xxxxx, xxxxx, xxxx

Andrew R. Casey

# The impact of the COVID-19 pandemic on academic productivity

A. R. Casey1,2 I. Mandel1,3,4 and P. K. Ray5 1School of Physics & Astronomy,
Monash University, Clayton 3800, Victoria, Australia
2ARC Centre of Excellence for Astrophysics in Three Dimensions (ASTRO-3D),
Australia
3OzGrav, Australian Research Council Centre of Excellence for Gravitational
Wave Discovery, Australia
4Institute of Gravitational Wave Astronomy and School of Physics and
Astronomy, University of Birmingham, Birmingham, B15 2TT, United Kingdom
5Department of Mathematics, Imperial College London, London, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

‘Publish or perish’ is an expression describing the pressure on academics to
consistently publish research to ensure a successful career in academia. With
a global pandemic that has changed the world, how has it changed academic
productivity? Here we show that academics are posting just as many
publications on the arXiv pre-print server as if there were no pandemic:
168,630 were posted in 2020, a +12.6% change from 2019 and $+1.4\sigma$
deviation above the predicted 162,577 $\pm$ 4,393. However, some immediate
impacts are visible in individual research fields. Conference cancellations
have led to sharp drops in pre-prints, but laboratory closures have had mixed
effects. Only some experimental fields show mild declines in outputs, with
most being consistent on previous years or even increasing above model
expectations. The most significant change is a 50% increase ($+8\sigma$) in
quantitative biology research, all related to the COVID-19 pandemic. Some of
these publications are by biologists using arXiv for the first time, and some
are written by researchers from other fields (e.g., physicists,
mathematicians). While quantitative biology pre-prints have returned to pre-
pandemic levels, 20% of the research in this field is now focussed on the
COVID-19 pandemic, demonstrating a strong shift in research focus.

###### keywords:

COVID, pre-print, academic productivity

Peer-reviewed publications are the most common measure of productivity in
academia. The arXivGinsparg:2011  (https://arxiv.org) is a distribution
service for research publications before they are printed in a journal (i.e.,
a pre-print). A pre-print on arXiv does not ensure that the contents have
passed peer-review, but most material on arXiv eventually goes through peer-
review because it is now standard in many research fields to post to arXiv
either during or after the peer-review process Lariviere:2014 . For this
reason, the number of pre-prints posted to arXiv approximates the number of
peer-reviewed publications written in these fields.

Here we make quantitative comparisons on the number of pre-prints posted to
arXiv before and during the pandemic. While we investigate the impact that the
COVID-19 pandemic has had on different fields of research, we highlight that
we are unable to identify or differentiate between authors and communities
whose productivity has been significantly harmed by the pandemic, and those
who were largely unscathed. The COVID-19 pandemic has impacted the population
in unequal waysNicola:2020 ; Chu:2020 ; Viglione:2020 ; Gewen:2020 ;
IbnMohammed:2021 ; King:2021 . Generational inequality, career stage, personal
circumstances, carer responsibilities, work environments, places of
employment, and many other factors, all significantly contribute to the
disproportionate and unequal impact the pandemic may have had on a scientist’s
capacity to conduct research. While this limited analysis does not address
these issues, it is important to consider that if a community has not yet
demonstrated a significant change in productivity, many researchers have faced
significant challenges and suffered physically, mentally, emotionally, and
professionally.

We retrieved metadata for 1,475,914 pre-prints posted to arXiv between 1 April
2007 and 31 May 2021. The metadata includes the creation date, research
field(s), title, author name(s), abstract, and other miscellaneous
informationClement:2019 . There is an increasing number of pre-prints posted
to arXiv each year in nearly every field (Figure 1). These long-term trends
are relatively predictable from year to year, allowing us to quantify any
change in academic productivity due to the COVID-19 pandemic. We used the
number of publications from January 2015 to December 2019 to predict the
expected number of monthly pre-prints in each field after January 2020. We
modelled the number of monthly pre-prints in each field with a straight line
to capture long-term (linear) trends, and used a Gaussian Process with a
quasi-periodic kernel functionRasmussen:2006 ; Ambikasaran:2014  to capture
seasonal variations (Figure 1). In nearly all research fields the number of
pre-prints posted since January 2020 agree excellently with the model
predictions (Table 1), indicating no collective impact on academic
productivity due to the COVID-19 pandemic. The total number of arXiv pre-
prints in 2020 across all fields (168,630) exceeds our model predictions
($162{,}577\pm 4{,}393$; $+1.4\sigma$).

Figure 1: There has been no significant change in monthly arXiv pre-prints
(black) since the COVID-19 pandemic began. The expected number of pre-prints
from our Gaussian Process model is shown in red, and the red region shows the
uncertainty in that expectation. The dashed light grey line shows predictions
from an autoregressive integrated moving average model.

Border closures and travel restrictions have forced many academic conferences
to be held online-only, or to be cancelled. Hosting conferences online creates
a mix of effects, as forgoing in-person experiences benefits those who could
not otherwise participate due to visa issues or travel costsGuinnessy:2021 .
The immediate impact of cancelling a conference is readily apparent in pre-
print counts in lattice physics (Figure 1; arXiv subject code hep-lat, see
Methods for explanation), where most pre-prints are posted around December
each year as conference proceedings from the International Symposium on
Lattice Field Theory. In 2020 the conference was
cancelledLatticeConferenceWebsite  and no accompanying pre-prints exist.
However, the impact on research from travel restrictions is likely to be much
longer than what is represented by the drop in lattice physics pre-prints.
Discussions at conferences or collaborative visits frequently spark new
research ideas that might lead to a publication many months or years later.

