# Microlensing constraints on clustered primordial black holes

Mihael Petač<EMAIL_ADDRESS>University of Nova Gorica, Center for
astrophysics and cosmology, Vipavska 11c, Ajdovščina, Slovenia  Laboratoire
Univers et Particules de Montpellier (LUPM), Université de Montpellier
(UMR-5299) & CNRS, Place Eugène Bataillon, F-34095 Montpellier Cedex 05,
France  Julien Lavalle<EMAIL_ADDRESS>Karsten Jedamzik
<EMAIL_ADDRESS>Laboratoire Univers et Particules de
Montpellier (LUPM), Université de Montpellier (UMR-5299) & CNRS, Place Eugène
Bataillon, F-34095 Montpellier Cedex 05, France

###### Abstract

The discovery of black-hole-binary mergers through their gravitational wave
(GW) emission has reopened the exciting possibility that dark matter is made,
at least partly, of primordial black holes (PBHs). However, this scenario is
challenged by many observational probes that set bounds on the relative PBH
abundance across a broad range of viable PBH masses. Among these bounds, the
ones coming from microlensing surveys lead to the strongest constraints in the
mass range from $\sim 10^{-10}$ to a few M⊙. The upper part of this range
precisely corresponds to the mass window inside which the formation of PBHs
should be boosted due to the QCD phase transition in the early Universe, which
makes the microlensing probes particularly important. However, it has been
argued that taking into account the inevitable clustering of PBH on small
scales can significantly relax or entirely remove these bounds. While the
impact of PBH clustering on the GW event rate has been studied in detail, its
impact on the microlensing event rate has not yet been fully assessed. In this
Letter, we address this issue, and show that clusters arising from
isocurvature perturbations, that originating from PBHs formed from Gaussian
initial curvature perturbations, do not alter the current microlensing
constraints, as they are not sufficiently compact.

††preprint: LUPM:22-002

##  I Introduction

The advent of gravitational wave (GW) astronomy [1] has opened a new window on
the Universe. In particular, the peculiar mass range inferred from the first
observations of binary-black-hole (BBH) mergers [1, 2, 3] has been interpreted
as a hint for a possible primordial-black-hole (PBH) origin of these events
[4, 5]. This revived the exciting idea that the still mysterious dark matter
(DM), which pervades the Universe and is currently the pillar of the theory of
structure formation [6, 7, 8], could be made, at least partly, of PBHs [9, 10,
4].

However, this scenario has to confront several observational constraints that
currently bound the PBH abundance along an extended range of masses (for
recent reviews, see e.g. [11, 12]). Among them, microlensing surveys provide a
powerful probe in a very special mass range for PBHs, from a fraction of up to
a few Solar masses [13, 14]. This mass range corresponds to PBHs that could
have been formed during the QCD phase transition, when a drop of pressure
decreases the overdensity threshold above which density fluctuations can
collapse directly to black holes (BHs) [15]. For approximately scale-invariant
pre-existing density perturbations, one expects thus a peak in the mass
function of PBHs around one Solar mass [16], which is within the reach of
Galactic microlensing surveys [17, 18, 19].

Current constraints limit PBHs to represent $f_{\rm pbh}\lesssim 0.1$ of the
total DM abundance in this mass range. However, these limits have been derived
assuming the PBHs are randomly distributed along a given DM density profile.
Throughout the literature it has been suggested that the inevitable clustering
of PBHs could significantly relax these bounds [20, 21, 22]. In this Letter,
we wish to explore in detail how such clustering affects the microlensing
limits. We shall see that for Gaussian inflationary perturbations they are
actually unlikely to change.

Where do these clusters come from? On large scales, the behavior of PBHs is
indistinguishable from that of any viable cold dark matter (CDM) particle
candidate [23]. However, on small scales there are two important differences
compared to the particle hypothesis, i.e. the macroscopic PBH masses and their
essentially vanishing initial peculiar velocities. For Gaussian perturbations
overdense regions exceeding the threshold for PBH formation are randomly
distributed in space [24, 25]. This subsequently leads to Poisson fluctuations
of PBH numbers in any given volume [26, 27]. These fluctuations represent the
dominant source of isocurvature perturbations on small scales and inevitably
seed the formation of PBH clusters already at very early times [28]. However,
due to small relaxation times, smaller clusters are subject to later
evaporation.

The Poisson fluctuations in PBH numbers and their vanishing initial velocities
lead to efficient formation of PBH binaries [28, 29, 30]. These binaries are
highly eccentric due the vanishing initial PBH peculiar velocities, and one
expects a large BBH merger rate if PBHs make a significant fraction of CDM. In
fact for $f_{\rm pbh}=1$, it was thought that the merger rate of PBHs should
be orders of magnitude larger than the one observed by LIGO/Virgo [31].
However, these estimates did not take into account the presence of dense
clusters, and PBH three-body interactions within, which significantly perturb
the orbits and thus coalescence times of binaries. Detailed studies found that
the merger rate could indeed be consistent with the observed one [32, 33], in
case that most of the PBH binaries are actually residing in clusters. As of
now, it seems that a $f_{pbh}=1$ with $M\sim M_{\odot}$ is marginally ruled
out due to the coalescence of isolated and highly eccentric binaries [34].

In this work we carefully examine the impact of PBH clustering on the
interpretation of microlensing searches for massive compact halo objects
(MACHOs). In our analysis, we pay particular attention to the differences in
the probability distribution for the expected number of events, as well as the
probability for recurring microlensing events, which were typically discarded
from the observational samples. For concreteness, we provide quantitative
results for the EROS-like microlensing survey [18] of the Large Magellanic
Cloud (LMC). However, the implications of our findings can be easily
generalized to other similar Galactic surveys, such as those of MACHO [35] and
OGLE [19], or even Subaru/HSC observations of the Andromeda galaxy [36].

##  II Microlensing searches for MACHOs

Microlensing surveys search for temporary brightening of distant sources
(e.g., stars) due to the passage of a MACHO near the line-of-sight (l.o.s.).
In essence, this phenomenon is a particular case of strong lensing, where the
angular separation between multiple images of the source is smaller than the
resolution of the telescope (i.e. of order of microarcseconds, hence the name
microlensing). The observed effect is the amplification of the source’s
brightness by a factor [37, 13]:

$\displaystyle A(u)=\frac{u^{2}+2}{u\sqrt{u^{2}+4)}}\,,$ (1)

where $u=R/R_{\rm E}$ is the displacement of the lens from the l.o.s. axis,
expressed in units of Einstein radius,

$\displaystyle R_{\rm E}=\sqrt{\frac{2r_{\rm S}(D_{\rm s}-D_{\rm l})D_{\rm
l}}{D_{\rm s}}}\,,$ (2)

where $r_{\rm S},D_{\rm s}$, and $D_{\rm l}$ are the Schwarzschild radius of
the lens, the distance of the observer to the source, and the distance of the
observer to the lens, respectively. The observed light curves are typically
fitted using the following parameterization of $u$ [38]:

$\displaystyle u(t)=\sqrt{u_{0}^{2}+\frac{(t-t_{0})^{2}}{t_{\rm E}^{2}}}\,$
(3)

where $u_{0}$ is the impact parameter of the lens, $t_{0}$ the time of maximum
amplitude, while $t_{\rm E}$ is the characteristic crossing time, i.e. $t_{\rm
E}=R_{\rm E}/v_{\perp}$ with $v_{\perp}$ being the lens transverse velocity
with respect to the l.o.s.

