# An observer’s perspective of the Unruh and Hawking effects

Satish Ramakrishna<EMAIL_ADDRESS>Department of Physics &
Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Road
Piscataway, NJ 08854-8019

###### Abstract

The Unruh effect is one of the first calculations of what one would see when
transiting between an inertial reference frame with its quantum field vacuum
state and a non-inertial (specifically, uniformly accelerating) reference
frame. The inertial reference frame’s vacuum state would not correspond to the
vacuum state of the non-inertial frame and the observer in that frame would
see radiation, with a corresponding Bose distribution and a temperature
proportional to the acceleration (in natural units).

In this paper, I compute the response of this non-inertial observer to a
single frequency mode in the inertial frame and deduce that, indeed, the
cumulative distribution (over the observer’s proper time) of frequencies
observed by the accelerating observer would be the Bose distribution with a
temperature proportional to the acceleration. The conclusion is that the Unruh
effect (and the related Hawking effect) is generic, in that it would appear
with any incoming incoherent state and the Bose distribution is obtained as a
consequence of the non-inertial frame’s motion, rather than some special
property of the quantum vacuum.

##  I Introduction

The Unruh effect refers to work done in 1976 by W. G. Unruh Unruh  which
examined the effect on particle detectors of an accelerated observer due to a
scalar photon field in its ground state as observed by an inertial observer.
The effect and its derivation from standard principles of quantum field theory
are well-known and are, now, material for textbooks and on-line presentations
about the effects of quantum field theory when applied to accelerated
observers and curved space-times.

As an aside, after the first version of this paper was circulated, I have
become aware of other work that have a similar approachPadma ; Milonni ;
Rosabal . The first two of these related papers start at the same initial
point and veer off in other directions. This paper considers, as a difference
to the other work, the Hawking effect as well as an experiment to verify the
effects in the laboratory.

We quickly describe the usual approachKempf , working in one-dimension, with
an inertial frame (coordinates $(t,x)$) and a frame uniformly accelerated in
the positive-x direction with acceleration $a$. The coordinates in the
accelerated frame are written as $(R,\tau)$. We use natural units i.e.,
$\hbar=1,c=1,G=1$ and the $+---$ metric signature, in all that follows. The
accelerated observer is supposed to start traveling in the negative-x
direction at the speed of light at $t=\tau=-\infty$ and end up traveling in
the positive-x direction at the speed of light at $t=\tau=\infty$, with the
position of the observer at $t=\tau=0$ being $x=e^{aR}$. This path is
displayed in Fig. 1 for two observers traveling with the same constant
acceleration.

Figure 1: Trajectories of two equally spaced (in their frame) accelerated
observers

The usual analysis begins with writing variables for the inertial frame
$(t,x)$ (usually, light-cone coordinates $u=t-x\>,v=t+x$ are used) while
Rindler variables $(\tau,R)$ are used in the accelerated reference frame, with
their light-cone coordinates ${\bar{u}}=\tau-R\>,{\bar{v}}=\tau+R$.

The equations connecting these variables are standard Kempf , i.e.,

$\displaystyle
x=\frac{1}{a}e^{aR}\cosh(a\tau)\>\>\>,\>\>\>t=\frac{1}{a}e^{aR}\sinh(a\tau)\>\>\>,\>\>\>u=-\frac{1}{a}e^{aR}e^{-a\tau}\>\>\>,\>\>\>v=\frac{1}{a}e^{aR}e^{a\tau}\>\>\>\>\>\>$
(1)

The Rindler patch is restricted, in this case, to the domain
$x>=0,-\infty<t<+\infty$ and the lines $t=\pm x$ are horizons that represent
all that the accelerating observer will ever see of the inertial frame. In
Fig.1, I show the constant-$R$ and constant-$\tau$ lines - the first are
hyperbolas, while the second are straight lines passing through the origin.
The horizon, in the Rindler coordinates, is described by
$R=\infty,\tau=\pm\infty$. The metrics in these spaces are