Experimental research projects often require specialised laboratories, or data
to be collected over many years. The pandemic has forced many laboratories to
close or operate with restricted accessPhysicsWorld ; ScienceMag , which could
lead to long-lasting delays in ongoing experiments or immediate drops in
publication rates in part due to difficulty in accessing completed (or nearly
completed) experiment data. There is some evidence of this in pre-print
numbers already, with 20% fewer pre-prints in high energy physics (hep-ex) in
2020 than expected by our model (observed 615; predicted 818 $\pm$ 94). But
declines are not ubiquitous across experimental fields. The closest comparable
field of phenomenology in high energy physics (hep-ph) showed no substantial
decline. Neither did nuclear experimental research (nucl-ex). And throughout
2020, research in condensed matter (cond-mat) saw a $1.8\sigma$ increase above
our model predictions (observed 16,188; predicted 15,325 $\pm$ 492).
Segmenting condensed matter pre-prints by research topic shows that most of
this increase (in cond-mat) was driven by a 30% increase in material science
pre-prints (cond-mat.mtrl-sci).

The field of quantitative biology (q-bio) research showed the largest increase
in pre-prints in 2020 above what is expected by our model. There were 2,790
quantitative biology pre-prints in 2020, 50% above the previous year,
representing a $+8\sigma$ deviation from our model predictions (predicted
1,914 $\pm$ 109, 876 fewer than observed). This increase is explainable by an
increase in COVID-19 related research, as there were 864 quantitative biology
pre-prints in 2020 with pandemic-related terms in their title or abstract (see
Methods), and just 39 in the decade prior. Indeed, in April 2020 nearly 60% of
the arXiv pre-prints in quantitative biology were related to the pandemic
(Figure 2). Pandemic-related pre-prints also appeared in other fields
(computer science, physics, statistics, and quantitative finance, all peaking
around April 2020), but pandemic-related pre-prints constituted fewer than 10%
of the total pre-prints in these fields. As of mid-2021 the number of monthly
pre-prints in quantitative biology has dropped to pre-pandemic levels.
Approximately 20% of current pre-prints in quantitative biology are related to
the ongoing pandemic, representing a strong shift in ongoing research focus.

Figure 2: Pre-prints related to the COVID-19 pandemic peaked in April 2020,
accounting for nearly 60% the quantitative biology (q-bio) pre-prints in that
month. Shown are five other research fields with the highest number of
pandemic-related pre-prints in 2020. Monthly pre-prints are shown on the left,
and the percent of pandemic-related pre-prints in that research field is shown
on the right.

An increase in quantitative biology\dagger\dagger$\dagger$Quantitative biology
is only a sub-field of biology, and most biology pre-prints are posted to
bioRxiv (https://biorxiv.org). research during a global pandemic is
unsurprising. However, the question arises if this increase was driven by
existing researchers in the field, or by authors from other fields entering
quantitative biology due to the pandemic. In 2020 there was a peak in the
number of new authors appearing for the first time in the quantitative biology
literature (Figure 3) while the number of new authors in all other fields
remained steady. The increase in pre-prints (and new authors) is driven by
small (1-4) groups of authors, where many had never posted pre-prints to
quantitative biology before.

Figure 3: The number of new authors appearing in arXiv pre-prints per month
(black) peaked in quantitative biology in 2020. Here we show new authors since
2010, but we used data since 2007 to determine if an author was new. The slow
change in new authors approximates the net number of new authors joining the
field. The number of new authors appearing per number of pre-prints is shown
in light blue, with identical limits on the right-hand y-axis. This highlights
authorship differences between fields (e.g., hep-ex and hep-lat) for large
experiments.

One quarter of the excess pre-prints posted to quantitative biology during
2020 were related to the pandemic and all authors had never appeared in the
quantitative biology literature (234 of the 876 excess pre-prints in q-bio). A
careful examination reveals that about half of these (45%; 107/234) were
written by mathematicians or physicists, with another 26% (61/234) led by
engineers, computer scientists, or economists. These pre-prints tend to focus
on modelling the COVID-19 outbreak using public data sets, rather than
quantitative biology research that requires more expert domain knowledge.
About 3% (8/234) were led by established biologists who have not used the
arXiv before and are now doing so presumably to help ensure that their
COVID-19 research is more widely available. This fraction is likely higher
among pre-prints where at least one author has appeared before in the
quantitative biology literature. This represents a genuine increase in new
researchers using the arXiv, and a boost to making quantitative biology
research more widely accessible.

Table 1: The number of arXiv pre-prints per primary research field in 2020 compared to 2019, and model predictions (see Methods). arXiv | Field of research | Pre-prints | Yearly | Predicted | Significance
---|---|---|---|---|---
code |  | (2020) | change (%) | (2020) |  ($\sigma$)
astro-ph | Astrophysics | 14,893 | $+3.4$ |  14,990 $\pm$ 653 | $-0.1$
cond-mat | Condensed Matter | 16,188 | $+7.0$ |  15,325 $\pm$ 492 | $+1.8$
cs | Computer Science | 54,808 | $+24.3$ |  51,673 $\pm$ 4,047 | $+0.8$
gr-qc | General Relativity and | 3,405 | $+12.4$ |  3,166 $\pm$ 154 | $+1.6$
| Quantum Cosmology |  |  |  |
hep-ex | High Energy Physics | 615 | $-19.8$ |  818 $\pm$ 94 | $-2.2$
| (Experiment) |  |  |  |
hep-lat | High Energy Physics | 394 | $-19.3$ |  475 $\pm$ 170 | $-0.5$
| (Lattice) |  |  |  |
hep-ph | High Energy Physics | 4,586 | $+1.3$ |  4,615 $\pm$ 329 | $-0.1$
| (Phenomenology) |  |  |  |
hep-th | High Energy Physics | 3,869 | $+0.9$ |  4,000 $\pm$ 291 | $-0.4$
| (Theory) |  |  |  |
math | Mathematics | 35,367 | $+5.5$ |  35,362 $\pm$ 1,177 | $+0.0$
math-ph | Mathematical Physics | 1,378 | $-6.3$ |  1,460 $\pm$ 95 | $-0.9$
nlin | Nonlinear Sciences | 885 | $+6.1$ |  822 $\pm$ 63 | $+1.0$
nucl-ex | Nuclear Experiment | 499 | $+2.9$ |  490 $\pm$ 61 | $+0.1$
nucl-th | Nuclear Theory | 1,212 | $-3.2$ |  1,234 $\pm$ 111 | $-0.2$
physics | Physics | 15,241 | $+14.3$ |  14,746 $\pm$ 506 | $+1.0$
q-bio | Quantitative Biology | 2,790 | $+49.0$ |  1,914 $\pm$ 109 | $+8.0$
q-fin | Quantitative Finance | 979 | $+23.1$ |  805 $\pm$ 55 | $+3.2$
quant-ph | Quantum Physics | 6,341 | $+13.4$ |  5,858 $\pm$ 266 | $+1.8$
stat | Statistics | 5,180 | $+20.2$ |  4,824 $\pm$ 490 | $+0.7$
| Total | $\mathbf{168{,}630}$ | $\mathbf{+12.6}$ | $\mathbf{162{,}577\pm 4{,}393}$ | $\mathbf{+1.4}$