The main quantity in predicting the observational signatures of MACHOs is the
differential event rate ${\rm d}\Gamma/{\rm d}t_{\rm E}$. The latter
quantifies the rate at which lenses enter the “microlensing tube”, defined as
the volume along the l.o.s. axis that leads to an amplification above a
certain threshold $A_{\rm T}\equiv A(u_{\rm T})$ (for more details see, e.g.,
[14] and Fig. 1 therein). ${\rm d}\Gamma/{\rm d}t_{\rm E}$ can be used to
compute the expected number of microlensing events in a given survey with
detection efficiency $\varepsilon(t_{\rm E})$ and total exposure $E$:

$\displaystyle\bar{N}=E\int{\rm d}t_{\rm E}\;\;\varepsilon(t_{\rm
E})\times\frac{{\rm d}\Gamma}{{\rm d}t_{\rm E}}(t_{\rm E})\,.$ (4)

In microlensing studies, it is commonly assumed that PBHs follow a smooth and
spherically symmetric Galactic halo density profile $\rho(r)$, where $r$
denotes the galactocentric distance, and have an isotropic Maxwellian velocity
distribution with a dispersion $\sigma(r)$. In that case, the differential
event rate can be computed by performing a single l.o.s. integral [14, 39].

However, as argued in [20, 21, 40, 22], these assumptions may no longer be
justified if PBHs are bound into compact clusters, but rather should be
applied to the clusters themselves and no longer to individual PBHs. This
argument was used to evade microlensing constraints on PBH dark matter simply
from the fact that (compact) cluster masses fall outside from the microlensing
target mass range. It is straightforward and intuitively clear to write down
the approximate condition for when a cluster should be treated as a single
lens (compact cluster) and when as a compilation of spatially correlated
individual lenses (extended cluster). Imagine a perfectly aligned observer-
lens-source event where the bending of light rays by the Einstein angle
$\Theta_{\rm E}=R_{\rm E}(M_{\rm cl})/D_{\rm l}$ is maximal, where $M_{\rm
cl}$ is the cluster mass. These light rays pass approximately $\Theta_{\rm
E}D_{\rm l}=R_{\rm E}(M_{\rm cl})$ from the center of the cluster and do not
enter it if its radius is $r_{\rm cl}\ll R_{\rm E}(M_{\rm cl})$. Therefore, a
cluster has to be treated as compact from the point of view of microlensing
surveys when $r_{\rm cl}\ll R_{\rm E}(M_{\rm cl})$. We will see that for
Gaussian perturbations this is not the case, as the clusters are orders of
magnitude larger than the corresponding Einstein radius, i.e. $r_{\rm cl}\gg
R_{\rm E}(M_{\rm cl})$. In such situations, characterized by low optical
depth, accounting only for the PBH that is the closest to the l.o.s. is
sufficient to model the net lensing effect [41].

##  III Some properties of PBH clusters

For Gaussian primordial curvature perturbations the probability for the
collapse into BHs is uniform in space and, therefore, results in initially
randomly distributed PBHs. This unavoidably seeds Poissonian isocurvature
perturbations on small scales, which later lead to the formation of PBH
clusters. The overdensities associated with scales that on average contain $N$
PBHs are of the size of $\delta(N)\sim\Delta N/N=1/\sqrt{N}$. In the radiation
dominated era, these fluctuations only grow logarithmically, however, after
matter-radiation equality, they enter the linear growth regime and collapse.
Throughout their subsequent evolution, smaller clusters evaporate, but are
themselves aggregated into more massive structures. The typical evaporation
time can be estimated as [42]:

$\displaystyle t_{\rm evap}\sim 140\,t_{\rm relax}\sim\frac{14\,N_{\rm
pbh}}{\log N_{\rm pbh}}\;t_{\rm cross}\,,$ (5)

where $t_{\rm relax}$ is the system relaxation time rewritten in terms of the
number of PBHs within the cluster, $N_{\rm pbh}$, and the typical crossing
time $t_{\rm cross}\sim r_{\rm cl}/v_{\rm cl}$ determined by the cluster size,
$r_{\rm cl}$, and its virial velocity, $v_{\rm cl}$. At present time, one
expects to find the largest abundance of PBH clusters with total masses
$M_{\rm cl}\sim 10^{4}\,M_{\odot}$, consisting of $\sim 10^{3}-10^{4}$
individual PBHs with masses $\sim 1-10\,M_{\odot}$, and with typical cluster
sizes $r_{\rm cl}\sim 10\,{\rm pc}$ [43, 32, 33]. Larger clusters, which are
less dense, are made up of these basic building blocks, i.e. the of $M_{\rm
cl}\sim 10^{4}\,M_{\odot}$ clusters, as these are the smallest structures that
have not yet evaporated.

##  IV Microlensing constraints for clustered PBH

As mentioned above, if PBHs live in compact clusters, one can readily rescale
the microlensing limits on the DM fraction in PBHs by considering single
lenses of mass $M_{\rm cl}$. This can completely lift the current microlensing
bounds on sub-Solar mass PBHs for $M_{\rm cl}\gg M_{\odot}$ [20]. However, the
condition for PBH clusters to act as individual lenses in surveys of the
Magellanic Clouds is that $r_{\rm cl}\ll{\rm 10\,au}(M_{\rm
cl}/M_{\odot})^{1/2}$ (for surveys targeting the Andromeda galaxy this ratio
is larger, but only by a factor of $\sim 20$). This criterion is definitely
not fulfilled for PBH clusters formed through the collapse of Gaussian
primordial adiabatic perturbations [30], whose typical features have been
recalled above. Such dense PBH clusters can only form in the early Universe
through some other mechanism (e.g. [44, 45, 46]), or different initial
conditions, such as non-Gaussian perturbations [47] (see also [48]). We
postpone a dedicated study of such scenarios to future work, however, we
anticipate that substantial fine-tuning of the corresponding PBH production
mechanism would be required to significantly relax the microlensing bounds,
while still obeying other observational constraints. Therefore, in the
following, we focus on the effect of extended PBH clusters on the microlensing
limits derived from the surveys of the LMC.

###  IV.1 Probability distribution for the number of observed events

To carry out the statistical analysis of the event rate, we resort to
dedicated Monte-Carlo (MC) simulations of microlensing signals produced by
clustered PBHs. While semi-analytical estimates are possible for sparsely
distributed sources, the crowded stellar fields of nearby galaxies make it
necessary to perform a numerical convolution of associated complex stellar
surface densities with the probability distribution of PBH cluster positions,
rendering it analytically intractable.