$\displaystyle(ds)^{2}=(dt)^{2}-(dx)^{2}$ (2)

in the inertial, Minkowski space and

$\displaystyle(ds)^{2}=e^{2aR}\bigg{(}(d\tau)^{2}-(dR)^{2}\bigg{)}$ (3)

in the accelerated observer’s coordinates. An accelerated observer standing at
a point $R$ in their space would trace out a hyperbola, whose equation would
be

$\displaystyle t^{2}-x^{2}=\frac{1}{a^{2}}e^{2aR}$ (4)

To quickly summarize, for a fixed $R$ (accelerated observer is stationary in
their own frame), the ‘outside’, i.e., inertial observer would see that
observer’s path as $x\propto\cosh(a\tau),t\propto\sinh(a\tau)$. This feature
will be useful for us in Section III.

As an aside, there are many interesting phenomena connected to the structure
of the Rindler coordinate system - one such is analyzed by Bell Bell .

The next step is to write the scalar photon field $\phi$ in the two light-cone
coordinates and establish a Bogolyubov transformation Unruh ; Kempf  that
relates the annihilation and creation operators in the two representations.
That immediately implies that the number operator in the mode-expansion used
by the accelerated observer will have a non-zero expectation value in the
vacuum state of the inertial observer. This expectation turns out, for every
frequency $\omega$ in the accelerated frame’s mode-expansion to have the
functional form $\sim\frac{1}{e^{\frac{2\pi\omega}{a}}-1}$. This thermal
radiation is seen by the accelerated observer (temperature $\frac{a}{2\pi}$),
despite the quantum field being in the vacuum state of the inertial observer.

##  II Motivation for an observer dependent approach

The current work was motivated by trying to directly consider the effect that
would be observed by the detector carried by the observer who is traveling
along a trajectory described by $R=0$. In order to understand what they are
seeing, it is useful to think of what exactly is a fluctuation the observer
could see and measure.

###  II.1 What is a vacuum fluctuation

It is a safe statement to say that a physical field fluctuation, i.e., not a
vacuum fluctuation, is one that persists (at least through its antecedents or
descendants) from $t=-\infty$ to $t=\infty$. In addition, if the expectation
of the number operator were calculated in the field state at a time-slice $t$,
one should get a non-zero answer.

Conversely, for a vacuum fluctuation, one needs both of these need to be
violated. The particle, or its antecedents and descendants should not persist
between $t=-\infty$ and $t=\infty$ and at any time slice in that reference
frame, the particle number should be non-zero.

As an illustration, a fluctuation that would qualify as a vacuum fluctuation
for the inertial observer and as physical fluctuations for the accelerated
observer is shown as a Feynman diagram in Fig. 2.

Figure 2: Dotted line - vacuum fluctuation in the inertial frame

Clearly, the vacuum fluctuation in the inertial frame meets the criteria to be
a physical field fluctuation in the accelerated frame. Indeed, if one of the
hyperbolic paths were that for the observer, the path of the field fluctuation
would terminate at the detector and such a fluctuation would be observed.
Depending upon the size of the vacuum fluctuation and exactly where it
intersects the path of the accelerated observer, the observer would see
different frequencies, since there is relativistic red-/blue-shift given the
traveling speed of the observer. From the distribution of the energies of
these vacuum fluctuations and their consequent persistence time, along with
the appropriate red-/blue- shift, we could compute carefully the distribution
of energies seen by the accelerated observer.

There is, however, a simple calculation one could do for one mode, with a
fixed frequency. We start as follows : the speed of the observers can be
computed from Equation (1), viz. $v=\tanh(a\tau)$ in the proper time of the
observer - interestingly, this is independent of $R$ which allows the
calculation to proceed easily.

Let’s now look at the effect of sending a single physical mode, frequency
$\omega_{0}$ from somewhere in the inertial frame, towards the accelerated
observer.

