# Effects of extending residencies on the supply and quality of family
medicine practitioners; difference-in-differences evidence from the
implementation of mandatory family medicine residencies in Canada.

Stephenson Strobel

###### Abstract

I examine the impacts of extending residency training programs on the supply
and quality of physicians practicing primary care. I leverage mandated
extended residency lengths for primary care practitioners that were rolled out
over 20 years in Canada on a province-by-province basis. I compare these
primary care specialties to other specialties that did not change residency
length (first difference) before and after the policy implementation (second
difference) to assess how physician supply evolved in response. To examine
quality outcomes, I use a set of scraped data and repeat this difference-in-
differences identification strategy for complaints resulting in censure
against physicians in Ontario.

I find declines in the number of primary care providers by 5% for up to nine
years after the policy change. These changes are particularly pronounced in
new grads and younger physicians suggesting that the policy change dissuaded
these physicians from entering primary care residencies. I find no impacts on
quality of physician as measured by public censure of physicians. This
suggests that extending primary care training caused declines in physician
supply without any concomitant improvement in the quality of these physicians.
This has implications for current plans to extend residency training programs.

JEL: I10, I18, I23, I28 Keywords: physician training; supply of physicians;
medical residency choice

##  1 Introduction

There is interest in extending the length of primary care medical residencies
to increase the quality of candidates graduating into independent medical
practice. In North America, policy makers are considering extending family
medicine residencies by an additional year (Fowler et al.,, 2022; Carek,,
2013; Douglass,, 2021; Woolever,, 2021). However, there has been little
evaluation of their impact on the supply or quality of physicians.

Extending residency might have multiple mechanisms that affect physician
supply. There is an obvious mechanical which occurs in the first cohort that
has to train for an additional year. When the policy occurs this delays
graduation but this disappears once physicians complete the new residency and
graduate into independent practice. This “cohort” effect should have impacts
that occur with the policy play out over the approximate change in length of
the residency.

However, extending residency programs could have other impacts through
altering residency choice. An additional year of training makes other, more
lucrative training programs look relatively appealing. This could induce
physicians to substitute away from primary care training towards other
programs. This “substitution” effect could potentially have longer-run effects
but they would also occur with the onset of the policy.

Finally, longer residency lengths could also change the composition of
physicians entering primary care which could also have longer run supply
effects. If older physicians with shorter work-lives are more likely to choose
primary care programs this would mean fewer years of work due to earlier
retirement. In contrast with cohort and substitution effects which produce
effects coincident with the policy, these composition effects could produce a
more pernicious long run supply shock. These would occur at the end of a
physicians work-life and would not necessarily be observable in data.

Supply effects aside, the stated goal of these primary care residency
extensions is to improve quality. If these interventions do cause declines in
physician supply, this might be acceptable if there is a trade-off in improved
physician quality. However, there is also little evidence on quality changes
resulting from program changes.

My contribution is two-fold. I estimate the effects of residency training
extensions on the supply of physicians. I examine a natural experiment where,
over a period of 20 years, Canadian provincial governments mandated two year
family medicine residencies in lieu of one year internships for primary care
providers. Prior to 1994 medical school graduates had the option to complete a
one year rotating internship to become a general practitioner or complete a
two year family medicine residency. After 1994 all graduates had to complete a
family medicine residency in order to practice in primary care. Using a
difference-in-differences (DiD) identification strategy I compare the supply
of family medicine practitioners relative to graduates of other specialty
programs, like internal medicine, who did not have changes made to their
residency lengths (first difference). I use the province-by-province roll out
to assess pre-post differences across these groups (second difference).

I then evaluate impacts on quality of residency graduates. Using the same DiD
framework comparing specialists to family medicine practitioners I restrict my
focus to the province of Ontario. I use a scraped dataset of complaints that
result in censures from the College of Physicians and Surgeons of Ontario
(CPSO). These censures include terms and conditions imposed on the physician,
suspensions and revocation of licence by the CPSO. I can evaluate effects of
the change in residency program length on physician quality for over 20 years
after the intervention.

I find a 5.8% overall decline in overall numbers of family medicine
practitioners after family medicine residencies become mandatory. This occurs
over nine years suggesting that a cohort effect cannot fully explain these
changes. While these declines in supply disappear by 10 years this is largely
in cohorts of physicians who are older and who graduate later. There are
persistent declines in cohorts of physicians who have just graduated and in
younger physicians suggestive that the policy resulted in primary care
physicians that have shorter work-life spans. I find no impact on the quality
of physicians as measured by censure by the CPSO. Taken together these results
suggest that increasing the length of residencies decreases the supply of
independent physicians without any impact on quality of the graduates as
measured by complaints resulting in censure.

###  1.1 Previous Literature

I contribute to two major literatures. The first is a medical literature on
estimating the supply of physicians. Given long training times and expensive
private and public investment (Government of Canada,, 2009), most governments
and medical organizations project physician supply decades in advance for
planning purposes (Bureau of Health Professions,, 2006; Dall et al.,, 2021). A
significant amount of policy and academic discourse also centres around how to
retain and improve physician supply through incentives (Ahmed and Carmody,,
2020; Golden et al.,, 2012; Chan,, 2002; Marchildon and Di Matteo,, 2023).
These policies usually involve increasing residency and medical school spots
and incentives to retain independent physicians.

Despite the importance placed on securing physician supply, there has been
little evidence on how residency length impacts it. Most residents and
residency program directors prefer current residency lengths but would
consider extensions depending on the scope and content of additional training
(Duane et al.,, 2002; Smits et al.,, 2006; Gopal et al.,, 2007; Sabey and
Hardy,, 2015; Eiff et al.,, 2019). There is a large discourse consisting of
expert opinion predicting the effects of residency extensions (Raîche,, 2009;
Carek,, 2013; Hopson et al.,, 2016; Douglass,, 2021; Woolever,, 2021;
Glauser,, 2022).

There is a similar lack of evidence on quality impacts of longer medical
residencies. What does exist suggests no association between residency length
and baseline applicant quality (Eiff et al.,, 2019) and no association between
residency length and test scores for national certification exams (Waller et
al.,, 2017). Outside of medicine, there is some evidence that reducing high
school length leads to poorer post-secondary outcomes (Krashinsky,, 2009;
Meyer and Thomsen,, 2016) and lower wages suggesting poorer productivity
related to reduced training (Krashinsky,, 2014).

This paper is the first to demonstrate evidence that extending medical
residency programs causes decreases in the supply of physicians. These supply
effects play out over 10 years after the policy changes. Moreover, based on
complaints resulting in censure, there is no improvement in outcomes with
longer residencies.

This substitution away from primary care residency training links these
results to a more general economic literature concerned with intertemporal
supply of labour. This literature is based on early models of consumption
smoothing (Fisher,, 1937). It has also been extended to examine labour supply
over the lifetime which depends on how much agents value leisure, home
production and how much their wage changes (Mincer,, 1962; Altonji,, 1986;
Mulligan,, 1998). Most important to training program selection is that human
capital formation is predicted to fall with age because there is less payoff
to further investment (Ghez and Becker,, 1975).

From this perspective, additional training time in primary care will make it
look less attractive relative to other specialties whose training lengths
remain constant. Physicians who switch to take on longer training will be
concentrated in younger graduates because they stand the most to gain in
earnings over their work-lives. The closest literature that finds similar
phenomenon suggests earnings potential can impact choice of university major
(Arcidiacono et al.,, 2012) and can more modestly influence university choice
(Delavande and Zafar,, 2019).