####  IV.1.1 Monte-Carlo simulations

Our MC simulations consist in drawing $N_{\rm cl}\sim{\rm Pois}(\lambda_{\rm
LMC})$ PBH clusters, whose positions and velocities are distributed according
to a chosen global model of the Galactic DM halo. The latter also defines
$\lambda_{\rm LMC}=M_{\rm l.o.s.}/M_{\rm cl}$, i.e. the expected number of PBH
clusters within the observed volume ($M_{\rm l.o.s.}$ corresponds to the DM
mass contained in a cone with the apex at the position of the observer and the
base in the approximate plane of the stellar source population, encompassing
the observed area in the sky). We additionally assume that all PBH clusters
have the same mass $M_{\rm cl}$, as justified above. For each of the clusters,
we draw a number of microlensing events from a Poisson distribution with the
appropriate mean. The latter is determined by computing the differential event
rate, ${\rm d}\Gamma$, from its standard expression (see, e.g., Eq. 8 in
[14]). We further assume that all PBHs have equal masses, $M_{\rm
PBH}=1\,M_{\odot}$, and that the velocity dispersion of PBHs inside a cluster
is negligible with respect to the cluster velocity in the Galactic halo, such
that the velocity distribution of the cluster members can be modeled as a
delta function picked at the cluster’s peculiar velocity $\vec{v}_{\rm p}$.
Thus, the differential event rate ${\rm d}\Gamma$ (integrated over angles) can
be written as:

$\displaystyle{\rm d}\Gamma=2\,n_{\rm PBH}(x)\,u_{\rm T}\,v_{\perp}\,R_{\rm
E}(x)\,{\rm d}x\;,$ (6)

where $n_{\rm PBH}(x)$ is the PBH number density at position $x$ along the
l.o.s. Consequently, the Einstein crossing time of each event associated with
this cluster is fixed to $t_{\rm E}=R_{\rm E}(x)u_{\rm T}/v_{\perp}$. The
corresponding differential event rate inside a cluster for a l.o.s with impact
parameter $b$ can then be expressed as the l.o.s. integral:

$\displaystyle\frac{{\rm d}\Gamma}{{\rm d}t_{\rm E}}(b,t_{\rm
E})=\frac{2v_{\perp}u_{\rm T}R_{\rm E}(x_{\rm cl})}{M_{\rm
PBH}}\,\delta\left(\frac{R_{\rm E}(x_{\rm cl})u_{\rm T}}{v_{\perp}}-t_{\rm
E}\right)$ $\displaystyle\times\int_{x_{\rm cl}-\Delta x}^{x_{\rm cl}+\Delta
x}{\rm d}x\;\rho_{\rm cl}\left(\sqrt{(x-x_{\rm cl})^{2}+b^{2}}\right)\,,$ (7)

where $x_{\rm cl}$ denotes the position of the PBH cluster along the l.o.s.,
$\Delta x\equiv\sqrt{r_{\rm cl}^{2}-b^{2}}$ the half-length of the l.o.s.
segment penetrating the cluster, and $\rho_{\rm cl}(r)$ the radial density
profile of the cluster (we assume that the PBH clusters are spherically
symmetric). The expected number of microlensing events for the entire cluster
is obtained by convoluting the above expression with the surface density of
stellar targets, $\Sigma_{\star}(\textbf{q})$, over the area of the cluster
(neglecting the small angle correction as $r_{\rm cl}\ll x_{\rm cl}$):

$\displaystyle\frac{{\rm d}\Gamma_{\rm cl}}{{\rm d}t_{\rm E}}(t_{\rm E})$
$\displaystyle=$ $\displaystyle\int_{0}^{r_{\rm cl}}{\rm d}b\,\,b\,\frac{{\rm
d}\Gamma}{{\rm d}t_{\rm E}}(b,t_{\rm E})$ (8) $\displaystyle\times$
$\displaystyle\int_{0}^{2\pi}{\rm
d}\varphi\,\,\Sigma_{\star}\left(\textbf{q}_{\rm
cl}-\textbf{q}(b,\varphi)\right)\;,$

where $\textbf{q}_{\rm cl}$ denotes the position of the cluster on the
celestial sphere and $\textbf{q}(b,\varphi)$ the displacement corresponding to
given values of $b$ and $\varphi$. This differential rate translates into the
expected number of lensing events induced by the $i^{\rm th}$ cluster:

$\displaystyle\bar{N}_{{\rm e},i}=t_{\rm obs}\,\int{\rm d}t_{\rm
E}\,\,\varepsilon(t_{\rm E})\times\frac{d\Gamma_{{\rm cl},i}}{dt_{\rm
E}}(t_{\rm E})\;\;.$ (9)

Finally, we draw the number of events due to each cluster, $N_{{\rm
e},i}\sim{\rm Pois}(\bar{N}_{{\rm e},i})$, and sum them up to obtain the total
number of events in a given simulation run:

$\displaystyle N_{\rm obs}=\sum_{i=1}^{N_{\rm cl}}N_{{\rm e},i}\;.$ (10)

Note that the stochastic distribution of PBH clusters in the observational
cone and the uneven distribution of target stars on the celestial sphere lead
to $N_{\rm obs}$ that does not trivially follow a Poisson distribution. This
explains the need for performing dedicated MC simulations.

####  IV.1.2 Results

For concreteness, here we apply our method to study the distribution of
microlensing events in the EROS-2 survey of the LMC [18]. In particular, we
adopt the corresponding target star distribution and detection efficiency,
while for the Milky Way DM halo we assume the same cored DM halo model – for
details see [18].

The MC simulations allow us to determining the probability distribution for
the total number of observed events, $P_{\rm obs}(N_{\rm obs})$, which we
report in Fig. 1 for various $M_{\rm cl}$ while keeping $r_{\rm cl}=10\,{\rm
pc}$ 111In the scenario where PBHs are formed from Gaussian curvature
perturbations, the adopted value of $r_{\rm cl}$ corresponds to clusters with
$M_{\rm cl}\sim 10^{4}\,M_{\odot}$, while more massive clusters are expected
to be larger and less dense, since $r_{\rm cl}\propto M_{\rm cl}^{5/6}$.
However, for $M_{\rm cl}\gtrsim 10^{7}\,M_{\odot}$ this would imply extremely
large clusters with high chance of spatial overlap, and, therefore, we
conservatively keep $r_{\rm cl}=10\,{\rm pc}$ for all the considered values of
$M_{\rm cl}$. Note that despite this choice even clusters with $M_{\rm
cl}=10^{8}\,M_{\odot}$ still acts as extended clusters.. First thing to note
is that the mean number of events, $\bar{N}_{\rm obs}\approx 23$, is always
the same, regardless of the cluster mass, as it only depends on the total
number of PBHs within the probed volume, i.e., $M_{\rm l.o.s.}/M_{\rm PBH}$.
From the same figure it can also be seen that $P_{\rm obs}(N_{\rm obs})$ is
virtually indistinguishable from the standard result obtained for unclustered
PBH if $M_{\rm cl}\lesssim 10^{6}\,M_{\odot}$. This comes from the fact that
for a large number of individual clusters, fluctuations about individual
l.o.s.’s average out and $P_{\rm obs}(N_{\rm obs})$ simply reduces to Poisson
distribution (for $M_{\rm cl}=10^{6}\,M_{\odot}$ one expects $N_{\rm cl}\sim
1300$ within the field of view). However, for higher cluster masses, $P_{\rm
obs}(N_{\rm obs})$ becomes increasingly skewed toward low $N_{\rm obs}$, but
with a thin tail reaching much larger values. For the highest PBH cluster mass
considered in this work, $M_{\rm cl}=10^{8}\,M_{\odot}$, $P_{\rm obs}(N_{\rm
obs})$ peaks at $N_{\rm obs}=8$, but has an extremely long tail, stretching to
hundreds of events. This would lead to very different exclusion limits on the
allowed fraction of DM in the form of PBHs. For example, one could not exclude
(at 95% confidence level) DM being fully made of PBH clusters of masses
$M_{\rm cl}=10^{8}\,M_{\odot}$ if more than 3 microlensing events were
detected, while for smoothly distributed PBH the threshold value shifts to 15.
This clearly demonstrates the importance of taking into the account
correlations along neighboring l.o.s.’s in the case of PBH clusters with
$M_{\rm cl}$ approaching $M_{\rm l.o.s.}$.