###  II.2 The effect on the accelerated observer

The accelerated observer would see the frequency of the mode as
$\omega_{obs}=\omega_{0}\sqrt{\frac{1-v}{1+v}}$, which is the usual
relativistic red-shift formula. $v$ is the speed of the observer away from or
towards the mode’s propagation direction (we are working in 1-dimension here).
In this case, since $v=\tanh(a\tau)$, we deduce that
$\omega_{obs}=\omega_{0}e^{-a\tau}$.

If the accelerated observer were to count wavefronts, i.e., the accumulated
phase over some period of proper time, one already has an answer for this
quantity from the inertial frame. In fact, this quantity should be invariant
regardless of which frame it is calculated in. It should be, over a time
interval $t_{0}$ to $t$, equal to $\omega_{0}\times(t-t_{0})$. This can be
written in the coordinates of the accelerating observer as
$\omega_{0}\int_{\tau_{0}}^{\tau}ds\>e^{-as}$. Hence, the single (positive
frequency) mode would be received as

$\displaystyle N_{0}e^{-i\omega_{0}\int_{\tau_{0}}^{\tau}ds\>e^{-as}}$ (5)

where we need to apply a UV cut-off for how closely the accelerated observer
can get to the horizon. $N_{0}$ is a normalization factor and is usually
chosen to be $\frac{1}{\sqrt{\omega_{0}}}$.

Another way of stating the above is that the waves in the inertial frame are
functions of $t\pm x$, which, when written in the accelerated observer’s
coordinates has the form shown.

This can be simplified to (where ${\cal C}$ is a phase).

$\displaystyle{\cal C}N_{0}e^{i\frac{\omega_{0}}{a}e^{-a\tau}}$ (6)

which is a chirp-type signal.

The next, and novel step, is to compute the Fourier transform of this
frequency-modulated signalnote . We can compute the positive (${\cal
F}(\Omega)$) and negative (${\cal G}(-\Omega)$) frequency components as below
(we treat $\Omega$ as positive),

$\displaystyle{\cal F}(\Omega)={\cal
C}N_{0}\int_{-\infty}^{\infty}d\tau\frac{1}{N_{\Omega}}e^{i\Omega\tau}\>e^{i\frac{\omega_{0}}{a}e^{-a\tau}}$
$\displaystyle{\cal G}(-\Omega)={\cal
C}N_{0}\int_{-\infty}^{\infty}d\tau\frac{1}{N_{\Omega}}e^{-i\Omega\tau}\>e^{i\frac{\omega_{0}}{a}e^{-a\tau}}$
(7)

Again, here, $N_{\Omega}$ is a normalization factor for the real-space
function and is chosen to be $\frac{1}{\sqrt{\Omega}}$. The integrals can be
computed using standard methodsKempf  -

$\displaystyle{\cal F}(\Omega)={\cal
C}\frac{N_{0}}{N_{\Omega}}\>\frac{1}{a}\>e^{\frac{\pi\Omega}{2a}}\>e^{i\frac{\Omega}{a}\ln(\frac{\omega_{0}}{a})}\Gamma(\frac{-i\Omega}{a})$
$\displaystyle{\cal G}(-\Omega)={\cal
C}\frac{N_{0}}{N_{\Omega}}\>\frac{1}{a}\>e^{\frac{-\pi\Omega}{2a}}\>e^{-i\frac{\Omega}{a}\ln(\frac{\omega_{0}}{a})}\Gamma(\frac{i\Omega}{a})$
(8)

so that

$\displaystyle|{\cal
F}(\Omega)|^{2}=\frac{\frac{2\pi}{a\omega_{0}}}{e^{\frac{-2\pi\Omega}{a}}-1}$
$\displaystyle|{\cal
G}(-\Omega)|^{2}=\frac{\frac{2\pi}{a\omega_{0}}}{e^{\frac{2\pi\Omega}{a}}-1}$
(9)

which is obtained for just one emitted mode and represents the central result
of the paper. One does not need to deal with a vacuum state with a myriad of
modes to see a thermal distribution - this is just the result of the
particular frequency modulation one gets with an accelerated observer. With
one mode, the detector would need to be analyzed after its entire journey is
complete to get a complete picture of the Bose distribution. The result is
generic and is related precisely to the existence of the horizon and would be
observed, as discussed below, to any incoherent set of waves sent towards the
accelerated observer.