However, very little evidence exists on how changes to medical training
programs influence specialty choice. Given public interest in physician
supply, it is important to understand how changes in policy directly effect
supply and how they may change composition in who becomes a primary care
physician. If older doctors are entering primary care in lieu of younger
physicians this means less years of labour supplied due to shorter work-life
spans. I find that evidence that this is the case.

##  2 Methods

###  2.1 Policy Context

Canadian medical students apply to residency programs through a centralized
matching service which allows them to rank their location and specialty
preferences (Lim and Khondker,, 2020). These include primary care specialties
as well surgical, diagnostic, and hospital based specialties. Physician
graduates would apply to all types of programs across Canada depending on
their location and specialty preferences. Prior to 1997, subspecialized
programs with limited positions like neurosurgery would participate in
specialized matches outside of the main match. Prior to 1994, medical schools
in Quebec did not participate in the centralized match (Banner,, 1993, 1995,
1997).

Provincial governments fund and create residency positions for medical
students. These are both based on historical and projected future needs. There
are other context specific reasons to fund residency spots such as emphasis on
French language physicians in Quebec and rural physicians in Northern Canada
(Dauphinee and Thurber,, 1997; Wilson and Bordman,, 2017). As a general rule
the number of residency positions exceeds the number of total medical
graduates although program or location specific restrictions exist. As an
example, during the 1995 match 101 positions across Canada went unfilled with
21 being in family medicine (Banner,, 1995).

Prior to 1994, medical students had two residency program routes to become
independent primary care practitioners. The first was completing a rotating
general internship of one year and then entering independent practice.
Alternately, a medical student could complete a family medicine residency
lasting two years. Both routes allowed physicians to practice independently
with the former route creating a general practitioner (GP) and the latter
route creating a family medicine specialist (FMD).

It is worth understanding the flexibility inherent in the rotating internship.
Prior to 1993, medical students could enter directly into a major speciality
like internal medicine or could spend a year in a rotating internship with a
similar first year schedule to all other residencies. This deferred choice for
a year and doctors who had completed the rotating residency could then re-
enter the match and apply for second year specialty positions. This provided a
year of experience for the physician to learn what specialties might be
suitable for them. In addition to re-entering the match, physicians had the
option of entering primary care practice without further training. Although it
continues to be possible to re-enter the match for another program after
completing a specialty, it became much less common after the rotating
internship was ended. From the period of 1985 to 1995, there was a 40%
reduction in the number of re-entry applicants to residency (Dauphinee and
Thurber,, 1997).

However, most provincial governments and the college of family physicians
considered a one year period of training after medical school inadequate.
Ending the rotating internship occurred on a provincial basis (Table 1) and by
1994, the rotating internship had concluded country-wide (Levitt and Klein,,
1991; Banner,, 1995; Chan,, 2002). These changes were grandfathered in for in-
practice primary care physicians so that general practitioners who graduated
before the policy change did not necessarily have to take additional training
and become family physicians.

###  2.2 Data

###  2.3 The Supply, Distribution and Migration of Physicians in Canada

I examine the impact of physician supply using provincial level data collected
by the Canadian Institute of Health Information. This is recorded annually for
each specialty type within a province. I exclude data prior to 1970 due to
documentation changes and data after 2003 because of large increases in
residency positions (Turriff et al.,, 2020).

This data contains information on the number of physicians and the physicians
per capita in each specialty. I am interested in the number of physicians per
capita that practice family medicine after the implementation of the policy. I
am able to distinguish between GPs who have completed the rotating general
internship and FMDs who completed a family medicine residency. I consider the
sum of these two categories as the number of primary care practitioners. These
data also contain information on the number of physicians by age cohort, and
graduation cohort which I exploit in extensions.

###  2.4 College of Physicians and Surgeons of Ontario Complaints

To examine the impact on complaints, I use data from the College of Physicians
and Surgeons of Ontario. The CPSO is the regulatory body that oversees the
professional behaviour of physicians within the province. The CPSO makes data
on all members public including any major actions against a physician that
result in censure. These data were scraped directly from the CPSO website in
2015 and they contain all individuals who have ever been registered from 1916
to 2015 including individuals who are no longer active.

These data include the specialty of the physician and any censure against the
physician. These censures, which are the outcome of interest, are whether a
license has had terms imposed on it, whether a license has been suspended, or
whether a license has been revoked. These do not include complaints which were
dismissed as these are not public information. In addition to these concrete
measures of a physicians quality, I also examine whether the physician has
resigned membership from the CPSO. This is occasionally used as a way to
settle a complaint without recording a negative outcome; however, there are
also benign reasons for resigning membership such as leaving the province to
practice in other jurisdictions.

All outcomes except this latter one are relatively rare in the sample of
individuals around the policy change (Table 4). Among the 30,823 physicians,
only 0.1% have had a license revoked, 1.9% have had terms and conditions
applied and 0.3% have been suspended. Family physicians are less likely to
have had terms and conditions applied to their license but more likely to have
resigned from membership.

###  2.5 Econometric Strategy

I use a regression adjusted difference-in-differences (DiD) estimator to
assess effects of implementation of residency changes on physician supply
(Sant’Anna and Zhao,, 2020; Callaway and Sant’Anna,, 2021). This approach
provides well defined treatment effect parameters as opposed to traditional
two-way fixed effects DiDs. I compare the primary care specialties affected to
all other physician specialties where training length did not change (first
difference). I make this comparison for 10 years pre-post based on the
province and the year the policy change occurred (second differences). I use a
sample of province-specialties such that

${(Y_{i,j,1},Y_{i,j,2}....,Y_{i,j,\tau},D_{i,j,1},D_{i,j,2}...,D_{i,j,\tau})}^{n}_{i=1}$
(1)

where $Y$ is the per-capita supply of physicians by specialty $i$ in province
$j$ in year $t$. A specialty-province is treated if $D_{i,j,t}=1$ in period
$t$. $G_{i,j,g}$ is equal to one when the province-specialty is treated at
time $g$ and is otherwise zero.

Using the never treated province-specialty groups as controls (ie. $C=1$), the
average treatment effect on the treated is estimated by:

$ATT^{nev}_{unc}(g,t)=E[Y_{t}-Y_{g-1}|G_{g}=1]-E[Y_{t}-Y_{g-1}|C=1]$ (2)

This average treatment effect is for treated province-specialties (ie. primary
care specialties) at time period $g$ in year $t$. These individual estimates
for a time $(g,t)$ are aggregated into a total overall effect for two, four
and ten years pre-post to estimate shorter and longer run effects of the
policy. I bootstrap standard errors because treatment occurs at the provincial
level and there is a few clusters issue (Cameron and Miller,, 2015; Roodman et
al.,, 2019).

DiD models assume that no other interventions occur simultaneously and that in
the absence of an effect, outcomes would have trended similarly. The latter
assumption can be tested by examining parallel trends in event analyses. Using
the above ATT, I can evaluate the impacts for specific points at specific time
points around adoption $g$. All estimates are relative to the year immediately
prior to implementation (ie. $g=-1$) and are plotted with 95% confidence
intervals. I limit the window to ten years before and after policy change.
DiDs are implemented via Rios-Avila et al. (Rios-Avila et al.,, 2022).

These events are important for two reasons. The first is a test on pre-trends
which provides evidence of causality. The second is that they suggest
mechanisms. If the effect of the policy change is purely through a mechanical
impact on the cohort of entering family physicians these should evolve over
the new period of residency length. If effects persist beyond this period, it
suggests that supply is impacted because of a decline in individuals entering
into family medicine residencies. This is consistent with a substitution
effect where physicians choose other specialties.