The pandemic has also created distinct challenges when interpreting trends in
arXiv data. For example, there was a large spike in papers submitted to the
stat category in June 2020 which one might assume was driven by COVID-related
work given what we have observed in q-bio. However, the number of COVID-
related papers posted in this period is modest, and the key factor appears to
be delayed paper submission deadlines for two large conferences (NeurIPS and
the Joint Statistical Meetings). Both conferences have had submission
deadlines in mid-May in recent years, but in 2020, their deadlines were moved
to early June.

The COVID-19 pandemic appears to have had minimal impact on collective
research productivity to date. In this study we have only focussed on the
quantity of publications as a measure for productivity, and not their quality.
While citations are the most used measure of impact of academic publications,
that metric becomes a more biased statistic when many related pre-prints are
all being posted nearly at the same timeFassin:2021 . Amongst individual
fields the largest decline is explainable by a cancelled conference, and the
few other drops in pre-prints are statistically insignificant. It’s plausible
that many of the pre-prints already posted are research projects that were
well underway before the pandemic. The relatively long timescales of academic
research would suggest that the full impact on academic productivity due to
the COVID-19 pandemic is yet to be seen. The approach applied here will be a
useful quantitative approach to return to in subsequent years to explore the
longer-term impact of the pandemic.

## Methods

### Grouping by research field

Pre-prints posted to the arXiv can be listed in a single field of research, or
cross-listed in multiple fields. A field is annotated by primary.SEC where the
primary field of research has the prefix, and the sub-field of research is
represented by a short suffix. For example, one pre-print may have a primary
field of research as stellar astrophysics (astro-ph.SA) and be cross-listed in
machine learning (stat.ML). These field(s) of research are supplied by the
corresponding author. It is a subjective decision whether to include more than
one field of research, or what those fields of research would be. For this
reason, throughout this work when we segment by research field we take the
primary parent research field provided and ignore any cross-listed fields of
research. When we segment pre-prints by sub-field, we similarly take the
primary sub-field provided and ignore any cross-listings.

A pre-print’s identifier is defined by the year and month that the pre-print
was posted, and the number of pre-prints already posted in that month (across
all fields). Pre-prints posted prior to April 2007 use a different identifier
scheme, and a different research field definition. For these reasons, we have
excluded pre-prints earlier than this date.

### Long-term modelling of monthly pre-print counts

We use monthly pre-print counts per primary research field from January 2015
until December 2019 to make predictions for the number of pre-prints expected
per month from January 2020 onwards. We use a straight line to model the mean
pre-prints over time, and a Gaussian Process with a quasi-periodic kernel
covariance function to capture seasonal periodicityRasmussen:2006 ;
Ambikasaran:2014 , and a white noise component. We fit the parameters of the
mean model and the kernel hyper-parameters simultaneously, but kept the
periodicity hyper-parameter fixed to one year. The predictions of the
expectation and variance in the number of monthly pre-print counts are
conditioned on the optimized hyper-parameters. We fit this model to monthly
pre-print counts for every primary research field. These predictions are
listed in Table 1. The expected total number of pre-prints, and the
uncertainty in that number, are summed from the models used in individual
research fields.

Our conclusions are reasonably insensitive to the choice of model. As an
alternative model we considered an autoregressive integrated moving average
(ARIMA) modelBoxJenkins:1970  with seasonal terms ($p=d=q=1$ for the non-
seasonal component, and $P=D=Q=1$ with $m=12$ for the seasonal component). The
predictions from the ARIMA model agree excellently with the mean predictions
from the Gaussian Process model. The largest differences between these models
is in hep-lat, where the ARIMA model predicts a higher peak number of pre-
prints than the Gaussian Process model. This is not unexpected: the peak
height of lattice physics pre-prints has been dropping for the last three
years, and this gradual decline is better captured by the Gaussian Process
model. If we adopted the ARIMA model then the drop in hep-lat pre-prints would
become more significant. Throughout this work we adopt the predictions from
the Gaussian Process model, unless otherwise specified.

### Uniquely identifying authors

The arXiv team has parsed given and family names from author strings for all
pre-prints. In general the author names have been appropriately parsed, even
given a wide set of input formats. We made only one correction to these parsed
names. When the author name was provided in the format of ‘Family-name First-
given-name I.’, without any commas, this was interpreted by the arXiv parser
such that the family name was ‘I.’ and the given names were ‘Family-name
First-given-name’. We identified situations like this by checking if the
parsed family name was just an initial, and if two given names were parsed. In
these situations, we correctly ordered the author name.

The arXiv metadata available to us do not include institutional affiliations,
or identifiers that would uniquely identify an author. There are two primary
ways that name confusion could impact our inferences. In the first scenario,
two people with the same name are amalgamated and treated as a single author
that is on average twice as productive (or more, for very common names) as
other authors. In the second scenario, an author will sometimes publish as ‘A.
B. Smith’, and other times publish as ‘A. Smith’. A careful exploration of the
data shows that this is a very frequent scenario, and if left uncorrected,
would appear as many ‘unique’ authors with half as many publications on
average.