To reiterate, a pure emitted mode can produce a thermal distribution of
received radiation purely due to kinematic transformation to the accelerated
observer’s reference frame. This has implications for other outstanding
problems in physics, including the black-hole information problemHawking2 .

###  II.3 Extension to multiple modes

If the inertial frame sent in a collection of incoherent modes with a
distribution $f(\omega)$ towards the accelerated observer, the consequent
radiation received would be just a sum over the respective $|{\cal F}|^{2}$ or
$|{\cal G}|^{2}$ for each mode, so we recover exactly the same distribution
for each, times a pre-factor. This is exactly the situation for vacuum
fluctuations, where the individual modes are uncorrelated with each other,
which is why one obtains the usual Unruh result.

If a coherent set of waves were sent, there are interference terms, related to
the phase-factors in Equation (8), which complicate the picture and prevent
the result from being a pure sum over the Bose distribution, except in some
limitsSatish2 .

##  III Extension to Hawking’s calculation

Hawking radiation is obtained through a similar calculation, with a non-flat
event-horizonKempf ; Hawking1 . The same considerations apply here, i.e., it
is generic and doesn’t apply to just the vacuum state, but to any general
state. Let us study this in detail.

We study the situation in two coordinate systems - the familiar Schwarschild
and Kruskal-Szekeres (KS) and write the regular Schwarzschild metric (with
$r_{S}=2M$)

$\displaystyle(ds)^{2}=(1-\frac{r_{S}}{r})(dt)^{2}-\frac{(dr)^{2}}{(1-\frac{r_{S}}{r})}-r^{2}(d\Omega)^{2}$
(10)

and the transition to the KS metric is achieved with

$\displaystyle
FOR:\>\>\>r>=r_{S}\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>$
$\displaystyle
T=\sqrt{\bigg{(}\frac{r}{r_{S}}-1\bigg{)}}\>\>e^{\frac{r}{2r_{S}}}\>\>\sinh\frac{t}{2r_{S}}$
$\displaystyle
X=\sqrt{\bigg{(}\frac{r}{r_{S}}-1\bigg{)}}\>\>e^{\frac{r}{2r_{S}}}\>\>\cosh\frac{t}{2r_{S}}$
$\displaystyle
FOR:\>\>\>r<r_{S}\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>$
$\displaystyle
T=\sqrt{\bigg{(}1-\frac{r}{r_{S}}\bigg{)}}\>\>e^{\frac{r}{2r_{S}}}\>\>\cosh\frac{t}{2r_{S}}$
$\displaystyle
X=\sqrt{\bigg{(}1-\frac{r}{r_{S}}\bigg{)}}\>\>e^{\frac{r}{2r_{S}}}\>\>\sinh\frac{t}{2r_{S}}$
(11)

The Schwarzschild metric in these coordinates is

$\displaystyle(ds)^{2}=\frac{32M^{3}}{r}e^{\frac{r}{r_{S}}}\>\bigg{(}(dT)^{2}-(dX)^{2}\bigg{)}+r^{2}(d\Omega)^{2}$
(12)

We see that for fixed $r$, the Hawking effect computation is analogous to that
for the Unruh effect, except for a visible difference in Equation (11), where
for $r>r_{S}$, there is a pre-factor $\sqrt{r/r_{S}-1}$, which is sub-dominant
compared to the exponential in $r$ for large $r$. There is also a difference
in how the $r$ space is mapped - the line $T=X$ is now $r=r_{S}$, not
$r=-\infty$ as in the Unruh example. However, in the limit of large $r$
($r\rightarrow\infty$), this is irrelevant.