I use a similar econometric strategy to assess whether physicians receive more
complaints as a result of the increase in residency length in Ontario. This
collapses to a canonical DiD with one point of treatment in 1994. This is
specified as

$c_{ijt}=\beta_{i}(D_{ijt}*Post_{ijt})+G_{ij}+T_{t}+\epsilon_{ij}$ (3)

where a dummy variable, $c$, is equal to one if a complaint has resulted in a
public censure for a physician $i$ in specialty $j$, who graduated in year
$t$. $D$ is equal to one for specialties that are treated which are total
family practitioners, FMDs and GPs. $Post$ is equal to one for these
specialties in Ontario after implementation and zero for other specialties. I
am interested in the effects of $\beta_{i}$; this is the overall difference-
in-differences effect on physician quality. $G$ is a fixed effect for the
specialty group. $T$ is a fixed effect for the year of graduation. $\epsilon$
is a bootstrapped error term.

Event regressions are specified as

$y_{ijt}=\alpha_{i}\sum^{10}_{-10}(D_{ijt}*Post_{ijt})+G_{ij}+T_{t}+\epsilon_{ij}$
(4)

$\alpha_{i}$ estimates effects for each individual dummy variable. These
dummies take a value of one for family medicine related specialty groups in
the years after 1994 when there was implementation of mandatory family
physician training in Ontario. They are zero otherwise. All estimates are
relative to the year immediately prior to implementation (ie. $t=-1$) and are
plotted with 95% confidence intervals. I limit the window to ten years before
and after policy change.

###  2.6 Extensions: Age of physician and physician year of graduation

The overall impact that I estimate occurs across all physicians regardless of
age or graduation year. Changes to this overall number of physicians could be
from new graduates, retirees, or physicians who migrate. The CIHI data has
information on the age and age cohorts of family physicians in practice in a
province in a given year which is suggestive of who is affected. As the
demographics of physician graduates skew younger (Lakoff et al.,, 2020), if
residency classes are affected this should increase the average age of primary
care providers. To demonstrate that this overall supply effect is likely
coming through changes to graduating residents I repeat the above DiD
estimation strategy using average age as an outcome. Similarly, I examine the
supply of “young” primary care physicians as defined as those under 40, and
“old” family physicians as defined as those 60 and older. I am also able to
perform a similar exercise by year of graduation. If this effect comes through
changes in residency, cohorts who are closer to medical school graduation
should be more affected.

###  2.7 Extensions: Consistency of effects across the four interventions.

Residency choice is a mixture of specialty preferences and location
preferences. In provinces that adopt the policy earlier, medical school
graduates could leave the province to complete a shorter residency. To assess
whether main effects are consistent I examine the impact of the policy changes
by year of adoption. By the 1994 cohort, there are no other provinces where
graduates could move in order to avoid longer residencies.

###  2.8 Extensions: Substituting towards other specialties

If declines in physician supply are purely through a cohort effect, then
supply will be constrained for a limited period but eventually rebound.
However, if the effect is through a substitution effect, supply may be
constrained for a much longer time with more pernicious effects. Two results
might suggest that medical students are substituting towards other programs.
First, if changes in primary care physician volumes occur more than two years
after implementation (ie. the length of that residency), it suggests that
delayed cohort effects are not the only mechanism. Second, specialties that
attract similar candidates as family medicine should see increases in their
graduating residents.

There is little evidence on the alternate preferences for individuals who are
admitted to family medicine programs. However there is guidance from medical
student groups which suggest specialties that are similar to family medicine
in emphasis. These are internal medicine, pediatrics, neurology, psychiatry,
and physical medicine (Lim and Khondker,, 2020). I repeat the above DiD and
event exercise with these as the treatment groups.

###  2.9 Extensions: Altering the control group

Along similar lines as the previous extension, I examine effects when changing
the control group of specialties. I change this to a group of specialties that
are thought to have relatively low substitutability with family medicine.
These are specialties that have a diagnostic and surgical focuses.

##  3 Results

###  3.1 Effects on physician supply

Figure 1 demonstrates the time-series of selected types of physicians over the
period of 1970 to 2003. Around 1994 when rotating internships are completely
phased out in Canada, the number of GPs begins to decline from 75 per 100,000
population. The number off FMDs increases and in 1994 is approximately 26 per
100,000 population. The overall number of primary care practitioners in 1994
is approximately 100 per 100,000 population.

Table 2 demonstrates the DiD effects estimated by total primary care
practitioners, FMDs, and GPs. There is an decrease in the overall supply of
family medicine practitioners but at statistically insignificant levels. This
effect is negative at marginally statistically significant levels for shorter
time horizons but zero for a 10 year horizon. At the 10% level the policy
causes a 1.56 and 1.89 total primary care provider reduction per 100,000
persons at two and four years respectively.

Figure 2 demonstrates the estimates from event analyses by type of physician
per 100,000 population. Pre-trends in all graphs are noisy but relatively flat
and stable suggesting that the parallel trends assumption holds well. In the
period after mandating increased training there are large declines in the
number of GPs and large increases in the number of FMDs. However, there is a
transient trough in supply about five years after policy implementation when
the net effect is negative and statistically significant. That trough
demonstrates a reduction in the number of total family practitioners of 5-6
per 100,000 population. As an alternate specification, I demonstrate effects
on the physician supply irrespective of population in figure A.1. These
estimates are much noisier but demonstrate similar point reductions for five
years after implementation. This decline in the overall number of primary care
practitioners is an approximately 20 per year difference between the pre-post
periods.

###  3.2 Extensions

Figure 3 demonstrates effects on the age of primary care practitioners. There
is an initial step increase in the average age of family practitioners. While
there are no increases in average age of family medicine specialists there are
increasing average ages of general practitioners. This is consistent with
blocked entry into this route to practice primary care. However the overall
average age of all primary care practitioners continues to increase. Event
studies of the median age show similar effects (Figure A.2).

These results are similar when examining the number of physicians by age
cohort (Figure 4). This policy does not affect cohorts of “old” physicians
until the end of the 10 year period of observation when younger general
practitioners are aging into the “old” cohort. However, the policy has a
relatively large and immediate effect on cohorts of “young” physicians.
Although noisily estimated, by the end of the 10 year period there are 200
fewer family medicine practitioners per year under the age of 40. The overall
10 year supply of these “young” MDs is reduced by 120 primary care
practitioners with an additional 58.57 family medicine specialists but 188
fewer general practitioners (Table 3).

Reinforcing this, declines in physician supply resulting from the policy are
through cohorts of more recent medical school graduates (Figure 5). Over 10
years, the number of primary care practitioners who graduated from medical
school within six years declines by 103 per year and this effect occurs
immediately suggesting that fewer overall recent grads are entering primary
care. This occurs through a decline in general practitioners by 131, with an
increase of 25.6 in the number of family medicine specialists (Table 3).

Figure 6 demonstrates effects by year of intervention. Short-run effects are
relatively consistent across all four intervention times. Pre-trends are
relatively stable in cohorts except for the 1989 intervention which occurred
in Quebec.

Figure 7 demonstrates the effects of the policy on substitute residency
programs. There are trends towards increases in these residencies over the 10
years after the policy change. These effects peak about five years after the
policy change which is the usual length of a specialist residency in Canada.
Table 2 also demonstrates the DiD estimates for these specialties.

These effects are stable to alteration of the control group. Figure A.3
demonstrates event analyses when the control group consists solely of surgical
and diagnostic specialties. Table A.1 demonstrates DiD estimates.

###  3.3 Effects on physician quality

Table 5 demonstrates the DiD estimates over a two, four and ten year horizon.
There are no significant effects save for a marginal increase in the
probability of resigning from Ontario membership in the cohorts graduating
after extensions of family medicine residencies are implemented. Figure 8
demonstrates the events of these effects. There are similar zero estimates on
probability of negative outcomes for graduating cohorts after the policy
change.