While the presented results were obtained for an EROS-like survey of the LMC,
we wish to emphasize that analogous conclusions are applicable also to other
microlensing surveys aimed toward the Magellanic Clouds, as well as the
Andromeda galaxy. Indeed, extended PBH clusters can significantly affect
$P_{\rm obs}(N_{\rm obs})$ only if their individual masses represent a large
fraction of the expected DM mass within the observational cone. Otherwise, the
fluctuations in the expected number of the events along individual l.o.s.’s of
each realization average out and the predicted signal becomes very similar to
that of smoothly distributed PBH.

Figure 1: Probability distribution functions for the number of observed
microlensing events $N_{\rm obs}$ in EROS-2-like survey of the LMC. The black
curve shows the standard result assuming smoothly distributed PBHs, while
color curves represent our results for PBHs bound in clusters of different
masses. The vertical red line shows the mean number of events, $\bar{N}_{\rm
obs}$, which is independent of PBH clustering.

###  IV.2 Recurrence of events

Microlensing events are extremely rare due to the smallness of the volume that
has to be crossed by a compact object to produce an observable amplification.
Therefore, multiple microlensing events from the same source are
conventionally attributed to its intrinsic variability rather than
microlensing. This view has been challenged in several works proposing that
highly clustered PBHs could lead to recurring amplifications along the same
l.o.s. [20, 21]. In the following, we critically reexamine this hypothesis,
first focusing on the effect of small separation distances between individual
PBHs in extremely compact clusters and, subsequently, on the possibility of
recurring events due to PBH binaries, which are expected to be efficiently
formed in the early Universe.

####  IV.2.1 Small PBH separation distances

For dense PBH clusters, the dynamical time is typically orders of magnitude
smaller than their ages. Therefore, they should be virialized and the total
energy of the cluster can be expressed as $E_{\rm cl}=-NM_{\rm pbh}v_{\rm
cl}^{2}/2=-G(N)^{2}/(4r_{\rm cl})$ [42], where $v_{\rm cl}$ is root-mean-
square velocity of an individual PBH of mass $M_{\rm pbh}$, and $N$ is the
number of PBHs within the cluster. This gives $v_{\rm cl}^{2}=GNM_{\rm
pbh}/(2\,r_{\rm cl})$. On the other hand, the separation distance between the
PBH can be estimated as $\Delta r\approx(V_{\rm cl}/N)^{1/3}=r_{\rm
cl}(4\pi/(3N))^{1/3}$. From this, the typical time required for a PBH to cross
$\Delta r$ reads:

$\displaystyle\Delta t\approx\frac{\Delta r}{v_{\rm
cl}}=\left(\frac{4\pi}{3N}\right)^{1/3}\left(\frac{2r_{\rm cl}^{3}}{GNM_{\rm
pbh}}\right)^{1/2}\;.$ (11)

By adopting a conservative approximation, $r_{\rm cl}\geq\sqrt{N}R_{\rm
E}(M_{\rm pbh})$, for which the extended-cluster condition is only marginally
fulfilled, we get:

$\displaystyle\Delta t\gtrsim 9.8\;{\rm yr}$
$\displaystyle\left(\frac{x(1-x)}{0.5}\right)^{3/4}\left(\frac{D_{\rm
s}}{55\;{\rm kpc}}\right)^{3/4}$ (12) $\displaystyle\times\left(\frac{M_{\rm
pbh}}{M_{\odot}}\right)^{1/4}\times\left(\frac{10^{4}}{N}\right)^{1/12}\;,$

where222Without impacting our conclusions below, for less dense clusters
$v_{\rm p}$ should be taken in Eq. (11). $x=D_{\rm l}/D_{\rm s}$. While this
is comparable to the duration of a typical microlensing survey, it is
important to note that most of the events would not produce light curves
consistent with Eq. (1) if $r_{\rm cl}\sim\sqrt{N}R_{\rm E}(M_{\rm pbh})$, as
the light rays would likely be affected by multiple PBHs. Therefore, in
reality, $\Delta t$ has to be significantly (orders of magnitude) larger than
the above value, making recurring events due to small PBH separation distances
very unlikely, even for clusters consisting of an overwhelming number of sub-
Solar mass PBHs.

####  IV.2.2 Binary PBHs

PBH binaries are efficiently formed in the early Universe around matter-
radiation equality. A large fraction of these binaries survive three-body
interactions in clusters [29, 30, 32] and could potentially lead to recurring
events [20, 21, 40]. We can derive a lower bound on the time separation of
microlensing events under the assumption that one of such pairs happens to lie
along a given l.o.s. aligned with its orbital plane. The latter can be simply
estimated as one half of the orbital period of a two-body system:

$\displaystyle\Delta t=\frac{T}{2}=\pi\sqrt{\frac{a^{3}}{G\mu}}\;,$ (13)

where $a$ is the orbital semi-major axis and $\mu$ is the reduced mass of the
PBH. To minimize $\Delta t$, we assume that both PBHs have the same mass,
$\mu=M_{\rm pbh}/2$, while we set $a\geq 2R_{\rm E}(M_{\rm pbh})$, so that the
single-lens assumption in Eq. (1) is still marginally applicable. This yields:

$\displaystyle\Delta t\geq 116\;{\rm yr}$
$\displaystyle\left(\frac{x(1-x)}{0.5}\right)^{3/4}\left(\frac{D_{\rm
s}}{55\;{\rm kpc}}\right)^{3/4}$ (14) $\displaystyle\times\left(\frac{M_{\rm
pbh}}{M_{\odot}}\right)^{1/4}\;.$

This is again very conservative, since only a fraction of PBHs should be in
the form of binaries with the orbital plane aligned along the l.o.s. and,
additionally, one should have $a\gg 2R_{E}$ to observe light curves that are
not distorted by the presence of the second object (such anomalies have
already been detected in a binary BH lens [49]). Therefore, we conclude that
PBH binaries are also very unlikely sources of recurring events, even for
$M_{\rm pbh}\ll M_{\odot}$.