We have taken a simple approach to address name confusion. We first define a
unique author by family name and the initial of the first given name (‘Family-
name, I.’), such that we intentionally group together authors that may share
the same initial of their first given name. While our approach to name
confusion is grossly simple, it is unlikely that these choices have any
substantial impact on our inferences. In general, this errs on the side of
name conflation and over-productivity, except for the relatively rare authors
who change their last names over their career. Any common name is likely to
appear in the literature early in the data set and will not impact the
conclusions we draw about how publishing changed in 2020. Indeed, we re-
produced the analysis and figures in this paper by taking all initials given
and it made no change to our conclusions. This is due, in part, because most
authors do not provide middle initials.

### Pre-prints related to the COVID-19 pandemic

We identified a pre-print as being related to the COVID-19 pandemic if the
title or abstract contained any of the following four (case-insensitive)
terms: ‘pandemic’, ‘COVID’, ‘SARS-CoV-2’, ‘lockdown’, or ‘coronavirus’. Before
2020, these terms rarely appear in the title or abstract of any pre-print on
arXiv (Figure 2).

No ethics approval was required for this work.

The entire dataset used in this article is from arXiv. The dataset is curated
by Cornell University and hosted by Kaggle (https://www.kaggle.com/Cornell-
University/arxiv). The dataset is updated weekly. We accessed this dataset on
01 June 2021.

All authors contributed to the discussions and have read and iterated upon the
text of the final manuscript.

The authors declare no competing interests.

A. R. C. is supported in part by the Australian Research Council through a
Discovery Early Career Researcher Award (DE190100656). Parts of this research
were supported by the Australian Research Council Centre of Excellence for All
Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number
CE170100013. I. M. is a recipient of the Australian Research Council Future
Fellowship FT190100574.

We thank Peter Skands (Monash University) and Ross Young (University of
Adelaide) for comments on publication trends in high energy physics.

The paper reflects the authors’ views and may not reflect those of their
employers, funding agencies, or affiliated institutions.

## References

  * (1) Ginsparg. P., ArXiv at 20, _Nature_ , 476, 7359, doi:10.1038/476145a  (2011).
  * (2) Lariviére, V., et al., arXiv E-prints and the journal of record: An analysis of roles and relationships, _Journal of the Association for Information Science and Technology_ , 65, 6, doi:10.1002/asi.23044  (2014).
  * (3) Nicola, M., et al., The socio-economic implications of the coronavirus pandemic (COVID-19): A review, _International Journal of Surgery_ , 78, 185–193, doi:10.1016/j.ijsu.2020.04.018  (2020).
  * (4) Yen-Hao Chu, I., Prima, A., Larson, H. J., Leesa, L. Social consequences of mass quarantine during epidemics: a systematic review with implications for the COVID-19 response, _Journal of Travel Medicine_ , 27, 7, doi:10.1093/jtm/taaa192  (2020).
  * (5) Viglione, G., Are women publishing less during the pandemic? Here?s what the data say. _Nature_ , 581, 356-366, doi:10.1038/d41586-020-01294-9  (2020).
  * (6) , Gewen, V., The career cost of COVID-19 to female researchers, and how science should respond. _Nature_ , 583, 867–869, doi:10.1038/d41586-020-02183-x  (2020).
  * (7) Ibn-Mohammed, T., et al., A critical analysis of the impacts of COVID-19 on the global economy and ecosystems and opportunities for circular economy strategies, _Resources, Conservation and Recycling_ , 164, 105169, doi:1016/j.resconrec.2020.105169  (2021).
  * (8) King, M. M., Frederickson M. E., The Pandemic Penalty: The Gendered Effects of COVID-19 on Scientific Productivity. _Socius_. January 2021. doi:10.1177/23780231211006977  (2021).
  * (9) Clement, C. B., Bierbaum, M., O’Keeffe, K. P., Alemi, A. A., On the Use of ArXiv as a Dataset, Preprint at https://arxiv.org/abs/1905.00075  (2019).
  * (10) Rasmussen, C. E., Williams, C. K. I., Gaussian Processes for Machine Learning, _MIT Press_ , ISBN 026218253X  (2016).
  * (11) Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D. W., O’Neil, M., Fast Direct Methods for Gaussian Processes, Preprint at http://arxiv.org/abs/1403.6015  (2014).
  * (12) Guinnessy, P. (Physics Today). 2021. _Physicists gathered virtually for APS March Meeting_. [online] Available at https://physicstoday.scitation.org/do/10.1063/PT.6.3.20210405a/full [Accessed 17 June 2021].
  * (13) Helmholtz-Institut für Strahlen- und Kernphysik Indico Service (Indico). 2021. _The 38th International Symposium on Lattice Field Theory (Lattice 2020)_. [online] Available at: https://indico.hiskp.uni-bonn.de/event/1/ [Accessed 3 June 2021].
  * (14) physicsworld (IOP Publishing). 2021. _Critical research hit as COVID-19 forces physics labs to close_. [online] Available at https://physicsworld.com/a/critical-research-hit-as-covid-19-forces-physics-labs-to-close/ [Accessed 4 June 2021].
  * (15) Science Magazine (AAAS). 2021. _Amid pandemic, Energy Department labs close to tens of thousands of users._ [online] Available at https://www.sciencemag.org/news/2020/03/amid-pandemic-energy-department-labs-close-tens-thousands-users [Accessed 4 June 2021].
  * (16) Fassin, Y. Research on Covid-19: a disruptive phenomenon for bibliometrics, _Scientometrics_ , 126, 5305–5319, doi:10.1007/s11192-021-03989-w  (2021).
  * (17) Box, G. E. P., Jenkins, G. M. Time series analysis: Forecasting and control. San Francisco: Holden-Day.  (1970).