##  4 Discussion

I leverage the phasing out of the rotating medical internship in Canada to
demonstrate the effects of a change in length of residency on physician supply
and quality. This effectively increased a primary care residency from one to
two years of training. This policy caused a transient decline in the total
supply of primary care practitioners. At its trough, this is a reduction by
5.8% of the overall number of practitioners. Current projections suggest that
supply of primary care specialists will need to increase by 7 to 20% by 2034
to keep up with demand (Dall et al.,, 2021). These results suggest that
extending family medicine residencies will cause an additional decline in
supply of physicians by the lower bound of these estimates.

These results do not occur solely because of a delays of the initially
affected cohort’s entry into independent practice. The measured decline is
statistically significant and negative for four years after intervention and
point estimates remain negative for nine years. This trough in supply is
several years longer than the length of a family medicine residency which is
not consistent with a pure cohort effect of lengthening the residency. While I
cannot observe direct substitution effects, there is some evidence consistent
with substitution towards other specialties occurring in historical CaRMS
reports from 1995. Prior to the transition, medical school graduates could
expect to match to one of their top three programs 80% of the time; in 1993,
this dropped to 70% suggesting that there was poorer matching success around
the time when the majority of provinces discontinued the rotating internship
(Banner,, 1995).

These policies also create a change in the composition of primary care
practitioners with possible long-run supply effects. Younger cohorts and
earlier graduates are the physicians most likely to avoid primary care after
the increase in program length. This creates a possible decline in long-run
supply. Primary care physicians trained after the change are older and so may
work fewer years during their career.

However, these results should be interpreted in the context of the other
changes that occurred with abolishing the rotating internship. This not only
extended the training required to become a credentialed primary care
practitioner but also meant reduced flexibility in program choice. In many
cases, the rotating internship was not an absorbing state and was used as a
way for a physician to test which specialty they would apply to in subsequent
re-entry matches. It was also often the case that general practitioners would
practice several years before going into a non-primary care specialty
(Dauphinee and Thurber,, 1997).

It is reassuring that policy effects are relatively consistent across
intervention years (Figure 6) especially when flexibility existed to switch to
provinces that retained the rotating internship; nevertheless, this provides
two points of caution in applying these results. First, extending primary care
residencies today would not be contaminated by these dual effects of
flexibility and extended length. It is ambiguous as to how this would change
results. Second, the implied effects that occur because of changes in the
work-life length of primary care physicians depends on how absorbing primary
care was for a general practitioner. More evidence on this point is needed.

These results also say nothing about access; it is possible that primary care
practices were able to absorb patients who could not access new family
practitioners. Moreover, residents entering family medicine programs treat
patients and often work very long hours (Pattani et al.,, 2014) which could
potentially increase access. Accounts of the policy change in 1994 also
suggest that this altered where patients could receive resident based care
implying changes in the distribution of care; it reduced trainee availability
in hospitals and improved access at community sites (Dauphinee and Thurber,,
1997).

This access to additional trainees is balanced against how autonomous the
resident is in patient care. Residents often perceive a lack of supervision
(Baldwin et al.,, 2010; Mieczkowski et al.,, 2021) which is consistent with
the resident having some ability to produce in parallel to independent
practitioners. However, in at least one resident setting, 30% of clinics
required faculty to see each patient treated by a resident and 42% had to
review each patient (Singman et al.,, 2017).

These overall impacts on supply may also reflect contaminated effects. The
majority of provinces abolished the one year internship in the 1990s when
other major changes were being made in physician training. Provincial policies
towards medical training were heavily influenced by reports that suggested
that there were too many medical school graduates (Marchildon and Di Matteo,,
2023). Provincial governments cut these positions by up to 10% in the years
after 1991 although this did not lead to any reduction in residency positions
(Dauphinee and Thurber,, 1997). This coincided with federal fiscal
retrenchment in 1995 which reduced federal health transfers to provinces. By
1997, this resulted in a physician shortage which triggered provincial
governments to provide more funding for training. This timing roughly
corresponds with these policy changes and might explain why physician supply
might rebound in the event analyses.

These declines in supply may be an acceptable trade-off if primary care
quality improves because of the longer training. To the extent that censure by
a licensing body represents physician quality, I find no improvements in
outcomes in Ontario caused by the policy change. More evidence on quality
improvement is needed as these results only speak to one possible avenue where
quality might be affected. This result though suggests that supply may be
impacted without improvements in quality.

In sum, these results show that extending the length of a family residency
program in Canada led to a reduction in the supply of primary care physicians.
Policy makers should consider these impacts on supply and quality in the
current debate on proposed extensions to family medicine training.