##  V Summary and conclusions

In this work we have carefully examined the impact of PBH clustering on the
interpretation of microlensing searches for MACHOs. Heuristically, it has
often been argued that microlensing constraints on DM in the form of PBHs can
be significantly relaxed, or even entirely removed, if one accounts for the
fact that PBHs are bound into clusters. However, our dedicated MC simulations
of EROS-2-like experiment show that microlensing limits can be lifted only if
clusters are extremely compact, $r_{\rm cl}\ll R_{E}(M_{\rm cl})$, or very
massive, $M_{\rm cl}\gtrsim 10^{8}\,M_{\odot}$. In the most commonly
considered scenarios, where PBHs are formed through the collapse of Gaussian
primordial curvature perturbations, both of these conditions are far from
being fulfilled. Similar conclusions also hold for other microlensing surveys
of the Magellanic Clouds, as well as of the Andromeda galaxy. While in models
that predict PBH formation from other initial conditions or through
topological defects it may be possible to produce sufficiently dense PBH
clusters to avoid the microlensing limits, further work is needed to establish
their survival to the present days.

Similarly, it has been argued that PBH clustering can lead to recurring
microlensing events which have been spuriously discarded from the
observational samples. However, using simple analytical estimates, we have
shown that recurring microlensing events due to dense PBH clusters or PBH
binaries should occur on significantly longer timescales than the typical
duration of microlensing surveys. Therefore, it is unlikely that any of the
observed recurring events were in fact of a PBH origin.

###### Acknowledgements.

We would like to thank Sébastien Clesse for interesting discussions during the
preparation of this work, as well as Anne Green and Matthew Gorton for
valuable feedback regarding the first e-print version of this work. We
acknowledge support from the ANR project ANR-18-CE31-0006 (GaDaMa), the
national CNRS-INSU programs PNHE and PNCG, and European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreement No 860881-HIDDeN. MP acknowledges partial support MP acknowledge
partial support from the ARRS grant no. P1-0031. The authors are grateful for
the computational resources and technical support provided by the IT
department of LUPM.

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Herbert Dingle and
“Science at the Crossroads”

Taha Sochi (Contact: ResearchGate)

London, United Kingdom

Abstract: In this article we pay tribute to Herbert Dingle for his early call
to re-assess special relativity from philosophical and logical
perspectives.[1] [1] [1] This tribute (which was originally a collection of
notes) was due to be published in 2022 which is the jubilee of Dingle’s book
“Science at the Crossroads” [1]. However, there were delays in editing and
finalizing this article (due to my busy schedule in the last two years).
However, we disagree with Dingle about a number of issues particularly his
failure to distinguish between the scientific essence of special relativity
(as represented by the experimentally-supported Lorentz transformations and
their formal implications and consequences which we call “the mechanics of
Lorentz transformations”) and the logically inconsistent interpretation of
Einstein (which is largely based on the philosophical and epistemological
views of Poincare). We also disagree with him about his manner and attitude
which he adopted in his campaign against special relativity although we
generally agree with him about the necessity of impartiality of the scientific
community and the scientific press towards scientific theories and opinions as
well as the necessity of total respect to the ethics of science and the rules
of moral conduct in general.

Keywords: Special relativity, Herbert Dingle, Lorentz transformations,
relativistic mechanics.

###### Contents

  1. 1 Introduction
  2. 2 The Concerns of Dingle
    1. 2.1 The Clock (or Twin) Paradox
    2. 2.2 The Role of Mathematics in Science
    3. 2.3 The General Attitude of Scientific Community
    4. 2.4 The General Attitude of Scientific Press
    5. 2.5 The Ethics of Modern Science
    6. 2.6 The Transformations of Lorentz
    7. 2.7 The Danger of Wrong Theory
  3. 3 The Approach and Attitude of Dingle Himself
  4. 4 Conclusions

##  1 Introduction

The controversy about the theory of special relativity dates back to the early
days of its appearance, where these controversies took initially the form of
“paradoxes” (e.g. the twin paradox, the barn-pole paradox and the twirling
pole paradox). Most of these controversies (as mostly formulated and embedded
in “paradoxes” and similar arguments) were targeting the logical foundations
and epistemological implications of special relativity. The essence of most of
these arguments and paradoxes against the theory is the apparent contradiction
between the requirement of the principle of relativity which denies the
existence of any privileged frame of reference (in uniform relative motion)
and the requirement of certain effects and implications of the theory to have
such a privileged frame of reference.

Although the majority of scientists (representing the mainstream physics)
accepted special relativity with many of them advocating it vigorously and
enthusiastically, there were a few prominent scientists who opposed it or
expressed their discomfort and doubt about it due to the aforementioned
logical and epistemological issues. One of these opponents to special
relativity was the English physicist and philosopher of science Herbert Dingle
who dedicated his last years trying to convince the scientific world that
special relativity is logically inconsistent and hence it is scientifically
wrong (and even dangerous). His effort (which is vividly documented in his
book “Science at the Crossroads” [1]) mostly took the form of communicating
with fellow scientists (either directly or indirectly through publication of
articles in scientific journals) trying to convince them of his views and
convictions which were largely based on the famous clock (or twin) paradox.

In this article we try (within our tribute to Herbert Dingle) to highlight the
main points and issues (or concerns) of Dingle in his vigorous campaign
against special relativity where we also present our assessment and position
towards the two sides of this dispute (or debate) noting that we agree with
Dingle on certain issues and disagree with him on other issues although we
generally sympathize with his views and campaign against special relativity
(which should explain our desire to pay tribute to him). In fact, we believe
that Dingle was not fairly treated and his views on certain issues deserve
better attention and reception from the scientific community of his time (as
well as the next generations of scientists including our present generation).

Our view about special relativity is outlined in our paper [2] and discussed
in detail in our book “The Mechanics of Lorentz Transformations” [3]. So, we
have no intention to go through these details in the present paper (i.e. we
advise the interested readers to refer to these documents for details).
However, we briefly summarize our view in the following paragraph. Our main
point against both parties (i.e. the proponents and the opponents of special
relativity) is that they both fail to distinguish between the formalism of the
theory (which is its real scientific essence represented by the Lorentz
transformations with their formal consequences and implications) and the
Einsteinian interpretation (which essentially represents the philosophical and
epistemological views of Poincare which Einstein adopted possibly with some
modifications that resulted in these logical inconsistencies).