Explicitly: In the Unruh effect, the space of $(t,x)$ is Minkowski, while
$(\tau,R)$ is the Rindler space. A pure mode in Minkowski leads to a chirp in
Rindler (as was shown in Section II), hence produces a thermal distribution in
the Rindler space, which observers infinitely far away would realize was a
consequence of their acceleration $a$. The temperature they would compute from
the distribution would be $\frac{a}{2\pi}$. Note that to do this, the
observers would need to collect particles and deduce the temperature from the
distribution obtained cumulatively when they got infinitely far away.

For the Hawking effect, $(T,X)$ is Minkowski for fixed $r$. And $(t,r)$, the
real space, is Schwarzschild, which appears related to the $(T,X)$ space by a
Rindler-type transformation, an approximation that gets better for large $r$.
Identically, a pure mode in Minkowski produces a chirp in real space (quasi-
Rindler for large $r$), which leads to a thermal distribution in this Rindler
space (where we look in the large $r$ region). And, to re-formulate the
argument in Section II, $X\propto\cosh({\cal A}t),T\propto\sinh({\cal A}t)$,
where ${\cal A}=\frac{1}{2r_{S}}$. Hence, in the $(t,r)$ space, which is the
coordinate system used by observers in real space, the quantity corresponding
to the acceleration (as in the calculation for the Unruh effect) would be
${\cal A}=\frac{1}{2r_{S}}$ (again, for large $r$). This leads to a
temperature that observers at infinity would observe, of $\frac{1}{4\pi
r_{S}}=\frac{1}{8\pi M}$, i.e., the usual Hawking result.

##  IV Discussion

The key points in the argument that the chirp observed by an accelerated
observer, which is a frequency set from $0$ to $\infty$ in spread and of the
particular functional form obtained in Section II, is required for the Bose
distribution to be realized. Some modifications to the form of this time-
domain behavior have been investigated and they lead to the Bose distribution
at leading order at least. Investigations are ongoingSatish2 , though since
effect is essentially a consequence of geometry, it is not surprising that it
is very stable.

This discussion has implications for how to experimentally verify the Unruh
and Hawking effect. One does not need to locate a suitable black-hole or look
for vacuum fluctuations. One can simply shoot a pure mode at an accelerated
detector, as in Fig . 3 and verify that the spectrum obtained is the Bose
distribution. A schematic of the experiment is indicated - positively charged
multi-level molecules (set up to be in a particular intermediate energy level)
are shot at high speed (close to the velocity of light) at a source of a
single frequency. An electric field decelerates, then accelerates the
molecules in the opposite direction to a high speed. Once an equilibrium
situation is set up, the field (and frequency source) is turned off - the
molecules drift at constant speed towards either end of the apparatus where
they are classified by the energy they deliver to the targets at the opposite
ends (deduced from spikes in electron current flow as they de-ionize). One
then obtains the statistics of what wavelength was absorbed at each point on
the path - essentially one has $\omega_{obs}$ at several points on the path. A
discrete Fourier transform can now be calculated and compared to the expected
distribution. In this situation, the molecules have a constant acceleration in
their rest frame owing to the transformation properties of the longitudinal
component (along the direction of travel) of the electric fieldJackson . We
would essentially be verifying the relativistic red/blue-shift phenomenon and
its consequent Bose spectrum, thereby verifying the Unruh and Hawking effects.

Figure 3: An experimental schematic

##  V Conclusions

We have shown explicitly that Unruh (and Hawking) radiation is generic and is
obtained as a general feature of a non-inertial, uniformly accelerated
reference frame. In addition, the thermal radiation could well emerge from
pure modes in a different reference frame.