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##  5 Tables and Figures

Province | Year of policy | Year of Effect
---|---|---
Alberta | 1976 | 1977
Quebec | 1988 | 1989
Saskatchewan | 1989 | 1990
British Columbia | 1993 | 1994
Ontario | 1993 | 1994
New Brunswick | 1993 | 1994
Prince Edward Island | 1993 | 1994
Nova Scotia | 1993 | 1994
Manitoba | 1994 | 1994
Newfoundland and Labrador | 1994 | 1994
Table 1: Year of policy change by province. Note that Manitoba and Newfoundland and Labrador implemented policies abolishing rotating medical residencies in 1994 but these affected the residency class of 1992-1993. | Primary Care Practitioners | Substitute Specialties
---|---|---
| Total | FMDs | GPs | Pediatrics | Neurology | Internal Medicine | Psychiatry | Physiatry
DiD Estimates (2 Years) |  -1.563+ |  1.468∗ |  -3.124∗ | 0.0114 | 0.0319 | -0.0601 | 0.0379 | 0.0303
| (-1.85) | (4.05) | (-3.78) | (0.13) | (0.70) | (-0.97) | (0.75) | (0.76)
DiD Estimates (4 Years) |  -1.891+ |  1.908∗ |  -3.972∗ | 0.0114 | 0.0277 | -0.0754 | 0.0429 | 0.0312
| (-1.83) | (4.70) | (-3.78) | (0.12) | (0.61) | (-1.19) | (0.81) | (0.78)
DiD Estimates (10 Years) | -0.288 |  3.912∗ |  -4.796∗ | 0.0968 | 0.0814 | -0.0610 | 0.0995 | 0.0785
| (-0.27) | (8.19) | (-4.45) | (0.84) | (1.45) | (-0.87) | (1.41) | (1.41)
$N$ | 8690 | 8690 | 8690 | 8350 | 8010 | 7704 | 7364 | 7024
Mean | 84.34 | 16.34 | 67.35 | 5.292 | 3.328 | 4.929 | 5.563 | 4.620
t statistics in parentheses
+ $p<0.10$, ∗ $p<0.05$
Table 2: DiD Estimates by type of physician. | 2 Year Effects | 4 Year Effects | 10 Year Effects
---|---|---|---
| Total | FMDs | GPs | Total | FMDs | GPs | Total | FMDs | GPs
Average Age |  0.290∗ | -0.174 |  0.552∗ |  0.362∗ | -0.107 |  0.701∗ |  0.495∗ | -0.126 |  1.123∗
| (2.10) | (-0.47) | (4.57) | (2.60) | (-0.39) | (5.04) | (2.57) | (-0.40) | (7.84)
Young MDs | -64.15 |  28.21∗ |  -94.55+ | -80.96 |  37.97∗ |  -122.2∗ |  -120.5+ |  58.57∗ |  -188.5∗
| (-1.34) | (2.09) | (-1.84) | (-1.40) | (1.96) | (-1.97) | (-1.82) | (2.22) | (-2.16)
Old MDs | -1.755 | -0.575 | -1.775 | 0.186 | 0.219 | -0.825 |  11.97+ |  2.210+ | 8.045
| (-0.35) | (-0.71) | (-0.41) | (0.03) | (0.39) | (-0.14) | (1.84) | (1.80) | (1.29)
Graduation - Fewer than 6 years |  -65.86+ |  20.86+ |  -87.08∗ |  -81.92+ |  25.58+ |  -108.1∗ |  -103.1∗ |  25.67+ |  -131.0∗
| (-1.72) | (1.93) | (-2.17) | (-1.71) | (1.67) | (-2.18) | (-1.98) | (1.80) | (-2.13)
Graduation - 6-10 years | -1.638 |  5.402∗ | -8.418 | -0.944 |  10.79∗ | -13.70 | -20.09 |  28.73∗ |  -54.58∗
| (-0.17) | (2.03) | (-0.99) | (-0.09) | (3.25) | (-1.31) | (-1.44) | (3.20) | (-2.23)
Graduation -11-15 years | 12.95 | 8.312 | 3.152 | 13.41 |  10.68+ | 0.715 |  22.10∗ |  21.02∗ | -3.771
| (0.91) | (1.54) | (0.62) | (1.08) | (1.67) | (0.14) | (2.46) | (2.34) | (-0.44)
Graduation -31-35 years | 13.47 | 0.675 | 12.19 | 17.68 | 1.292 | 15.60 |  35.58+ | 6.513 |  27.81∗
| (1.36) | (0.61) | (1.21) | (1.57) | (0.76) | (1.49) | (1.86) | (1.24) | (1.97)
t statistics in parentheses
+ $p<0.10$, ∗ $p<0.05$
Table 3: Effects by age and graduation cohort. Figure 1: Time series of selected physician categories. Lines indicate dates of provincial changes to residency program length. Figure 2: Event analysis for family medicine practitioners per 100,000 persons. Point estimates are displayed with bootstrapped 95% CI. Figure 3: Event analysis for the average age of primary care practitioners. Point estimates are displayed with bootstrapped 95% CI. Figure 4: Event analysis for “young” and “old” primary care practitioners.“Young” MDs are any physician under the age of 40 whereas ”old” MDs are any physician over the age of 60. Point estimates are displayed with bootstrapped 95% CI. Figure 5: Event analysis for example graduation cohorts. Point estimates are displayed with bootstrapped 95% CI. Figure 6: Event analysis by year of intervention. Point estimates are displayed with bootstrapped 95% CI. Figure 7: Event analysis for substitute specialties to family medicine. Point estimates are displayed with bootstrapped 95% CI. | Mean - FMD | Mean - Other MDs | p-value
---|---|---|---
License revoked | 0.00124 | 0.000824 | 0.254
Terms and conditions applied | 0.0171 | 0.0208 | 0.0192
Suspension of license | 0.00336 | 0.00282 | 0.403
Resigned from membership | 0.164 | 0.0905 | 1.74e-84
N | 13709 | 17059 |
Table 4: Summary statistics and t-tests for key quality outcomes by whether the physician is in a primary care or any other specialty. | 2 year effect | 4 year effect | 10 year effect
---|---|---|---
Revoked | 0.00144 | 0.0000814 | 0.000222
| (1.23) | (0.06) | (0.45)
Suspension | -0.000503 | 0.000805 | 0.000827
| (-0.19) | (0.37) | (0.40)
Terms and Conditions | 0.00248 | -0.000810 | -0.00317
| (0.61) | (-0.25) | (-1.18)
Resigned from membership | 0.0246+ | 0.0143 | 0.00763
| (1.93) | (1.23) | (0.76)
$N$ | 30,703 | 30,703 | 30,703
t statistics in parentheses
+ $p<0.10$, ∗ $p<0.05$
Table 5: Effects of increased primary care residency length on quality
measures in Ontario. Figure 8: Event analysis for quality measures in Ontario
for primary care practitioners. Point estimates are displayed with
bootstrapped 95% CI. Note that revocation is such a rare quality outcome that
estimates cannot be produced for several years.

##  Appendix A Additional Tables and Figures

Figure A.1: Event analysis for total number of family medicine practitioners. Point estimates are displayed with bootstrapped 95% CI. Figure A.2: Event analysis for the median age of primary care practitioners. Point estimates are displayed with bootstrapped 95% CI. Figure A.3: Event analysis for total number of family medicine practitioners per 100,000 population relative to an altered control group. This control group consists of surgical and diagnostic residency specialties. Point estimates are displayed with bootstrapped 95% CI. | Family Medicine Practitioners
---|---
| Total | FMDs | GPs
DiD Estimates (2 Years) |  -1.546+ |  1.485∗ |  -3.108∗
| (-1.82) | (4.00) | (-3.74)
DiD Estimates (4 Years) |  -1.865+ |  1.934∗ |  -3.946∗
| (-1.80) | (4.72) | (-3.78)
DiD Estimates (10 Years) | -0.234 |  3.966∗ |  -4.742∗
| (-0.22) | (8.09) | (-4.40)
$N$ | 8690 | 8690 | 8690
Mean | 84.34 | 16.34 | 67.35
t statistics in parentheses
+ $p<0.10$, ∗ $p<0.05$
Table A.1: DiD Estimates by type of physician. The control group is altered to
specialties with lower substitutability and includes surgical and diagnostic
specialties.


# Exact solution for a traversable wormhole in a curvature-coupled
antisymmetric background field

R. V. Maluf<EMAIL_ADDRESS>Universidade Federal do Ceará (UFC),
Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  C. R Muniz
<EMAIL_ADDRESS>Universidade Estadual do Ceará (UECE), Faculdade de
Educação, Ciências e Letras de Iguatu
Av. Dário Rabelo, s/n, 63.502-253, Iguatu-CE, Brazil.

(August 30, 2024)

###### Abstract

In this work, we study a traversable wormhole sourced by an ideal matter fluid
with an antisymmetric 2-tensor background field coupled to gravity in a
scenario of spontaneously broken Lorentz symmetry. Contrary to employed in the
literature, we use a nonminimal curvature-coupling term
$B^{\mu\nu}B^{\kappa\lambda}R_{\mu\nu\kappa\lambda}$ which incorporates all
three kinds of Lorentz-violating coefficient for the pure-gravity sector of
the minimal standard-model extension. We find that the wormhole is non-
asymptotically globally flat and determine the allowed parameters of the
theory, showing that the matter fluid must be necessarily anisotropic. We also
analyze the energy conditions, checking their validity range and comparing
them with those predicted by general relativity.

Lorentz symmetry breaking, Antisymmetric 2-tensor field, Traversable wormhole,
Standard-Model Extension

##  I Introduction

The possibility of Lorentz symmetry breaking at the Planck scale is supported
by some fundamental theories, including strings Kostelecky:1988zi ;
Kostelecky:1991ak , loop quantum gravity Gambini:1998it ; Bojowald:2004bb ,
noncommutative spacetimes Carroll:2001ws , and Horava-Lifshitz gravity
Horava:2009uw . A theoretical framework that can describe the low energy
effects resulting from such symmetry breaking is the Standard-Model Extension
(SME) proposed by Colladay and Kostelecký in the late 1990s Colladay:1996iz ;
Colladay:1998fq . The SME incorporates CPT- and Lorentz symmetry violation
terms in all the usual standard model sectors. It was later expanded to take
gravity into account as an effective theory, capable of making predictions
that can be tested or verified observationally within current technological
limits Kostelecky:2003fs . Since its construction, the SME has been subject to
several theoretical studies and experimental tests that made it possible to
raise tight constraints on Lorentz invariance in nature. An updated
compilation of all upper bounds can be found in Kostelecky:2008ts .