In brief, we distinguish between the logically inconsistent theory of special
relativity as an epistemological interpretation of the scientifically sound
formalism of Lorentz transformations (which is the authentic scientific
essence of the so-called “special relativity”). So, while we accept the
formalism of the theory (which in our view is no more than the experimentally
supported Lorentz transformations) we reject the epistemological and
philosophical content of the theory since we regard it as an interpretation to
the formalism. In this regard, we compare special relativity to the
interpretations of quantum mechanics (such as the Copenhagen interpretation
and the hidden-variable interpretation), and compare Lorentz transformations
(with all their implications and consequences which form the formalism of the
so-called “special relativity”) to the formalism of quantum mechanics.[2] [2]
[2] The duality of formalism-interpretation of scientific theories is fully
investigated and discussed (within the context of quantum mechanics) in our
book “The Epistemology of Quantum Physics” [5] which the interested readers
should refer to. In fact, this is the motive behind our labeling of “special
relativity” as “the mechanics of Lorentz transformations” (which is the title
of our book that we referred to earlier).

##  2 The Concerns of Dingle

In the following subsections we outline the main concerns (which are mostly
presented in the form of arguments, challenges, warnings, etc. or embedded
within such forms) that Dingle presented in his campaign against special
relativity (as depicted and outlined in his book “Science at the Crossroads”
[1]).

###  2.1 The Clock (or Twin) Paradox

This is the most famous of all “paradoxes” that were invented and proposed to
challenge the theory of special relativity. In fact, it was proposed (and
actually was the subject of discussion and controversy in the scientific
community which includes some of the most prominent physicists of their time)
since the early days of the appearance of special relativity. This paradox is
very well known and it is fully explained and discussed in many textbooks
about the relativity theories (see for instance [6, 3]) and hence we are not
going to present it or discuss it in this short paper.

However, this paradox is at the heart of Dingle’s arguments against special
relativity (as presented in detail in his book) and hence we should outline it
from Dingle’s perspective. The following excerpt from Dingle’s book outlines
this paradox (and should actually summarize his main challenge to special
relativity):[3] [3] [3] In fact, this excerpt is just a sample representing
other similar excerpts that can be quoted to represent Dingle’s argument and
viewpoint.

According to the theory, if you have two exactly similar clocks, A and B, and
one is moving with respect to the other, they must work at different rates (a
more detailed, but equally simple, statement is given on pp. 45-6, but this
gives the full essence of the matter), i.e. one works more slowly than the
other. But the theory also requires that you cannot distinguish which clock is
the ‘moving’ one; it is equally true to say that A rests while B moves and
that B rests while A moves. The question therefore arises: how does one
determine, consistently with the theory, which clock works the more slowly?
Unless this question is answerable, the theory unavoidably requires that A
works more slowly than B and B more slowly than A - -which it requires no
super-intelligence to see is impossible. (End of quote)

We totally agree with Dingle about his challenge to special relativity from
this perspective. In fact, we have discussed this in detail in our book [3].
However. this should be a challenge to the epistemological interpretation of
special relativity and not to the formalism of Lorentz mechanics (as Dingle
apparently wants). In fact, there is nothing in the entire formalism of Lorenz
mechanics (i.e. neither in the Lorentz transformations nor in their direct or
indirect formal consequences and implications) that indicates such a logical
absurdity or allows it. So in brief, although Dingle is completely right in
his challenge to this logical and epistemological aspect of special
relativity, he should be wrong in his seeming attempt to discredit the
formalism of Lorentz mechanics (which is the scientific essence of what is
called “special relativity”). Also see $\S$ 2.6.

###  2.2 The Role of Mathematics in Science

We quote in this regard the following excerpt from Dingle’s book (referring
the readers to his book for more quotes like this):

What I believe to be the basic misconception of modern mathematical physicists
— evident, as I say, not only in this problem but conspicuously so throughout
the welter of wild speculations concerning cosmology and other departments of
physical science – is the idea that everything that is mathematically true
must have a physical counterpart; and not only so, but must have the
particular physical counterpart that happens to accord with the theory that
the mathematician wishes to advocate. (End of quote)

We generally agree with Dingle’s view about the role of mathematics in
physical sciences and his opposition to the excessive mathematization of
modern science.[4] [4] [4] In fact, we disagree with Dingle about some details
with regard to this issue. Science, after all, is an observational-
experimental enterprise and hence it should rely firstly and mostly on first
hand observation of nature and experimenting on it. Yes, the behavior of
nature can generally be mathematized through modeling and formulation but this
does not mean that nature necessarily follow our mathematical models and
formulations (at least in their entirety and with all their consequences). So,
the right course of action is to observe and experiment first, and this should
be followed by trying to put the results of our observations and experiments
in mathematical models and forms (if possible).

The danger of excessive mathematization of science is that it leads to
illusions and fantasies since mathematics is full of artifacts and abstract
objects that do not correspond to physical reality (e.g. we have imaginary
solutions, singularities, infinities, senselessly-negative quantities, etc.).
In fact, the obsession of modern scientists with mathematics and
mathematization has led to some disastrous consequences such as wasting huge
resources on investigating trivial or illusory things and the emergence of
bogus branches of science and scientific thinking. The theories of relativity
(especially general relativity) should take part of the blame for these
dangerous trends in modern science since they are over-mathematized or require
excessive mathematization to digest some of their nonsensical consequences and
implications.[5] [5] [5] We refer the readers to our books about the
relativity theories (see [3] and [4]) for more details.

###  2.3 The General Attitude of Scientific Community

This is another concern of Dingle that we generally agree with. In brief,
there is a considerable amount of prejudice and discrimination among the
scientific community when it comes to the assessment and treatment of
scientific theories and opinions where some are favored on the basis of non-
scientific criteria and non-scholar grounds, and this is particularly true
with regard to the relativity theories which enjoy special treatment in which
the scientific and professional standards and rules (including some ethical
codes and principles) are generally ignored or marginalized or sidelined due
to the magic influence of Einstein and the huge propaganda in favor of him and
his theories and views (which sometimes lead even to intimidation and bullying
as well as other forms of thuggish behavior).[6] [6] [6] The following extract
from Dingle’s book outlines this issue: The fact that, nevertheless, a request
for assistance in restoring integrity in science can be read as a request to
‘try to distinguish between the two sides’ on a particular scientific point I
can only regard as one more example of the evil spell cast by the word
‘relativity’ — a word that immediately reduces the mental power of even
leading physicists to impotence and is the greatest stumbling-block to my
efforts to bring home to them the extreme seriousness of the state to which we
have been reduced. Apart from that word-magic, there is nothing in the whole
course of events which I related which might not have happened if
‘crystallography’ had been substituted for ‘relativity’: it is just a
historical accident that Einstein’s theory caused, or showed up, the
corruption. (End of quote)
We also quote the following extract: Science no longer refuses to tolerate the
neglect of any anomaly; it refuses to tolerate anything but neglect of a most
outstanding anomaly. It no longer fears only prejudice and preconception; it
fears to the point of terror a particular threat to its prejudices and
preconceptions, and does everything in its power to suppress such a threat.
Its criteria of truth — if that word can still be used in connection with it —
are no longer reason and experience, but strict conformity to a theory … etc.

