by a prescription which uses only the infinitesimal form of the
transformation. If the $\kappa$ of the infinitesimal coordinate transformation
can be directly related to the temperature associated with the Rindler horizon
at $X=T$, then it is not surprising if we get the horizon thermality using
only the infinitesimal transformation. But this is not the case; one does not
expect any relation between the parameter $\kappa$ in the infinitesimal
coordinate transformation and the temperature associated with the Rindler
horizon at $X=T$ using full Rindler transformation, which makes our result
nontrivial and interesting.

We have tried to give detailed reasoning as to why this happens and how it
connects up with the broader picture of degrees of freedom etc. We will now
study some generalizations of this approach to curved spacetimes.

##  5 Generalization to curved spacetime

The above ideas can be generalized to a wide class of curved, static
spacetimes with horizon. One such class is described by metrics of the form

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}~{}.$
(44)

with $f(r=r_{H})=0$ defining the location of the horizon and the surface
gravity given by $\kappa=f^{\prime}(r_{H})/2\neq 0$. It is well known that
(see e.g., page 343 of Ref. [9]) all such metrics allow a transformation to
Kruskal-like coordinates with

$\displaystyle\kappa T=e^{\kappa r_{*}}\sinh(\kappa t);\quad\kappa X=e^{\kappa
r_{*}}\cosh(\kappa t)~{},$ (45)

where the tortoise coordinate $r_{*}$ is defined by the relation
$dr_{*}=[dr/f(r)]$. Comparison with the transformation between inertial and
Rindler coordinates shows that the $T,X$ coordinate system is similar to
inertial coordinates and $t,r$ coordinate system is similar to Rindler
coordinates. In fact, the $T,X$ coordinate system is a freely falling
coordinate system near the horizon and allows us to define locally inertial
frames in that region.

Using the Taylor series expansion in $\kappa$ of the coordinate
transformations in Eq. (45) we can define the infinitesimal version of the
transformations and obtain $q^{a}=x^{a}-X^{a}$. We then define the Noether
potential for this $q^{a}$ and evaluate the Noether integral as before. The
straight forward but somewhat long computation (see Appendix A) now gives the
result that

$\int d^{2}\Sigma_{ab}\;J^{ab}=-t\,\frac{\kappa A_{\perp}}{8\pi}$ (46)

to the lowest order in $\kappa$.

The rest of the interpretation is similar. The $T,X$ coordinate system and the
associated freely falling observers near the horizon do not attribute any
special properties to the horizon just as the inertial observers do not
attribute any thermal properties to the null surface $X=T$. We however know
that the static observers outside the horizon attribute a heat content and
entropy to the horizon. The above result shows that infinitesimal coordinate
transformation from the Kruskal-like coordinates to Schwarzschild-like
coordinates captures the essence of this phenomena.

In the case of flat spacetime $N^{a}_{bc}$ was zero in the inertial frame and
non-zero in the Rindler frame. In this case Kruskal-like coordinates we have,
in general, non-zero values for connection at an arbitrary event. However,
near the horizon since Kruskal coordinates reduce to a freely falling frame,
we have $N^{a}_{bc}\approx 0$ near the horizon in the $T-X$ plane in Kruskal
coordinates. The coordinate transformation to Schwarzschild-like coordinates
now generates $N^{a}_{bc}$ near the horizon, the integral over which leads to
the result in Eq. (46). We believe this provides a fairly local description of
the heat content of the horizons and an alternate way of interpreting the
results even in the well known cases like the Schwarzschild metric, de Sitter
metric etc..

A natural question to ask would be how general these results are. We describe
below one possible direction of analysis and some preliminary results leaving
further details for a future investigation. Consider a spacetime described by
a metric of the form

$ds^{2}=-\Omega(u,v,x_{\perp})dudv+dx_{\perp}^{2}~{},$ (47)

where $u=T-X$, $v=T+X$ are the standard null coordinates. We will consider the
spacetime around the null surface $u=0$. Consider an infinitesimal coordinate
transformations: $\bar{u}=u+\xi^{u}(u,v,x_{\perp})$ and
$\bar{v}=v+\xi^{u}(u,v,x_{\perp})$, such that