The experiments devoted to checking for the effects could well use a single
mode, replace the ensemble average by an ergodic average and verify these
rather straightforward results.

##  VI Data Disclosure

Data sharing is not applicable to this article as no new data were created or
analyzed in this study. No conflicts exist.

##  VII Acknowledgments

Very detailed discussions and helpful criticism from Scott Thomas were
especially useful and I thank him for this and other rather educational
excursions. Dr. Achim Kempf’s rather clear discussions were inspiring. The
hospitality of the Rutgers Physics Department and the NHETC is also gratefully
acknowledged.

## References

  * (1) Unruh, W. G. Notes on black-hole evaporation Phys. Rev. D 14, 4 (1976)
  * (2) A rather good introduction is by Achim Kempf, available at https://uwaterloo.ca/physics-of-information-lab/teaching/quantum-field-theory-cosmology-amath872phys785-w2022
  * (3) Bell, J. S.  How to teach special relativity Progress in Scientific culture 1(2) (1976), pp. 1–13. Reprinted in J. S. Bell: Speakable and unspeakable in quantum mechanics (Cambridge University Press, 1987), chapter 9, pp. 67–80.
  * (4) Srinivasan, K. , Sriramkumar, L. and Padmanabhan, T. Plane waves viewed from an accelerated frame: Quantum physics in a classical setting Phys. Rev. D 56, 6692-6694 (1997) & Padmanabhan, T. Gravity and the Thermodynamics of Horizons, gr-qc/0311036
  * (5) Alsing, P., Milonni, P. Simplified derivation of the Hawking-Unruh temperature for an accelerated observer in vacuum Am.J.Phys. 72 (2004) 1524-1529
  * (6) Rosabal, J. A. New Perspective On The Unruh Effect Phys. Rev. D 98, 056015 (2018)
  * (7) Ramakrishna, Satish in preparation
  * (8) Hawking, S. W. Particle Creation by Black Holes Commun. Math. Phys. 43 (1975) 199; Erratum: Commun. Math. Phys. 46 (1976) 206.
  * (9) The $a\rightarrow 0$ limit has to be taken right at the start, when we calculate $\omega_{obs}$, since for $a=0$, there are no horizons - the phenomenon is not continuously connected to the non-zero $a$ case.
  * (10) Hawking, S. W. The Information Paradox for Black Holes arXiv:1509.01147 [hep-th]
  * (11) Jackson, J. D. Classical Electrodynamics 3rd edition, Wiley ISBN: 978-0-471-30932-1


significantly dimmer. To give a concrete example, covering the entire area
between 1–1.5 $R_{500}$ of NGC 5846444for the $R_{500}$ value in Panagoulia et
al. (2014) would require no less than 423 observations with the
$Athena$/X-IFU.

Two future missions currently under study promise to address this issue by
offering capabilities that are complementary to those of $Athena$: the Hot
Universe Baryon Surveyor (HUBS) (Cui et al., 2020), which is a project of the
Chinese National Space Administration (CNSA), and the JAXA-led Super DIOS
("Diffuse Intergalactic Oxygen Surveyor", Sato et al. 2020) mission. Both have
expected launch dates in the 2030s. By prioritising a shorter mirror focal
length (which implies a smaller on-axis effective area) but using larger
pixels and covering a larger field of view (of order $\sim$1 deg2), these
future satellites will be able to survey a wider area of the sky more
efficiently while maintaining a high spectral resolution ($\Delta E\lesssim 2$
eV). Due to the shorter focal length and their planned deployment in low-Earth
orbit which both help to minimise the detector background, these missions are
optimised to enable detailed high-resolution spectroscopy of the faint
outskirts of nearby groups and clusters of galaxies, among several other
science topics. In Fig. 15, we summarise the capabilities and advantages of
planned X-ray IFUs over the next $\sim$decade. Remarkable to note is that the
Super DIOS concept plans to employ nearly an order of magnitude more TES
pixels than $Athena$ with developments of a new TES readout system, enabling a
10–15 arcsec resolution over 0.5-1 deg2; while HUBS has a similar number of
pixels as $Athena$ (and therefore a poorer spatial resolution of $\sim$1
arcmin over a 1 deg2 FoV), it carries a central 12 by 12 arcmin sub-array with
an energy resolution of 0.6 eV, that will be the first detector to reach a
resolving power of $E/\Delta E>1000$ around the Fe-L complex.