In fact, the episode of Dingle in his campaign against special relativity
should highlight the dogmatic attitude of the supposedly enlightened
scientific community towards certain “untouchable” elements in science (or
“holy” beliefs) which makes discussion or opposition to these elements a
“taboo” and could lead to the “excommunication” of whoever dares to challenge
them. This “religious” component in modern science is one of the many
disgraces and shames of modern science and should be addressed. This dogmatic
attitude is reflected in many aspects and activities of science and
scientists, e.g. in the choice of teaching and research topics, in the
determination of what should and should not be accepted for publication in the
scientific press, in the giving of grants and awards, in the allocation of
honors and prestige, and so on. Although this unscientific attitude is not
limited to special relativity, special relativity is one of the prominent
examples of “holy” elements that enjoy special treatment in modern science
(since it has a “special place” in the heart of the mainstream scientists).

However, we should note that the situation has improved in modern times
(compared to the time of Dingle) with the emergence of more theories,
experiments, observations, individual scientists, opinions (… etc.) which are
not in favor of special relativity and its principles and propositions. For
instance, the emergence of Bell’s theorem in quantum mechanics (and what
followed of quantum entanglement experiments) shook some of the foundations of
special relativity. Also, novel experiments and observations related to the
speed of light put some question marks on the dogmatic belief of special
relativity about the speed of light as the ultimate and invariant physical
speed.

###  2.4 The General Attitude of Scientific Press

This is another concern of Dingle that we generally agree with. Scientific
press (especially the prestigious ones) generally welcomes only certain types
of papers which are supportive of the mainstream (or orthodox) theories and
opinions and rejects (generally without consideration) any paper that
advocates or sympathizes with opposite (or non-orthodox) theories and
opinions, and this is particularly true when it comes to the relativity
theories where any paper that questions these theories or challenges them is
usually (and automatically) rejected as a form of heresy or pseudo-science or
crackpot idiocy.

The obvious result is that only certain views and trends in science are heard
and sponsored and hence only certain views and trends are encouraged to grow
and prosper since the opposite views and trends will naturally die or weaken
because very few scientists have the determination to continue their work when
this work has no chance (or very little chance) to be in the public space.
This form of unnatural selection is not only against the spirit and ethics of
science but it is also against the interest of science and humanity since
valuable ideas and contributions can be lost as a result of this form of
unfair discrimination.

However, the situation generally improved in the recent times thanks to the
emergence and wide availability of pre-print repositories and social networks
(professional as well as non-professional) which enable anyone to publish his
work. However, the attention and seriousness that published work enjoys when
it is published in the mainstream and prestigious media (e.g. the high impact
journals) still favors views and opinions that represent and accept (or inline
with) the mainstream science, and this is certainly a type of prejudice and
favoritism that should be addressed and fixed.

As indicated earlier, this discriminatory attitude is not limited to the
scientific press but it can be seen and felt in every department of modern
science such as awards, grants, academic posts, scientific venues (like
conferences) and so on. In fact, the attitude of scientific press is just one
of many forms and examples of discrimination and unfair treatment that
characterize all the departments of modern science (noting that we limited our
attention to the attitude of scientific press because that is what concerned
Dingle in his book).

###  2.5 The Ethics of Modern Science

One of the most important (and “most true”) concerns of Dingle (which he
documented in detail in his book, sometimes explicitly and directly and
sometimes implicitly and indirectly) is the lack of transparency and the
absence of respect to the rules of ethics and morality in many practices of
modern scientists. Unfortunately, science in modern times (and possibly even
in old times) seems to be no more than a way for making a living and getting
financial gains, reputation, prestige, awards, and so on. So, the search for
truth and service to humanity (which are supposed to be, or at least should
be, the main purpose of science) are of no concern to many (if not most)
scientists.

In my view, the dodgy practices and unethical behavior of scientists of the
kind indicated and highlighted by Dingle is just one type of the many types of
immorality (and possibly the slightest type of immorality) of modern science
and scientists. In fact, modern science and scientists are embroiled in many
other types of immorality (and even criminality) such as experimenting on
animals (or rather torturing and killing them in the name of science) and
developing lethal weapons (especially weapons of mass destruction and weapons
designed to target civilians or necessarily damage civilians and civilian
infrastructures). So, despite the alleged ethical codes and rules of morality
which are supposedly adopted by “respected” scientific institutions and
bodies, the reality is very different especially when we note that even if all
these alleged codes of conduct are implemented literally they do not satisfy
even the minimum requirements of real ethical conduct and moral behavior
because immorality and criminality are at the very essence of some of these
“scientific” practices and disciplines. Anyway, this is not our main concern
in this paper (but we intend to come back to this issue in the future).

###  2.6 The Transformations of Lorentz

Dingle seems to reject not only special relativity as an interpretation
(demonstrated for instance in some of its epistemological aspects and
consequences) as we do, but he rejects even its formalism. Although we do not
believe that the formalism of Lorenz mechanics (which is based on the
transformations of Lorentz as postulates in our view) is a final theory and it
may not be a complete theory, we believe that the accumulated observational
and experimental evidence should qualify this formalism to be regarded as an
accepted scientific theory with the rejection of the illogical epistemological
interpretation that is attached to it (as represented by special relativity).

In fact, Dingle seems to put question marks even on Maxwell’s electrodynamic
theory which proved to be very successful over its long history (in theory as
well as in application and practice). This view of Dingle is understandable
when we note the intimate relationship between Lorentz transformations and
Maxwell’s electrodynamic theory. Although it is possible and acceptable (in
principle) to replace Maxwell’s electrodynamic theory within a “global” and
radical change (or rather revolution) in modern physics, it seems unwise (as
well as impractical) to target Maxwell’s theory initially and in this manner
because of a logical inconsistency in special relativity. In other words, we
can easily dismiss special relativity and get rid of it without sacrificing
the formalism of Maxwell’s equations or the Lorentz transformations, and this
should not only be the easiest (or even the very easy) thing to do but also
the most wise option. In fact, it is the option that is most consistent with
the spirit and tradition of science and how it should develop and progress.

###  2.7 The Danger of Wrong Theory

The following excerpt from Dingle’s book should summarize one of his major
concerns about special relativity:

The second reason for the publication of this book is a practical one.
Directly or indirectly - at present chiefly the latter, though none the less
inseparably - special relativity is involved in all modern physical
experiments, and these are known to be attended by such dangerous
possibilities, should something go wrong with them, that the duty of ensuring
as far as possible that this shall not happen is imperative. It is certain
that, sooner or later, experiments based on false theories will have
unexpected results, and these, in the experiments of the present day, may be
harmless or incalculably disastrous. In these circumstances an inescapable
obligation is laid on experimental physicists to subject their theories to the
most stringent criticism. As this book will show, their general practice is to
leave such criticism to mathematical theorists who either evade or ignore it,
and the possible consequences are evident and unspeakably menacing. This alone
would compel the publication of the facts here revealed. (End of quote)[7] [7]
[7] We should refer the readers to Dingle’s book [1] for more quotes in this
regard.