$\displaystyle\xi^{u}(u,v,x_{\perp})=A_{1}+uA_{2}+u^{2}A_{3}+\dots;$
$\displaystyle\xi^{v}(u,v,x_{\perp})=B_{1}+uB_{2}+u^{2}B_{3}+\dots~{},$ (48)

where $A_{i}$ and $B_{i}$ are functions of $(v,x_{\perp})$ and we have done a
Taylor series expansion around $u=0$. If the null congruence associated with
the surfaces $u=$ constant is $l^{a}$, then the non-affinity parameter is
defined by the relation $l^{a}\nabla_{a}l_{b}=\lambda l_{b}$. This leads to
$\lambda(u,v,x_{\perp})=\partial_{v}\ln\Omega$ (see Appendix B for details).
We can also do a Taylor series expansion of $\lambda$ as

$\displaystyle\lambda(u,v,x_{\perp})$ $\displaystyle=$
$\displaystyle\lambda_{0}(v,x_{\perp})+uC_{1}(v,x_{\perp})+u^{2}C_{2}(v,x_{\perp})+\dots~{},$
(49)

Repeating the analysis described in the previous sections, one can show that
we again get the result:

$\int d^{2}\Sigma_{ab}\;J^{ab}=-t\,\frac{\kappa A_{\perp}}{8\pi}$ (50)

provided the coefficients of Taylor series satisfies a particular condition
(see Eq. (B.9) of Appendix B). Here $\kappa$ is an average over the horizon of
the surface gravity given by

$\kappa=\int_{u=0}\frac{\sqrt{\sigma}d^{2}x}{A_{\perp}}\lambda_{0}(v,x_{\perp})$
(51)

This shows that our results will hold for a wide class of infinitesimal
transformations near a null surface provided the condition in Eq. (B.9) of
Appendix B holds. The physical meaning of this condition is unclear and
deserves further analysis.

##  6 Discussion

We provided an explicit example which throws more light on the issues raised
in Sec. 1 related to observer dependence of thermal phenomenon. The central
idea is: (i) to introduce a freely falling coordinate system ($X^{a}$) near
the horizon and a more standard coordinate system ($x^{a}$) appropriate to
observers who attribute thermality to the horizon; (ii) determine the
infinitesimal coordinate transformations connecting these two coordinate
systems and thereby determine the vector field $q^{a}=x^{a}-X^{a}$ and (iii)
study the integral over a codimension-2 cross-section of the horizon of the
Noether potential $J^{ab}$ appropriate for $q^{a}$. We found that one can
determine the heat content $H_{\rm sur}=TS=E-F$ of the horizon in terms of
these quantities and also relate it to the surface term of the gravitational
action. The Christoffel symbols vanish in the freely falling frame but are
non-zero in the static coordinates natural to the horizon. This difference is
closely related to the degrees of freedom contributing to the heat content and
entropy of the horizon and remarkably enough, can be captured just from
studying the infinitesimal version of the coordinate transformation.

These results are easy to interpret and understand in the case of flat
spacetime where the freely falling coordinates are globally defined static
coordinates. We showed that the results also generalize, in a straight forward
manner, to the standard spherically symmetric static horizons including
Schwarzschild, de Sitter spacetimes etc. One possible direction of further
work will be to study the infinitesimal coordinate transformation near a
general null surface as indicated at the end of last section.

## Acknowledgments

The research of one of the authors (TP) is partially supported by a J.C. Bose
research grant of DST, India.

##  Appendix A  Static, spherically symmetric spacetimes

We express the Kruskal-like metric using the coordinates ($T,X$) which have
dimensions of length as:

$\displaystyle ds^{2}=f(r)e^{-2\kappa
r_{*}}(-dT^{2}+dX^{2})+r^{2}d\Omega^{2}~{},$ (A.1)

with the transformations:

$T=\frac{e^{\kappa r_{*}}}{\kappa}\sinh(\kappa t);\,\,\
X+\frac{1}{\kappa}=\frac{e^{\kappa r_{*}}}{\kappa}\cosh(\kappa t)~{}.$ (A.2)

Expanding Eq. (A.2) in a Taylor series in $\kappa$ and keeping the terms upto
${\cal{O}}(\kappa)$, we find

$T=t+\kappa r_{*}t;\,\,\,\ X=r_{*}+\frac{1}{2}\kappa t^{2}+\frac{1}{2}\kappa
r_{*}^{2}~{}.$ (A.3)

The inverse transformations are

$t=T-\kappa XT;\,\,\,\ r_{*}=X-\frac{1}{2}\kappa T^{2}-\frac{1}{2}\kappa
X^{2}~{}.$ (A.4)

So the components of the generator $q^{a}\equiv x^{a}-X^{a}$ for this
transformation are