###  4.5 Arcus

Arcus is a proposed NASA Medium Explorer mission that aims to significantly
improve our spectroscopic capabilities using soft X-ray gratings. With
spectral resolution $R>$2500 between 12-50 $\mathring{A}$, the Arcus proposal
possesses the ability to resolve the absorption of a diverse range of metal
species in the extended halos of galaxies, groups, and clusters along sight
lines toward distant quasars (Smith et al., 2016). X-ray grating spectroscopy
provides a critical complementary probe to planned micro-calorimeters by
having $\sim 10\times$ greater resolution and probing the diffuse hot gas
comprising the majority of missing metals and baryons (Cen & Fang, 2006;
Wijers et al., 2019). Arcus is designed to achieve a 10-fold increase in
sensitivity (i.e. square root of effective area times resolution) over
existing grating spectrometers on Chandra and XMM-Newton.

According to cosmological simulations (Wijers et al., 2020), the IGrM has the
greatest potential to show a rich variety of ion species. O VIII should be the
most detected species within group virial radii, while O VII should be
detected in lower mass groups at $M_{500}\la 10^{13.5}M_{\odot}$. In its main
mission, Arcus should detect the $T=10^{5.4-6.8}$ K IGrM in at least 10 group
haloes at a 3 m$\mathring{A}$ equivalent width threshold (at $5\sigma$) for O
VII and O VIII. Interestingly, the Fe XVII should be detectable within
$R_{500}$ for groups, probing gas up to $\approx 10^{7}$ K for at least 5
group haloes. Integrating longer on the brightest X-ray quasars opens up the
possibility to detect more species using a 1 or 2 m$\mathring{A}$ detection
threshold, including Ne IX and Ne X. C V and C VI (and maybe N VII) may also
be detectable in the group outskirts for deeper observations probing gas down
to $10^{5}$ K. Sensitive UV absorption line spectroscopy brought about by the
Cosmic Origins Spectrograph on Hubble discovered that the objects with the
richest and most diverse set of species probing gaseous halos are star-
forming, $L^{*}$ galaxies (Werk et al., 2014). It is expected that groups will
be the analogous richly detected, multi-species objects to be probed via X-ray
absorption line spectroscopy.

###  4.6 Lynx

The $Lynx$ mission concept (Özel, 2018) proposed to the US 2020 Decadal Survey
is designed to carry a new generation of X-ray telescopes enabling a sub-
arcsecond resolution over a 22${}^{\prime}\times$22′ field of view and an
effective area of 2 m2 at 1 keV. It is designed to be equipped with three
complementary instruments. An active pixel array (the high-definition X-ray
imager, HDXI) that would provide wide-field CCD-like spectral imaging, but
with 0.3′′ pixels to take advantage of the exquisite spatial resolution. The
$Lynx$ X-ray Microcalorimeter (LXM) which would bring 3 eV spectral resolution
on 1′′ spatial scales over a 5′ field of view (0.3 eV in a ultra-high
resolution sub-array). Finally an X-ray grating spectrometer (XRG) with an
effective area of 4000 cm2 and resolving power greater than 5000 to exploit
the better spectral resolution of gratings for point sources in the soft
energy band. The key improvements $Lynx$ will bring if approved concern the
ability to resolve the metal abundance structure of the gas near and outside
the virial radius of groups at low redshift by combining a study of emission
and absorption against background AGNs. $Lynx$ will also push the realm of
enrichment studies in the IGrM to the z=2-3 range where strong trends are
expected. $Lynx$ is specifically targeting observations of high-redshift
groups down to a mass scale of $M_{500}=2\times 10^{13}$ $M_{\odot}$ at z $>3$
(https://www.lynxobservatory.com/report, Gaskin et al., 2019).