#  Possible connection between time-splitting parameter and surface tension in
Casimir-type problems __

Iver Brevik  Department of Energy and Process Engineering, Norwegian
University of Science and Technology, N-7491 Trondheim, Norway

###### Abstract

We discuss four different, though related, fundamental topics related to the
Casimir effect: 1) We suggest that the application of Casimir theory to real
dielectric materials, thus implying the atomic spacing as a course-grained
length parameter, makes it natural to assume that this parameter is of the
same order of magnitude as the QFT time-splitting parameter multiplied with
the velocity of light. 2) We show that application of Casimir theory to a
thick fluid shell (apparently a closed mechanical system), leads actually to
an unstable situation if not extra mechanical forces, typically surface
tension forces, are brought into consideration. 3) We analyze how the presence
of a radial Casimir repulsive pressure modifies the filling process of a
spherical vacuum hole in an infinite fluid (the Reynolds problem), with the
result that a bounce occurs at a finite though very small radius. 4) As a
comment on an apparently similar situation in general relativity, we consider
the gravitational collapse of a singular shell. It might seem natural to allow
for the presence of a repulsive Casimir pressure in this case also, thereby
obtaining a bounce-like situation again. However, we have to conclude that
such a procedure implies an omission of the Casimir field’s gravitational
energy, and is therefore hardly tenable, although it is in our opinion worth
mentioning.

##  I Introduction. Analysis of a dielectric ball

From a fundamental viewpoint, delicate issues appear when the quantum field
theory (QFT) of the Casimir effect is applied to a dielectric condensed body.
As QFT is basically a theory derived for vacuum surroundings, there is no
length scale involved (looking away from the extreme Planck scale of
$10^{-33}~{}$cm). Related to the absence of such a scale, there appear
infinities in the theory, which are regularized in various ways. One often
makes use of a time-splitting parameter $\tau$, which after a Wick rotation
implies a splitting in imaginary time. It is natural to ask: is there a
physical significance of this parameter? As we will argue, it is possible to
suggest that there is a physical connection between $\tau$ and the interatomic
spacing of a condensed dielectric medium.

Consider in this context, as a very typical example, the Casimir theory of a
nonmagnetic dielectric ball of radius $a$. This theory was worked out by
Milton back in 1980 [1] by using the time-splitting method (cf. also the
related exposition in Ref. [2]). We confine ourselves here to give only the
essentials. The material in the ball is assumed nonmagnetic and is for
simplicity also assumed to be dilute, $\varepsilon-1\ll 1$, where
$\varepsilon$ is the permittivity. The surroundings are vacuum. The stress $P$
on the surface $r=a$ is calculated as the difference between the $rr$
electromagnetic stress tensor components on the outside and the inside. Some
calculation gives the expression

$P=-\frac{(\varepsilon-1)^{2}\hbar c}{16^{2}\pi
a^{4}}\left[\frac{16}{\delta^{3}}+\frac{1}{4}\right],\quad\delta=\frac{\tau
c}{a}.$ (1)

Both terms are negative, corresponding to an attractive force. We shall be
interested in the cutoff dependent part only. Calling this $P_{\rm cutoff}$,
we have

$P_{\rm
cutoff}=-\frac{(\varepsilon-1)^{2}}{16\pi}\frac{\hbar}{ac^{2}}\frac{1}{\tau^{3}}.$
(2)

A striking property of this expression is that the parameter $\tau$ (dimension
seconds) becomes naturally related to the surface tension coefficient $\sigma$
(dimension N/m), the latter known from ordinary hydrodynamics.

The following estimate is illustrative: Forget about Casimir effect for a
while, and start from the expression (2) as it stands. Require that this
inward pressure is equal to the inward surface pressure following from
Laplace’s formula, i.e.,

$\frac{(\varepsilon-1)^{2}}{16\pi}\frac{\hbar}{ac^{2}}\frac{1}{\tau^{3}}=\frac{2\sigma}{a}.$
(3)

This estimate gives us information about how $\tau$ is related to ordinary
physical quantities. First, we observe the connection

$\tau\propto\sigma^{-1/3},$ (4)

which is in this special case independent of the radius $a$ (it is not a
general property). We may here insert $\sigma=0.073~{}$N/m as for a water-air
surface, and choose a reasonable value $\varepsilon-1=0.01$ for the diluteness
parameter. For the quantity $\tau c$, the path travelled by light during the
time $\tau$, we thus get

$\tau c\sim 1~{}\rm{\AA},\quad\tau\sim 10^{-19}~{}\rm{s}.$ (5)

This is the point we wished to emphasize: the minimum distance $\tau c$
corresponds to typical interatomic distances. Is this merely a numerical
coincidence? It might be so, but we think it is worth noticing that the time-
splitting parameter $\tau$ coming from QFT becomes so naturally linked to the
natural ’lattice spacing’ in condensed matter physics.

##  II On the stability of a closed mechanical system under the influence of
Casimir forces

We go on with a related example, also that apparently simple, but nevertheless
being of fundamental interest. Let us consider the force balance for a closed
(an insulated) mechanical system when subjected to Casimir forces. Adopt the
same fluid sphere as above, but now with a spherical hole of radius $r=a$ in
it. The outer radius is renamed as $r=b$. The fluid body is accordingly an
annular spherical region bounded by the surfaces $a$ and $b$. For $r<a$ and
$r>b$ there is a vacuum.

We will calculate the Casimir force on the two spherical surfaces. To
circumvent problematic counter terms, we will assume that the fluid medium has
a permittivity $\varepsilon$ and a permeability $\mu$ such that

$\varepsilon\mu=1.$ (6)

For this kind of medium, called an isorefractive medium (cf., for instance,
Refs. [3, 4, 5]) the refractive index is unity. As a slightly more complicated
situation we may take the inner medium 1 to have material constants
$\varepsilon_{1},\mu_{1}$ and outer medium 2 to have material constants
$\varepsilon_{2},\mu_{2}$, both cases satisfying the condition (6).