$q^{T}=-\kappa XT;\,\,\,\ q^{X}=-\frac{1}{2}\kappa T^{2}-\frac{1}{2}\kappa
X^{2}~{}.$ (A.5)

For the metric in (A.1), the essential component of the Noether potential,
using (A.5), is found to be:

$\displaystyle J^{TX}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\Big{[}\frac{2e^{2\kappa r_{*}}}{f(r)}\kappa
T+\frac{e^{4\kappa r_{*}}}{f^{2}(r)}\Big{\\{}\kappa
XT\partial_{X}\Big{(}f(r)e^{-2\kappa r_{*}}\Big{)}$ (A.6) $\displaystyle+$
$\displaystyle\Big{(}\frac{1}{2}\kappa T^{2}+\frac{1}{2}\kappa
X^{2}\Big{)}\partial_{T}\Big{(}f(r)e^{-2\kappa
r_{*}}\Big{)}\Big{\\}}\Big{]}~{},$

and the unit normals are

$n_{a}=\Big{(}0,\sqrt{f(r)}e^{-\kappa r_{*}},0,0\Big{)};\,\,\,\
m_{a}=\Big{(}\sqrt{f(r)}e^{-\kappa r_{*}},0,0,0\Big{)}~{}.$ (A.7)

Therefore,

$\displaystyle d^{2}\Sigma_{TX}=-d^{2}x\sqrt{\sigma}\frac{1}{2}f(r)e^{-2\kappa
r_{*}}~{},$ (A.8)

and so

$\displaystyle d^{2}\Sigma_{TX}J^{TX}$ $\displaystyle=$
$\displaystyle-d^{2}x\sqrt{\sigma}\frac{\kappa
T}{16\pi}-\frac{d^{2}x\sqrt{\sigma}}{32\pi}\frac{e^{2\kappa
r_{*}}}{f(r)}\Big{[}\kappa XT\partial_{X}\Big{(}f(r)e^{-2\kappa r_{*}}\Big{)}$
(A.9) $\displaystyle+$ $\displaystyle\Big{(}\frac{1}{2}\kappa
T^{2}+\frac{1}{2}\kappa X^{2}\Big{)}\partial_{T}\Big{(}f(r)e^{-2\kappa
r_{*}}\Big{)}\Big{]}~{}.$

Now in the following we shall show that near the horizon ${\cal{H}}$, the
second and the last term in Eq. (A.9) are of the order $\kappa^{2}$ and
$\kappa^{3}$, respectively. The second term can be expressed as

$\displaystyle\frac{e^{2\kappa r_{*}}}{f(r)}\Big{(}\kappa
XT\Big{)}\partial_{X}\Big{(}f(r)e^{-2\kappa
r_{*}}\Big{)}=\Big{(}\frac{\partial_{X}f(r)}{f(r)}-2\kappa\partial_{X}r_{*}\Big{)}\kappa
TX~{}.$ (A.10)

Since $X=X(r_{*},t)$,

$\displaystyle\frac{\partial_{X}f(r)}{f(r)}$ $\displaystyle=$
$\displaystyle\frac{1}{f(r)}\Big{[}\frac{\partial r^{*}}{\partial
X}\frac{\partial}{\partial r_{*}}+\frac{\partial t}{\partial
X}\frac{\partial}{\partial t}\Big{]}f(r)$ (A.11) $\displaystyle=$
$\displaystyle\frac{1}{f(r)}\frac{\partial r^{*}}{\partial X}\frac{\partial
f(r)}{\partial r}\frac{\partial r}{\partial r_{*}}$ $\displaystyle=$
$\displaystyle f^{\prime}(r)\Big{(}1-\frac{1}{2}\kappa
X+{\cal{O}}(\kappa^{2})\Big{)}~{},$

where in the last step (A.4) has been used. So, near the horizon

$\displaystyle\frac{\partial_{X}f(r)}{f(r)}\kappa
TX|_{\cal{H}}={\cal{O}}(\kappa^{2})~{}.$ (A.12)

Similarly, $2\kappa(\partial_{X}r_{*})(\kappa TX)={\cal{O}}(\kappa^{2})$.
Therefore the second term in (A.9), near the horizon, is of the order
$\kappa^{2}$. A similar analysis shows that the third term in Eq. (A.9) is of
the order $\kappa^{3}$. So both the terms in (A.9) can be ignored giving:

$\displaystyle
d^{2}\Sigma_{TX}J^{TX}|_{\cal{H}}=-d^{2}x\sqrt{\sigma}\frac{\kappa
T}{16\pi}=-d^{2}x\sqrt{\sigma}\frac{\kappa t}{16\pi}~{},$ (A.13)

upto linear order in $\kappa$. This has been used in computing the integration
in Eq. (46).