In this context, it is of deep interest to explore the consequences of Lorentz
violation in the gravitational scenario involving compact objects like black
holes Casana:2017jkc ; Lessa:2019bgi ; Ding:2020kfr ; Maluf:2020kgf  and
topologically non-trivial structures as wormholes Ovgun:2018xys ;
Lessa:2020imi  as well as in cosmology Bertolami:2005bh ; Capelo:2015ipa ;
Maluf:2021lwh  . These objects originally were found as solutions for the
field equations of general relativity, unraveling unexpected connections
between two remote regions of the spacetime Einstein:1935tc ; Misner:1957mt ;
Ellis:1973yv . Wormholes can also occur in spacetimes with arbitrary
dimensions and in several topologies (See Dias:2010uh , and references
therein). They usually do not satisfy the energy conditions of the general
relativity, being necessary some exotic matter as a source, like the Casimir
energy Morris:1988tu ; Garattini:2019ivd ; Jusufi:2020rpw ; Alencar:2021ejd .
Nevertheless, classical and quantum-modified theories of gravity can change
this feature, which has been widely discussed in the recent literature
Richarte:2007zz ; Richarte:2009zz ; Cuyubamba:2018jdl ; Fu:2018oaq ;
Alencar:2021enh . Still concerning such objects, a class of modified theories
as those embodying background fields coupled to the gravity, with
spontaneously Lorentz symmetry breaking, were analyzed in Refs. Ovgun:2018xys
; Lessa:2020imi .

Thus, in this letter, we will study a traversable wormhole immersed in an
antisymmetric 2-tensor background field coupled to gravity and sourced by a
matter fluid in a scenario of spontaneously Lorentz symmetry breaking
Altschul:2009ae . Differently from that was employed recently in Lessa:2020imi
, we will use another coupling to the antisymmetric field with the curvature,
which plays the role of a pseudo-electric one. We will show that the matter
fluid can be ideal; however, it must be anisotropic. We will also determine
the allowed parameters of the theory, analyze the energy conditions and verify
where they are satisfied or violated, compared with that predicted by general
relativity.

##  II The Einstein-bumblebee gravity with an antisymmetric tensor

Here we present and discuss the main aspects of the model adopted in this
work. The chosen action involves the Einstein-Hilbert term of general
relativity with the presence of an antisymmetric 2-tensor
$B_{\mu\nu}=-B_{\nu\mu}$, and it is given by Altschul:2009ae

$S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2\kappa}R-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}-V+\frac{\xi_{1}}{2\kappa}B^{\kappa\lambda}B^{\mu\nu}R_{\kappa\lambda\mu\nu}+\mathcal{L}_{M}\right],$
(1)

where $\kappa=8\pi G_{N}$ such that $G_{N}$ is the Newtonian gravitational
constant, and the coupling constant $\xi_{1}$ (with mass dimension
$[\xi_{1}]=M^{-2}$ in natural units) represents a nonderivative gravitational
coupling to $B_{\mu\nu}$ that is linear in the curvature. The field-strength
tensor $H_{\mu\nu\lambda}$ associated with $B_{\mu\nu}$ is defined by

$H_{\mu\nu\lambda}=\partial_{\mu}B_{\nu\lambda}+\partial_{\lambda}B_{\mu\nu}+\partial_{\nu}B_{\mu\lambda},$
(2)

with $H_{\mu\nu\lambda}$ being invariant under the gauge transformation
$B_{\mu\nu}\rightarrow
B_{\mu\nu}+\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu}$. Also, the
potential $V$ is responsible for triggering spontaneous Lorentz violation
inducing a nonzero vacuum value $\left\langle
B_{\mu\nu}\right\rangle=b_{\mu\nu}$ , and $\mathcal{L}_{M}$ handles the
ordinary matter content that will be specified later.

Our motivation in choosing the theory defined by the action (1) is twofold.
First, the coupling involving the constant $\xi_{1}$ can generate all three
Lorentz-violating coefficient fields of the SME, usually denoted as $u$,
$s^{\mu\nu}$, and $t^{\kappa\lambda\mu\nu}$ Bailey:2006fd . In particular, the
$t$-coefficient can only be expressed in terms of the field $B_{\mu\nu}$ when
contracted with the Riemann curvature tensor from the $\xi_{1}$ coupling
Altschul:2009ae . Second, the phenomenological role of the $t$-coefficient is
an intriguing issue still little explored in the literature since previous
investigations involving solutions of black holes Lessa:2019bgi  and wormholes
Lessa:2020imi  did not take into account this type of coupling present in (1).
Hence, our main focus of this letter is to determine the existence of a
wormhole solution modified by the $\xi_{1}$ coupling, responsible for
producing a nonzero vacuum value for $t^{\kappa\lambda\mu\nu}$.

The equations of motion for gravity can be obtained from (1) by varying with
respect to $g_{\mu\nu}$ and keeping the other fields fixed. Thus, we get

$G^{\mu\nu}=\kappa(T_{M})^{\mu\nu}+\kappa(T_{B})^{\mu\nu}+(T_{\xi_{1}})^{\mu\nu}.$
(3)

On the left side in (3), we have the usual Einstein tensor
$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$, while on the right side are
represented the energy-momentum tensors due to the matter content
$(T_{M})_{\mu\nu}$ and the contributions originating due to $B_{\mu\nu}$ field
that appear from the kinetic and potential terms $(T_{B})_{\mu\nu}$ and the
nonminimal coupling $(T_{\xi_{1}})_{\mu\nu}$, respectively.

For $B$-terms, we find explicitly

$\displaystyle(T_{B})^{\mu\nu}$
$\displaystyle=\frac{1}{2}H^{\alpha\beta\mu}H_{\
\alpha\beta}^{\nu}-\frac{1}{12}g^{\mu\nu}H^{\alpha\beta\gamma}H_{\alpha\beta\gamma}$
$\displaystyle-g^{\mu\nu}V+4B^{\alpha\mu}B_{\alpha}^{\ \nu}V^{\prime},$ (4)

where for simplicity we have assumed the dependence of the potential $V$ with
respect to $B_{\mu\nu}$ through the form

$V\equiv V(B_{\mu\nu}B^{\mu\nu}-x),$ (5)

with $x$ being a real number representing the vacuum value of the invariant

$x\equiv\left\langle B_{\mu\nu}B^{\mu\nu}\right\rangle=\left\langle
g^{\alpha\mu}\right\rangle\left\langle g^{\beta\nu}\right\rangle
b_{\alpha\beta}b_{\mu\nu},$ (6)

and the prime ′ means derivative with respect to the potential argument. Note
that $\left\langle g^{\mu\nu}\right\rangle$ is the vacuum value to the inverse
metric. For our present purpose, we can assume the $B_{\mu\nu}$ field and the
metric are frozen in their vacuum values so that

$B_{\mu\nu}=b_{\mu\nu},\ \ \ \ g_{\mu\nu}=\left\langle
g_{\mu\nu}\right\rangle,$ (7)

and the vacuum conditions $V=V^{\prime}=0$ are guaranteed.