From Ref. [3] we have for the Casimir pressure on the surface $r=b$,

$P(b)=\frac{0.09235}{8\pi}\frac{\hbar
c}{b^{4}}\left(\frac{\mu_{12}-1}{\mu_{12}+1}\right)^{2}\left[1+0.311\frac{\mu_{12}}{(\mu_{12}+1)^{2}}\right],$
(7)

where $\mu_{12}\equiv\mu_{1}/\mu_{2}=\mu$. An important property of this
expression is that it is invariant under the substitution
$\mu_{12}\rightarrow\mu_{21}$. We may thus apply the expression (7) to find
the pressure $P(a)$ on the inner surface $r=a$ also, only by substituting $b$
with $a.$ Both pressures are outward directed.

Consider a narrow cone corresponding to a solid angle $d\Omega$, so that a
surface $a^{2}d\Omega$ is crossed out at $r=a$ and $b^{2}d\Omega$ is crossed
out at $r=b$. The Casimir radial force on the fluid within the cone, assuming
for simplicity the medium to be dilute, is

$F_{\rm Casimir}=\frac{0.09235\hbar
c}{32\pi}(\mu-1)^{2}\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}\right)d\Omega.$ (8)

When summing up the forces acting on the actual fluid body, we have in
principle to consider the electrostrictive force also. This force will however
be without influence here, since it is a gradient force. To see this, it is
sufficient to give the expression for the force density in the usual case of a
nonmagnetic medium,

${\bf f}_{\rm
elstr}=\frac{1}{2}\varepsilon_{0}{\bf\nabla}\left(E^{2}\rho\frac{d\varepsilon}{d\rho}\right).$
(9)

As written, this expression assumes a nonpolar medium, so that $\varepsilon$
is a function of $\rho$ only. The derivative $d\varepsilon/d\rho$ may be
calculated via the Clausius-Mossotti relation.

The electrostrictive force is outward directed at $r=a$ and inward directed at
$r=b$. These two opposing forces press the fluid together, but the total
radial force $F_{\rm elstr}$ on the actual fluid body is zero, in view of
Gauss’ law. (In order to detect the electrostrictive effect, one has in
general to do local pressure experiments.)

In fluid mechanics it is physically legitimate to assume ideal conditions, so
that there is no viscosity to counteract fluid motion. We thus become enfaced
with a paradoxical situation: the Casimir force should in principle be able to
accelerate the fluid outwards. The sphere should fly apart. There must
accordingly be extra forces, not included so far, that make the system stable.
Quite evidently, it is the surface tension that is the remedy here. The
surface pressure is $2\sigma/r,\,r=a,b$ on the two surfaces, both pressures
acting inward. Multiplying with the appropriate surface areas, it follows that
the total radial force on the considered volume from surface stresses is
$2\sigma(a+b)d\Omega$.

Let us require the two kinds of forces to balance each other out, thus

$\frac{0.09235\hbar
c}{32\pi}(\mu-1)^{2}\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}\right)=2\sigma(a+b).$
(10)

When $a\ll b$ we get approximately

$\frac{\hbar c}{640\pi}(\mu-1)^{2}\frac{1}{a^{2}b}=\sigma.$ (11)

Take $\sigma=0.073$ N/m as before, and take $|\mu-1|=0.01,\,b/a=10$.
Remarkably enough, this leads to results of the same order of magnitude as
already given above, in Eq. (5). Realistic input values for the surface
tension as well as for the other parameters give results in rough agreement
with condensed matter atomic dimensions.

The Casimir force thus cannot stand ’alone’ when acting on a closed dielectric
system; it needs the cooperation with conventional effects in fluid mechanics
to make the situation physically consistent. Note that the closeness of the
system is an essential point here. The usual situation with two parallel
plates exposed to an attractive force is different as it is an open system: an
external force is needed to hold the plates in place.

##  III On the filling of a spherical hole in a fluid, in the presence of the
Casimir force

We now enter a class of dynamical problems for which the Casimir force comes
into play. In this section we consider an infinite isotropic dielectric fluid
of density $\rho$, with a central hole of radius $r=a$ in it. If the pressure
at infinity is $p_{\infty}$, there will be a force aiming to fill the hole.
For times $t<0$ we assume the situation is static, via the help of external
stabilizing agencies. At $t=0$ these agencies are removed and the hole starts
filling. We regard this problem as a Newtonian one, in the sense that
relativistic effects are omitted. The fluid is assumed to be nonviscous and
incompressible so that the velocity of sound is infinite.

For simplicity we assume the fluid to be isorefractive such as before, so that
it satisfies the condition (6). Thus within the hole
$\varepsilon_{1}=\mu_{1}=1$, while on the outside $\varepsilon_{2}=1/\mu_{2}$.
In view of the already noted invariance of the expression (7) under the
interchange $\mu_{12}\rightarrow\mu_{21}$, we can use this expression to
calculate the surface pressure $P(a)$ in the initial static period $t<0$.
Moreover, we can use the same expression to calculate the pressure $P(R(t))$
at later times $t>0$ also, where $R(t)$ is the instantaneous radius of the
hole (note that $R(0)=a$). The physical reason for this is that the
characteristic time for establishment of electromagnetic equilibrium, roughly
of order $R(t)/c$, is much smaller than the characteristic time scale for the
fluid motion.

We now write the surface pressure in the form

$P(R(t))=\frac{C}{8\pi R^{4}(t)},$ (12)

where $C$ is a constant. The Euler equation for the velocity $v=v(t)$ at an
arbitrary point in the fluid is

$\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial
r}=-\frac{1}{\rho}\frac{\partial p}{\partial r},$ (13)

where $p=p(r,t)$ is the pressure. Continuity of the radial flow implies that

$r^{2}v(r,t)=F(t),$ (14)

where $F(t)$ is an unspecified function (this can also be seen from the
incompressibility condition $\nabla^{2}\phi=0$, with $\phi$ the velocity
potential). Thus we can write

$\frac{\dot{F}(t)}{r^{2}}+\frac{\partial}{\partial
r}\left(\frac{1}{2}v^{2}\right)=-\frac{1}{\rho}\frac{\partial p}{\partial r}.$
(15)

For a fixed value of $t$ we integrate this equation from $R(t)$ to infinity,

$-\frac{\dot{F}(t)}{R(t)}+\frac{1}{2}V^{2}(t)=\frac{p_{\infty}}{\rho}-\frac{P(R(t))}{\rho},$
(16)

where $V(t)=\dot{R}(t)$ is the velocity of the surface. It is here assumed
that the velocity at infinity is equal to zero.