##  Appendix B A more general spacetime

For the metric Eq. (47) we find that:

$d^{2}\Sigma^{uv}=\frac{\sqrt{\sigma}}{\Omega}d^{2}x$ (B.1)

and

$\displaystyle
J_{uv}=-\frac{\Omega}{2}\Big{[}A_{2}-\partial_{v}B_{1}+2uA_{3}-u\partial_{v}B_{2}+{\cal{O}}(u^{2})\Big{]}$
$\displaystyle-\frac{1}{2}\Big{[}(A_{1}+uA_{2}+{\cal{O}}(u^{2}))\partial_{u}\Omega-(B_{1}+uB_{2}+{\cal{O}}(u^{2}))\partial_{v}\Omega\Big{]}~{}.$
(B.2)

Therefore,

$\displaystyle
d^{2}\Sigma^{uv}J_{uv}=-\frac{\sqrt{\sigma}}{2}d^{2}x\Big{[}A_{2}-\partial_{v}B_{1}+2uA_{3}-u\partial_{v}B_{2}+{\cal{O}}(u^{2})\Big{]}$
$\displaystyle-\frac{\sqrt{\sigma}}{2}d^{2}x\Big{[}(A_{1}+uA_{2}+{\cal{O}}(u^{2}))\partial_{u}\ln\Omega-(B_{1}+uB_{2}+{\cal{O}}(u^{2}))\partial_{v}\ln\Omega\Big{]}~{}.$
(B.3)

In the following we will show that $\partial_{v}\ln\Omega$ is related to the
non-affinity parameter corresponding to the null surface $u=$ constant. Since
$u=0$ is the null surface, the normal to it is a null vector. The explicit
expression for the null vector $l^{a}$ in a ($u,v,x_{\perp}$) coordinate
system is:

$\displaystyle l^{a}=(0,1,0,0);\,\,\,\ l_{a}=(-\frac{\Omega}{2},0,0,0)~{}.$
(B.4)

The non-affinity parameter $\lambda$ is defined by the relation
$l^{a}\nabla_{a}l_{b}=\lambda l_{b}$. This leads to
$\lambda(u,v,x_{\perp})=\partial_{v}\ln\Omega$ so that:

$\displaystyle\ln\Omega=\int\lambda(u,v,x_{\perp})dv~{}.$ (B.5)

Expressing since $\lambda(u,v,x_{\perp})$ in terms of the expansion in Eq.
(49) we get:

$\displaystyle\partial_{u}\ln\Omega=\int\partial_{u}\lambda(u,v,x_{\perp})dv=D_{1}(v,x_{\perp})+uD_{2}(v,x_{\perp})+\dots~{},$
(B.6)

where $D_{1}(v,x_{\perp})=\int C_{1}(v,x_{\perp})dv$, etc. Therefore (B.3)
leads to

$\displaystyle
d\Sigma^{uv}J_{uv}=-\frac{1}{2}d^{2}x\Big{[}A_{2}-\partial_{v}B_{1}+2uA_{3}-u\partial_{v}B_{2}+{\cal{O}}(u^{2})\Big{]}$
(B.7)
$\displaystyle-\frac{1}{2}d^{2}x\Big{[}(A_{1}+uA_{2}+{\cal{O}}(u^{2}))(D_{1}+uD_{2}+\dots)$
$\displaystyle-$
$\displaystyle(B_{1}+uB_{2}+{\cal{O}}(u^{2}))\lambda\Big{]}~{}.$

In the near null surface limit, the above reduces to

$\displaystyle d^{2}\Sigma^{uv}J_{uv}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{\sigma}}{2}d^{2}x\Big{[}A_{2}-\partial_{v}B_{1}+A_{1}D_{1}-B_{1}\lambda_{0}\Big{]}~{}.$
(B.8)

Now to obtain the required value of the Noether charge given in Eq. (50), we
need to impose the following condition on the coefficients:

$\displaystyle\Big{[}A_{2}-\partial_{v}B_{1}+A_{1}D_{1}-B_{1}\lambda_{0}\Big{]}_{(u=0)}=v|_{(u=0)}\lambda_{0}=2T\lambda_{0}\simeq
2t\lambda_{0}~{}.$ (B.9)

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