Finally, the contributions due to nonminimal gravitational coupling is

$\displaystyle(T_{\xi_{1}})^{\mu\nu}$
$\displaystyle=\xi_{1}\left(\frac{1}{2}g^{\mu\nu}B^{\alpha\beta}B^{\gamma\delta}R_{\alpha\beta\gamma\delta}+\frac{3}{2}B^{\beta\gamma}B^{\alpha\mu}R_{\
\alpha\beta\gamma}^{\nu}\right.$
$\displaystyle+\frac{3}{2}B^{\beta\gamma}B^{\alpha\nu}R_{\
\alpha\beta\gamma}^{\mu}+\nabla_{\alpha}\nabla_{\beta}B^{\alpha\mu}B^{\nu\beta}$
$\displaystyle+\left.\nabla_{\alpha}\nabla_{\beta}B^{\alpha\nu}B^{\mu\beta}\right).$
(8)

The equations of motion for the antisymmetric tensor field are also obtained
from the action (1). By varying this action concerning $B_{\mu\nu}$, and this
time keeping the metric and matter fields fixed, we have

$\nabla_{\alpha}H^{\alpha\mu\nu}=4V^{\prime}B^{\mu\nu}-\frac{2\xi_{1}}{\kappa}B_{\alpha\beta}R^{\alpha\beta\mu\nu}.$
(9)

Note that here we are explicitly disregarding any type of coupling between the
matter fields and $B_{\mu\nu}$. Such a possibility may imply changes in the
conservation of the conventional matter currents ant it is beyond our present
scope.

##  III Static spherical wormhole solution with spontaneous Lorentz breaking

The Morris-Thorne geometry which represents a static and spherically symmetric
wormhole solution is described by the line element Morris-Thorne ;
Morris:1988tu

$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left(1-\frac{\Omega(r)}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}{\theta}d\phi^{2},$
(10)

where $\Phi(r)$ is called the redshift function, admittedly everywhere finite
to avoid event horizons or singularities, and $\Omega(r)$ is the shape
function of the wormhole. Additional conditions on $\Phi(r)$ and $\Omega(r)$
are required for a transversable wormhole solution. One of them is the
existence of the minimum radius: $\Omega(r_{0})=r_{0}$, where $r=r_{0}$ is the
radius of the throat of the wormhole. Another important requirement is the
flare-out condition at the throat: $\Omega(r)<r$, while
$\Omega^{\prime}(r_{0})<1$. Also, the tidal gravitational forces must be very
small, i.e., $\left|\Phi\right|\ll 1$. For the remainder of this work, we will
neglect the tidal force, assuming $\Phi(r)=0$. With this simple choice, we can
determine an exact wormhole solution, as we will see below.

For the matter-energy content, we adopt a perfect fluid such that the energy-
momentum tensor for the matter has the $(T_{M})^{\mu}_{\ \
\nu}=\mbox{diag}(-\rho,p_{r},p_{\theta},p_{\phi})$. It is worth emphasizing
that the perfect fluid is not assumed to be isotropic since the radial and
lateral pressures are not a priori equal.

Now let us configure the Lorentz-violating field. Following the Refs.
Lessa:2019bgi ; Lessa:2020imi , we will restrict ourselves to the pseudo-
electric configuration in which the field $B_{\mu\nu}$ is frozen in its vacuum
expectation value $b_{\mu\nu}$, whose explicit form is given by

$b_{\mu\nu}=b_{10}=-b_{01}=\frac{a}{\sqrt{1-\frac{\Omega(r)}{r}}}.$ (11)

In this way, the background field $b_{\mu\nu}$ has a constant norm,
$b_{\mu\nu}b^{\mu\nu}=-2a^{2}$, where $a$ is a real and positive parameter.
This setup preserves the spherical and static spacetime symmetry. Moreover,
according to relations (2) and (11), the field strength $H_{\mu\nu\lambda}$ is
identically null, and it can be explicitly verified that the equations of
motion for $B_{\mu\nu}$ expressed in (9) are automatically satisfied.

Indeed, assuming the vacuum conditions, i.e., $V^{\prime}=0=V$ and the vev
(11) for $B_{\mu\nu}$, the relevant components obtained from Eq. (9) are
$-\frac{4\xi_{1}}{\kappa}b_{01}R^{01\mu\nu}$. On the other hand, for the
metric (10) with the zero tidal condition ($\Phi=0$), it can be shown that the
nonzero components of the Riemann curvature tensor are

$R_{1212}=-\left(\frac{\Omega-r\Omega^{\prime}}{2r-2\Omega}\right),\ \
R_{1313}=\sin^{2}\theta R_{1212},\ \ R_{2323}=r\sin^{2}\theta\Omega,$ (12)

and also those obtained by the allowed permutations of indices. Thus, it is
clear that the $b_{01}R^{01\mu\nu}$ term is identically null for the
configuration adopted by us.

After these preliminary considerations, the extended Einstein equations (3)
result in the following nonvanishing components:

$\frac{\Omega^{\prime}}{r^{2}}-\frac{\lambda}{r^{3}}\left(r\Omega^{\prime}+\Omega-2r\right)-\kappa\rho=0,$
(13) $\frac{\Omega}{r^{3}}+\kappa p_{r}=0,$ (14)
$\frac{1}{2}\left(\Omega^{\prime}-\frac{\Omega}{r}\right)+\kappa
r^{2}p_{\theta}=0,$ (15)
$\sin^{2}\theta\left[\frac{1}{2}\left(\Omega^{\prime}-\frac{\Omega}{r}\right)+\kappa
r^{2}p_{\phi}\right]=0,$ (16)

where we have defined the Lorentz-violating parameter $\lambda=2\xi_{1}a^{2}$.
Note that the signature of the Lorentz violation only appears in the temporal
part (13) of the field equations.

From Eq. (14), one can express the radial pressure in terms of the shape
function as

$p_{r}=-\frac{\Omega}{\kappa r^{3}}.$ (17)

Assuming the equation of state: $p_{r}=\omega\rho$, where $\omega$ is a
dimensionless real parameter, we insert the relation (17) into Eq. (13) to get

$(1-\lambda)r\Omega^{\prime}+\left(\frac{1}{\omega}-\lambda\right)\Omega+2\lambda
r=0.$ (18)

Solving the above equation, we obtain the solution for the shape function as
follows

$\Omega(r)=\frac{1}{1+(1-2\lambda)\omega}\left[-2\lambda\omega
r+(1+\omega)r_{0}\left(\frac{r_{0}}{r}\right)^{\frac{1-\lambda\omega}{(1-\lambda)\omega}}\right],$
(19)

and as before-mentioned $\Omega(r_{0})=r_{0}$.

Inserting Eq. (19) into Eq. (17), we get the energy density

$\rho(r)=\frac{1}{\omega\kappa\left(1+\omega-2\lambda\omega\right)r^{3}}\left[2\lambda\omega
r-(1+\omega)r_{0}\left(\frac{r_{0}}{r}\right)^{\frac{1-\lambda\omega}{(1-\lambda)\omega}}\right].$
(20)

It is easy to verify that the solutions (19) and (20), together with the
equation of state to $\rho$ and $p_{r}$, satisfy the temporal (13) and radial
(14) components of the Einstein equations. However, these solutions do not
verify the other components (15-16) to the isotropic condition, i.e.,
$p_{r}=p_{\theta}=p_{\phi}$, except for the usual case when $\lambda=0$. Thus,
it is impossible to determine a specific value for $\omega$ that preserves the
isotropy of the perfect fluid and still allows the Lorentz breaking. For
wormhole solutions in the context of bumblebee gravity models, this
anisotropic condition is a new result and differs from those already obtained
in the literature Ovgun:2018xys ; Lessa:2020imi .