We now apply Eq. (14) at the instantaneous surface,

$R^{2}(t)V(t)=F(t).$ (17)

so that in view of $d/dt=Vd/dR$ we can rewrite the first term in (16) as

$-\frac{\dot{F}}{R}=-2V^{2}-\frac{1}{2}R\frac{dV^{2}}{dR},$ (18)

so that (16) reduces to

$\frac{dV^{2}}{dR}+\frac{3V^{2}}{R}=\frac{C}{4\pi\rho
R^{5}}-\frac{2p_{\infty}}{\rho R}.$ (19)

With an integrating factor $\exp{\int(3/R)dR}=R^{3}$ we can integrate this
equation analytically,

$V^{2}=\frac{a^{3}}{R^{3}}\left[\frac{2p_{\infty}}{3\rho}\left(1-\frac{R^{3}}{a^{3}}\right)-\frac{C}{4\pi\rho}\frac{1}{Ra^{3}}\left(1-\frac{R}{a}\right)\right].$
(20)

where we have made use of the initial condition $V(0)=0$.

Taking the square root of this expression, using $V=dR/dt$, we can by
integration over $R$ follow the time dependence of the filling process. For
our purpose it is however of greater interest to analyze the surface velocity
in the limit of small $R/a$. We see that the velocity becomes zero at a radius
$R=R_{\rm min}$, where

$R_{\rm min}=\frac{3C}{8\pi p_{\infty}a^{3}}.$ (21)

The collapse of the fluid bounces off at this radius. Quantitatively, from (7)
and (12) we get

$C=0.09235\hbar cf(\mu_{12})=2.9\times 10^{-27}f(\mu_{12}),$ (22)

where $f(\mu_{12})$ is the material-dependent function in (7). If we
approximate this function by unity, and choose as input values
$p_{\infty}=1~{}$mPa, $a=1~{}$mm, we find

$R_{\rm min}\approx 3~{}\rm{\AA}.$ (23)

At this value the hydrodynamical description of the fluid as a continuous
medium is however questionable. The sensitivity of $R_{\rm min}$ with respect
to $a$ is very pronounced. If we had inserted for instance $a=1~{}$cm, $R_{\rm
min}$ would have been very low and the continuum model would definitely have
been lost.

Let us summarize the physics of this process: the filling of the hole is
initiated by the pressure $p_{\infty}$, and the time dependence of $R(t)$ is
further dependent on the constants $\rho$ and $\mu_{2}$. The very existence of
the bounce is however due to the Casimir repulsion, i.e., a quantum mechanical
effect.

In the absence of the Casimir modification, the solution to the above problem
can be found in Landau-Lifshitz [6]. It was originally solved by Rayleigh in
1917.

##  IV Gravitational collapse of a singular shell, in the presence of the
Casimir force

As an analogy to the example in the previous section, we will now consider the
gravitational collapse of a singular spherical shell when account is taken of
the repulsive Casimir force. A shell is a very simple material system but
nevertheless able to demonstrate main features of the collapse process when
the Casimir surface pressure is brought into play, in a very simple way..

Let the shell whose gravitational (Schwarzschild) ) mass is $M$, start off the
collapse at zero velocity when the radius is infinity (we now put $c$, as well
as Newton’s gravitational constant $G$, equal to 1). The collapsing shell
surface separates spacetime into two regions with different extrinsic
curvature tensors $K_{ij}$. The time-dependent shell radius is $R=R(\tau)$,
where $\tau$ is proper time.

The external geometry is Schwarzschild,

$ds^{2}=-\left(1-\frac{2M}{R}\right)dt^{2}+\left(1-\frac{2M}{R}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(24)

while the inner geometry is flat,

$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\Omega^{2}.$ (25)

In this section we let $\sigma$ be the proper surface mass energy, and
$u^{\alpha}$ the four-velocity of the matter. The surface energy-momentum
tensor can the be written as

$S^{\alpha\beta}=\sigma u^{\alpha}u^{\beta},$ (26)

where $S^{\alpha\beta}$ is a result of integrating the local interior energy-
momentum tensor in the shell, $T^{\alpha\beta}$, in the normal direction $\bf
n$ across the shell,

$S^{\alpha\beta}=\int_{-\eta}^{\eta}T^{\alpha\beta}dn,$ (27)

with $\eta=0^{+}$.

From the equation of motion ${{S_{i}}^{m}}_{|m}=0$ (see below) we obtain
$(\sigma R^{2})_{,\tau}=0$, so that $4\pi R^{2}\sigma$ is a constant. In view
of our initial condition $\dot{R}=0$ at $R=\infty$, the constant becomes equal
to the Schwarzschild mass [7],

$4\pi R^{2}\sigma=M.$ (28)

The four-acceleration for particles on the shell is [7, 8, 9, 10]

$D{\bf u}/d\tau=u^{i}u^{j}_{|i}{\bf e}_{j}+K_{ij}u^{i}u^{j}{\bf n},$ (29)

where $u^{j}_{|i}$ is the covariant derivative with respect to the three-
geometry of the hypersurface swept out by the two-dimensional shell. The basis
vectors of this surface are ${\bf
e}_{i}=(\partial/\partial\tau,\partial/\partial\theta,\partial/\partial\varphi)$.
The orthogonality relations are

${\bf e}_{i}\cdot{\bf e}_{j}=g_{ij},\quad{\bf e}_{i}\cdot{\bf n}=0,\quad{\bf
n\cdot n}=1.$ (30)

We will look for the projection of the four-acceleration $\bf a$ onto $\bf n$.
From Eq. (29),