##  5 Concluding Remarks

Throughout this review, we have seen how crucial the case of galaxy groups is
in order to complete our understanding of the journey of metals – from stars
and supernovae to the largest scales of our Universe. Groups are in fact a
unique piece of the puzzle to relate chemical enrichment at (sub-) galactic
scales and at cluster scales (Sect. 2.4), making IGrM abundance studies
particularly valuable in this respect. As we have also discussed however, such
studies are still in their pathfinder steps, and enrichment in groups remains
much less explored (hence, understood) than in clusters. The reasons are
diverse, and include notably on the observational side: (i) the intrinsic
faintness of the IGrM with respect to the ICM, making observational studies
more demanding in terms of exposure time; (ii) the lack of well defined
samples of groups in the X-ray band, marking again a stark contrast with
respect to clusters; (iii) our lack of spectral knowledge of the Fe-L complex
(ruling almost all the X-ray emission of the IGrM) and of the likely multi-
temperature structure of the gas (Sect. 2.1). As we have seen in Sect. 4, in
the next years each of the above limitations will be tackled by higher quality
data in terms of sizes of the sample of groups to be targeted and available
high-throughput and high-resolution spectra. Those data will be coupled with
an improved theoretical understanding of the spectral modeling. It is also
essential to continue the efforts to improve the accuracy of the theoretical
SN yields and the measurements of SN rates as a function of the cosmic time.
On the simulation side, the main challenge will be to reproduce key stellar
and gaseous properties on both galactic and Mpc scales, simultaneously
including micro-scale physics and the large-scale cosmological context (Sect.
3.1 and 3.2). With the ongoing exponential high-performance computing
advancements, we are getting closer to a quantum leap in terms of a single
high-resolution hydrodynamical cosmological simulation reaching this goal,
thus providing more detailed predictions for the regime of galaxy groups. It
is a reasonable bet to forecast the coming era as a golden age of maturity for
the study of metal abundances in galaxy groups.

The review is based on public data and/or published papers. The results
collected from literature and used to generate some of the figures in this
review can be found at this link https://doi.org/10.5281/zenodo.5011831

“This research received no external funding” F.G.: lead author, Sect. 1, 2.2;
A.S: Sect. 2.1, 2.3.2, 2.3.3; F.M.: Sect. 2.3.1, 2.5; V.B.: Sect. 3.2; M.G.:
Sect. 3.1; K.S and K.M: Sect. 2.4 with contribution from F.G. F.G., A.S., F.M.
and K.S contributed to Sect. 4. All authors contributed to Sect.5. All authors
have read and agreed to the published version of the manuscript.

###### Acknowledgements.

We thank the two anonymous referees for helpful reports which improved the
quality of this review. We would like to thank Ben Oppenheimer for providing
the section of the future mission Arcus and for useful discussions. We would
like to thank David Buote, Silvano Molendi, Simona Ghizzardi, Fabrizio
Brighenti, William Mathews for comments and suggestions. F.G acknowledges
financial contribution from INAF "Call per interventi aggiuntivi a sostegno
della ricerca di main stream di INAF". AS is supported by the Women In Science
Excel (WISE) programme of the NWO, and acknowledges the World Premier Research
Center Initiative (WPI) and the Kavli IPMU for the continued hospitality. SRON
Netherlands Institute for Space Research is supported financially by NWO. MG
acknowledges partial support by NASA Chandra GO8-19104X/GO9-20114X and HST
GO-15890.020-A grants. The authors declare no conflict of interest. yes

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