Nevertheless, we can still assume $p_{\theta}=p_{\phi}$ and use Eqs. (15) and
(19) to fix the lateral pressure, thus

$p_{\theta}(r)=p_{\phi}(r)=\frac{(1+\omega)r_{0}}{2\omega\kappa(1-\lambda)r^{2}}\left(\frac{r_{0}}{r}\right)^{\frac{1-\lambda\omega}{(1-\lambda)\omega}}.$
(21)

As expected, at the Lorentz invariant regime, i.e., $\lambda\rightarrow 0$, we
recover the isotropic case for $\omega=-1/3$, such that
$\Omega(r)=r^{3}/r_{0}^{2}$ and $p_{r}=p_{\theta}=p_{\phi}=-1/\kappa
r_{0}^{2}$, which represents a spacetime of constant curvature Cataldo:2016dxq
.

One can easily see that the radial metric component
$g_{rr}=\left(1-\frac{\Omega}{r}\right)^{-1}$ diverge at $r=r_{0}$, as is
expected for any wormholes. However, our wormhole solution is non-
asymptotically flat when $r\rightarrow\infty$:

$\lim_{r\rightarrow\infty}\frac{\Omega(r)}{r}\Rightarrow-\frac{2\lambda\omega}{1+\omega-2\lambda\omega}+\frac{1+\omega}{1+\omega-2\lambda\omega}\lim_{r\rightarrow\infty}\left(\frac{r_{0}}{r}\right)^{1+\frac{1-\lambda\omega}{(1-\lambda)\omega}}.$
(22)

From the above result, it is clear that the first term does not depend on $r$,
while the second term is vanished if

$1+\frac{1-\lambda\omega}{(1-\lambda)\omega}>0,$ (23)

otherwise the metric diverges at infinity. Generally speaking, the geometry of
wormholes is asymptotically flat, at least in the general relativity context.
Here we are dealing with a scenario of extended gravity, which comprises a
classical background field which spontaneously violates the local Lorentz
symmetry, and this explains the non-flatness at infinity. A similar result
involving a wormhole solution in the presence of a vector background field was
obtained in Ref. Ovgun:2018xys . Such solutions have analogous asymptotic
behavior to the spacetimes having topological defects. In fact, by a simple
transformation of coordinates we can rewrite the wormhole metric at the
asymptotic limit in the form,

$ds^{2}=-dt^{2}+d\tilde{r}^{2}+\left(1+\frac{2\lambda\omega}{1+\omega-2\lambda\omega}\right)\tilde{r}^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}),$
(24)

where we take

$\tilde{r}=\left(1+\frac{2\lambda\omega}{1+\omega-2\lambda\omega}\right)^{-1/2}r.$
(25)

The above result show that the asymptotic geometry corresponds to a global
monopole Barriola:1989hx , provided the condition (23) is satisfied. Aligning
this condition with the flare-out one, $\Omega(r)<r$ for all $r>r_{0}$, gives
us the parameter space configuration shown in Figure 1, in which the allowed
values for $\lambda$ and $\omega$ are exhibited, in thermal colors. Notice
also that phantom-like fluids ($\omega<-1$), for little values of $\lambda$
can generate the wormhole under inspection.

Figure 1: Parameter space ($\omega$,$\lambda$) associated to the modified
wormhole solution due to Lorentz breaking. The colored regions indicate the
allowed values of these parameters for the formation of the wormhole. Hot
(cold) colors indicate great (small) values for the Lorentz breakdown
parameter, $\lambda$.

From solution (19), we can calculate the Ricci and the Kretschmann scalars.
Their values at $r=r_{0}$, are respectively,

$\left.R\right|_{r=r_{0}}=-\frac{2(1+\lambda\omega)}{(1-\lambda)\omega}\frac{1}{r_{0}^{2}},$
(26)

and

$\left.K\right|_{r=r_{0}}=\frac{2+4\omega+(6-8\lambda+4\lambda^{2})\omega^{2}}{(1-\lambda)^{2}\omega^{2}r_{0}^{4}}.$
(27)

These quantities show that the wormholes are free of singularities at the
throat, guaranteeing thus their traversability, for all $\lambda\neq 1$.

Lastly, we now analyze the energy conditions for the perfect anisotropic fluid
supporting our modified wormhole solution. We depict in Figure 2 some
quantities taken from the energy density and pressures as functions of the
radial coordinate, in order to verify the regions where the Null Energy
Conditions (NEC, $\rho+p_{i}\geq 0$), Weak Energy Conditions (WEC, $\rho\geq
0$, $\rho+p_{i}\geq 0$), Strong Energy Conditions (SEC, $\rho+p_{i}\geq 0$,
$\rho+\sum p_{i}\geq 0$), and Dominant Energy Conditions (DEC, $\rho\geq 0$,
$-\rho\geq p_{i}\geq\rho$) are satisfied. We can notice that NEC, WEC, and SEC
are entirely obeyed for

$r\geq
r_{c}=\left(\frac{2\omega\lambda}{\omega+1}\right)^{\frac{\omega(1-\lambda)}{2\omega\lambda-w-1}}r_{0},$
(28)

which is confirmed in the left panel of Figure 2.

On the other hand, those conditions as well as DEC are fully satisfied for
$r\geq r_{d}$, which cannot be analytically determined. Notice however that
$r_{d}>r_{c}$, and this can be clearly seen in the right panel of Figure 2.
Thus, we conclude that only in the neighborhood of the wormhole throat all the
energy conditions are violated. We also notice that on returning to general
relativity by making $\lambda=0$, they leave be satisfied at all $r>r_{0}$.
The present analysis shows us that even an ordinary fluid, like
electromagnetic radiation ($\omega=1/3$), can contribute to the formation of
the wormhole due to the presence of a background field that violates the
Lorentz symmetry, behaving effectively like exotic matter nearby the wormhole
throat.

Figure 2: Energy density, pressures, and their combinations associated to the
matter fluid, as functions of the radial coordinate, for $\lambda=0.1$,
$\omega=1/3$, and $r_{0}=20$, in natural units.

##  IV Conclusion

In this work, we have studied traversable wormhole solutions with an
antisymmetric 2-tensor background field coupled to gravity, with spontaneous
Lorentz symmetry breaking. We have used a gravity coupling to that field still
not employed in the literature, which by its turn plays the role of a pseudo-
electric field, finding a traversable wormhole solution non-asymptotically
globally flat. To ensure that all components of the modified Einstein
equations are satisfied, we relax the isotropy assumption for the matter
content and assume that the equation of state $p=\omega\rho$ is valid only for
radial pressure. The solution obtained is then used to fix the lateral
pressures. Also, we have established a parameter space where this must occur,
together with the flare-out condition, for a perfect fluid sustaining the
wormhole. Also, we have identified that even a phantom-like fluid can be a
source for this modified wormhole solution. Furthermore, the matter fluid
needs to be anisotropic, compatible with the nonvanishing vev of the
background field itself. Such a feature was not found in the case examined in
Ref. Lessa:2020imi . Furthermore, we have found that all the energy conditions
are satisfied beyond a critical radius. That is different from what happens in
general relativity, in which those conditions leave to be valid everywhere. We
have shown that such a violation occurs only near the wormhole throat and
verifies that an ordinary fluid, as electromagnetic radiation, can be a source
to this structure, behaving, thus, as an exotic fluid, due to the presence of
the antisymmetric background field. As a future perspective, we intend to
study the influence of the pseudo-magnetic sector of the antisymmetric
background field, together with the matter fluid in building a traversable
wormhole.

## Acknowledgments

The authors thank the Fundação Cearense de Apoio ao Desenvolvimento Científico
e Tecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES), and the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Grants no 307556/2018-2 (RVM) and PRONEM
PNE-0112-00085.01.00/16 (CRM) for financial support.

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