${\bf a}=K_{ij}u^{i}u^{j}{\bf n}.$ (31)

From the equation of motion one can derive [7]

${\bf a}^{+}+{\bf a}^{-}=0,$ (32)

which generalizes the equation ${\bf a}=0$ for a free particle, and also

${\bf a}^{+}-{\bf a}^{-}=4\pi\sigma{\bf n}.$ (33)

This is the conventional theory for the collapse. Can this theory be
generalized, in the sense that we allow for a repulsive Casimir force on the
spherical surface in the same way as we did in the previous section? Let us
start from the pressure $P$ in the same form as before,

$P=\frac{C}{8\pi R^{4}},$ (34)

where now $C$ has the value corresponding to a singular shell [11],

$C=0.09235\hbar c,$ (35)

in physical units. It might seem possible to modify the equation of motion by
multiplying the mean acceleration $({\bf a}^{+}+{\bf a}^{-})/2$ by $\sigma$
and put it equal to the Casimir pressure,

$\frac{1}{2}\sigma({\bf a}^{+}+{\bf a}^{-})=\frac{C}{8\pi R^{4}}{\bf n}.$ (36)

Such a procedure was actually followed in Ref. [12], leading to the following
equation of motion,

$(1+\dot{R}^{2})^{1/2}=1-\frac{C}{2MR}+\frac{M}{2R}\frac{1}{1-C/2MR},$ (37)

which satisfies ${\dot{R}}=0$ at $R=\infty$.

The presence of the factor $C$ means that the collapse bounces off at a
minimum radius $R=R_{\rm min}$, where

$R_{\rm min}=\frac{C}{2M}.$ (38)

At first sight, this result is striking and points towards the existence of a
gravitational analogy to the bouncing effect discussed in the previous
section. The argument, consisting in the use of Eq. (36) has however the
drawback that it neglects the gravitational influence from the Casimir energy.
We therefore cannot regard the formalism as physically satisfactory after all.
We confine us here only to mentioning this point without going into further
detail. If we put $C=0$ in Eq. (37), we get

$(1+\dot{R}^{2})^{1/2}=1+\frac{M}{2R},$ (39)

which implies that the collapse occurs down to $R=0$.

There are several papers discussing the importance of Casimir energy in a
gravitational context under various circumstances; cf., for instance, Refs.
[13, 14], and in the present context, Ref. [15].

##  V Summary

By means of some examples, we have in this paper elucidated how the
application of the Casimir force to ordinary dielectric matter can be more
delicate than usually assumed. Let us summarize:

1\. The QFT formalism, although not containing any length scale in itself
(Planck scale omitted), will when applied to a condensed medium be confronted
with a natural length, namely the interatomic spacing. In Sec. I we pointed
out a possible physical interpretation of the conventional mathematical time-
splitting parameter $\tau$ as being of the same order as the interatomic
transit time. A numerical estimate supported this suggestion.

2\. The thick fluid shell model considered in Sec. II, at first sight
apparently a stable mechanical system (gravity omitted), needs under influence
from the Casimir force assistance from extra forces to secure stability. The
natural choice of such a local force is the surface tension. For simplicity,
we assumed the medium to be of the so-called isorefractive kind. A numerical
estimate with realistic input parameters gave the same order of magnitude
result as in Sec. I.

3\. In Sec. III we analyzed the filling of a vacuum hole in an infinite fluid
(the Reynolds problem), when taking into account the radial repulsive Casimir
force, with the result that the collapse becomes halted. There occurs a
Casimir-induced bounce at a finite small radius $R_{\rm min}$. The bigger the
initial radius $a$ of the hole, the smaller value is obtained for $R_{\rm
min}$.

4\. In Sec. IV we commented briefly on an analogous situation from general
relativity, namely the collapse of a singular spherical shell. A simple
application of the standard repulsive Casimir pressure to the gravitational
equation of motion (36) led to an equation of motion for the radius $R$ that
implied a bounce, thus in apparent qualitative agreement with the collapse
considered in Sec. III. However, as such a procedure ignores the gravitational
influence of the Casimir energy, we have to doubt its physical reality.

## Acknowledgment

I acknowledge the very valuable comments of K. A. Milton on preliminary
versions of this manuscript.

## References

  * [1] K. A. Milton, Ann. Phys. (NY) 27, 49 (1980).
  * [2] J. S. Høye and I. Brevik, Phys. Rev. A 95, 052127 (2017).
  * [3] I. Brevik and H. Kolbenstvedt, Ann. Phys. (NY) 143, 179 (1982).
  * [4] I. Brevik and H. Kolbenstvedt, Ann. Phys. (NY) 149, 237 (1983).
  * [5] S. Å. Ellingsen, I. Brevik and K. A. Milton, Phys. Rev. E 80, 021125 (2009).
  * [6] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, 1966, Sec. 10, Problem 7.__
  * [7] A. P. Lightman, W. H. Press, R. H. Price and S. A. Teukolsky, Problem Book in Relativity and Gravitation (Princeton University Press, 1975), Problem 21.10.
  * [8] W. Israel, Nuovo Cimento B 44, 1(1966); 48, 463 (E) (1967).
  * [9] W. Israel, Phys. Rev. 153, 1388 (1967).
  * [10] C. M. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (San Francisco, 1973), page 555.
  * [11] K. A. Milton, L. L. DeRaad, Jr. and J. Schwinger, Ann. Phys. (NY) 115, 388 (1978).
  * [12] I. Brevik and H. Kolbenstvedt, Nuovo Cimento B 82, 71 (1984).
  * [13] K. A. Milton, K. V. Shajesh, S. A. Fulling and P. Parashar, Phys. Rev. D 89, 064027 (2014).
  * [14] V. D. Coronell, A. E. Gonçalves, M. C. Baldiotti and R. C. Batista, Eur. Phys. J. C 82, 50 (2022).
  * [15] D. Vick, Nuovo Cimento B 94, 54 (1986).