In fact, this is one of the strangest claims of Dingle because if “special
relativity” is a scientifically correct theory then it should already be
“tested” in the laboratory of Universe and hence any new (human-made)
experiment based on it should not lead to such disastrous consequences (as
Dingle claims) more than any other experiment (based on other correct theory)
do. On the other hand, if “special relativity” is a scientifically incorrect
theory then any danger from experiments based on it should be an illusion
because the theory should not lead to the consequences that the theory is
supposed to produce. So, any danger from such experiments should originate (if
so) from other sources of danger which all scientific experiments share (i.e.
any scientific experiment can go wrong because of its bad design or because of
its bad theoretical formulation or because of many other things like these).
In fact, someone may argue exactly the opposite by claiming that false
theories should be safer than correct theories because the expected effects of
false theories should be null (or void).

##  3 The Approach and Attitude of Dingle Himself

Although we appreciate the enthusiasm of Dingle in his campaign against
special relativity and his attempt to convince and persuade the scientific
world of its falsehood and “danger”, we generally disagree with him about the
approach that he followed and the attitude that he adopted in this campaign.
We note for example that he personalized the dispute in some occasions and
gave it the taste and flavor of a fight instead of keeping it as a theoretical
scientific debate. Moreover, he was relentlessly insistent and repetitive in
his demands trying to force his views and convictions and impose his wishes on
other people who disagree with him or do not share his views and convictions.

However, we should forgive and excuse Dingle in this regard considering that
special relativity in his view was not only a theoretical or logical “danger”
to science but it is a physical danger to humanity and its physical safety and
existence. The factor of age should also be considered and respected since a
man of his age (at that time) can easily lose patience and temper and become
somewhat detached from reality. In my view, the episode of Dingle in his
campaign against special relativity should be seen as an amusing and
entertaining story that should add a touch of humor and humanity on “cold” and
“emotionless” science instead of being treated as an example of naughtiness
and misbehavior and remembered with bitterness and discomfort (as reflected in
the attitude of some opponents of Dingle whether those who were engaged in
that dispute or those who expressed their opinion about it later on).

Anyway, Dingle’s book contains a considerable amount of nonsense and useless
arguments. It was better for him to present his views about special relativity
(as well as about the integrity of science, scientists, scientific press, the
ethics of science and so on) in a more professional way and academic manner
instead of this talkative and vulgar way.

##  4 Conclusions

We summarize the main issues discussed or implicated in the present article in
the following bullet points:

$\bullet$ In our view, “special relativity” (as it is commonly known) is a
combination of the experimentally-endorsed formalism of Lorentz
transformations and the logically-inconsistent interpretation of Einstein
(which originally belongs to Poincare with the possibility of Einstein
introducing some modifications or elaborations on it).[8] [8] [8] In fact,
Einstein’s version of the theory may not represent the views of Poincare
exactly (since we have no sufficient knowledge about some details of
Poincare’s views). So, it is possible that the views of Poincare may not be
logically inconsistent or they may not share the same logical inconsistencies
as those in the interpretation of Einstein. In other words Einstein’s version
of “special relativity” may not be exactly the same as Poincare’s version of
“special relativity”. So, while we criticize and reject the logically-
inconsistent interpretation of Einstein we think that the formalism of the
theory (which we call “the mechanics of Lorentz transformations”) is generally
acceptable theory (although it is possibly a limited and interim theory).

$\bullet$ In our view, the main failure in Dingle’s thesis is his failure to
distinguish between the formalism of special relativity (which essentially is
no more than Lorentz transformations with their implications and consequences)
and its interpretation as an epistemological paradigm whose main purpose is to
make sense of the formalism. In other words, Dingle was unable to distinguish
between the scientific essence of the theory of relativity (i.e. the formalism
of Lorentz transformations) which is experimentally supported and the
philosophical (or epistemological) interpretation of Einstein which is
logically inconsistent.

$\bullet$ Despite Dingle’s failure to distinguish between the logically
inconsistent interpretation of special relativity and the scientifically sound
formalism of Lorentz transformations (which is the real scientific essence of
what is commonly called “special relativity”), he should still get the credit
for highlighting some important issues about the theory of special relativity,
and for this reason we pay our tribute to him in this article. Interestingly
and fortunately, there is more acceptance to the challenges to special
relativity these days than fifty years ago and Dingle has obviously
contributed positively to this shift in the attitude towards special
relativity.

$\bullet$ Dingle’s work and activities (which are mostly documented or
indicated in his book “Science at the Crossroads”) related to his opposition
to special relativity highlight a very important issue, that is the resistance
(with stubbornness) of the “mainstream” science and scientists to listen to
arguments and consider ideas and theories that run outside their “stream” or
against it. In fact, non-scientific (and even nasty) methods and tactics are
used (shamelessly) by some scientists to suppress such opposition voices and
this is especially true when it comes to the theories of relativity where
Einstein is commonly regarded as a “holy” figure or symbol of science.

$\bullet$ We generally agree with Dingle about several other issues which
include: the excessive mathematization of science, the general attitude of
scientific press (or rather the general attitude of almost all institutions
and departments of modern science), and the unethical practices in science.

$\bullet$ We reject the fears of Dingle about potential disaster happening as
a result of accepting special relativity and experimenting on it. We also
reject the depiction of his campaign against special relativity as a campaign
to protect the public and save humanity. Most of his fears were baseless (as
particularly related to special relativity) although warning of potential
dangers of any scientific theory of serious experimental impacts and
consequences is generally good.

$\bullet$ We also disagree with Dingle on many of his detailed accounts and
analyses. However, we did not go through these due to the limits on the size
of the paper and their relative insignificance (e.g. because they are out of
scope or because of their similarity with the issues that we already discussed
or indicated).

$\bullet$ There seems to be some confusion among the experts on relativity
(including some of those who communicated and engaged with Dingle) between
classical physics and classical logic where the rejection (or modification) of
some classical physical concepts (like Newtonian time and space) seems to
justify to those who reject these concepts to reject or sideline the rules of
logic which should be universally and unequivocally accepted and respected to
avoid nonsensical thinking and argumentation.

## References

  * [1] H. Dingle.  Science At the Crossroads.  Martin Brian & O’Keeffe, first edition, 1972.
  * [2] T. Sochi.  Special Relativity: Scientific or Philosophical Theory?  arXiv:1610.05640, 2016.
  * [3] T. Sochi.  The Mechanics of Lorentz Transformations.  CreateSpace, first edition, 2018.  ISBN: 9781727118483.
  * [4] T. Sochi.  General Relativity Simplified & Assessed.  Amazon Kindle Direct Publishing, first edition, 2020.  ISBN: 9798670614641.
  * [5] T. Sochi.  The Epistemology of Quantum Physics.  Amazon Kindle Direct Publishing, first edition, 2022.  ISBN: 9798844338182.
  * [6] R. D’Auria; M. Trigiante.  From Special Relativity to Feynman Diagrams.  Springer, second edition, 2016.