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Starts With A Bang — November 5, 2021
Astro2020: Astronomy's bright future revealed in game-changing decadal report
Big dreams and big telescopes are back at last, but everything depends on sufficiently funding NASA, the NSF, and the DOE.
This artist's concept shows the geometry of a space telescope aligned with a starshade, a technology used to block starlight in order to reveal the presence of planets orbiting that star. From tens of thousands of kilometers away, the starshade and the telescope must achieve and maintain perfect alignment to enable direct exoplanet imaging. Compared to a coronagraph, a starshade's optics are superior, but far fewer systems can be probed in any given amount of time.
(Credit: NASA / JPL-Caltech)
Astronomy needs a combination of space-based and ground-based missions across all wavelengths for new discoveries.
The report recommends a mix of small, medium, and flagship missions, and it refuses to skimp on the science.
The big winners are exoplanets, gravitational waves, and cosmology, but without public funding, the entire plan will fall apart.
Ethan Siegel
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Every ten years or so, the National Academies conducts a decadal survey, where after soliciting recommendations and white papers from across the disciplines of astronomy and astrophysics, they release a report that charts the course for the next decade of the field. Previously, decadal surveys have led to NASA's great observatories and generations of space-based science missions, from Hubble to Spitzer to Chandra to James Webb and beyond. While many of these missions have been tremendous successes, there have also been setbacks: cost overruns, budget mismanagement, and the infamous flawed mirror that almost rendered Hubble a near-total failure.
No more. With the release of the decadal report that charts the course for astronomy and astrophysics from 2023-2033 — known as Astro2020 — those setbacks are guaranteed to be a thing of the past. For the first time in history, not only has a truly balanced mission portfolio been advanced by the latest decadal survey, but it both paves a realistic and optimistic path forward for astronomy across all wavelengths and even beyond light, as well as putting robust safeguards in place to prevent the types of cost overruns that have plagued previous large missions. For the first time, practically every astronomer sees a positive path forward for both the field in general as well as their specific sub-field. So that you don't have to read the 600+ page report yourself, here's what everyone should know.
The various plans and initiatives from the Astro2020 decadal survey touch on a wide variety of initiatives and pathways, ensuring a successful short-term and long-term future for astronomy and astrophysics. (Credit: National Academies / Astro2020 decadal survey)
A vision for the future
If you want to know what we can discover within the science of astronomy and astrophysics, you have to invest in facilities. That includes telescopes, instruments, support infrastructure, ongoing science operations, personnel, and much more. All of those things, if we want those facilities and their findings to be available to everyone, require federal funding across both ground-based and space-based initiatives. In order to maximize the value of what we invest in, the decadal survey solicited white papers from the astronomical community, and the community didn't disappoint; for Astro2020, more than 500 white papers were submitted.
Synthesizing as many of them together as possible, the committee identified three major themes for the scientific vision of the upcoming decade.
Pathways to Habitable Worlds. The goal of this is to identify and characterize, as best as possible, as many Earth-like exoplanets as we can. That includes direct imaging and taking spectra of their atmospheres, potentially paving the way to discovering the first planet beyond our Solar System with life on it.
New Windows on the Dynamic Universe. For most of history, astronomy focused only on light. But astrophysical sources also emit gravitational waves and particles, and we have started to detect them. By bringing observations that span the electromagnetic spectrum together with cosmic rays, neutrinos, and gravitational waves, we can hope to understand multi-messenger events and, perhaps, even probe the relic backgrounds from cosmic inflation.
Drivers of Galaxy Growth. We have been able to detect and measure the cosmic web, including the gas within it, but only coarsely. By probing never-before-revealed details, including how that gas collapses, condenses, and infalls to form stars and grow galaxies, we can learn how our Universe "grew up" in unprecedented fashion.
To learn how galaxies grow and evolve over cosmic time, including an understanding of how gas infalls and gets expelled from galaxies, we will need to develop a suite of multiwavelength observatories, missions, and facilities. All of this is possible in the plan laid out in the Astro2020 decadal. (Credit: National Academies / Astro2020 decadal survey)
The next flagship mission
The previous decadal surveys set the course for our current path and are the reason why the James Webb Space Telescope will launch this December, followed by the construction and launch of the Nancy Grace Roman Telescope — a wide-field version of Hubble with vastly superior instrumentation — later in the 2020s. Four missions were selected as finalists for this decadal survey: HabEx, LUVOIR, Lynx, and Origins.
For the very first time, none of them were chosen.
Instead, the next mission was described in terms of its science goals, rather than in terms of the technical specifications that the telescope facilities would possess. The Astro2020 report recommends that the next flagship mission be what I would call an "ultra-Hubble," an all-purpose infrared, optical, and ultraviolet observatory with ~250 percent the resolution of Hubble and more than six times Hubble's light-gathering power. It should be larger and more powerful than even the largest and most expensive HabEx proposal, but a little bit smaller, less expensive, and with a faster timetable to completion than any of the LUVOIR proposals. This should be the next flagship mission after the Nancy Grace Roman Telescope, but it is being approached as never before.
The prospect of detecting and characterizing the atmosphere of a true Earth-like planet — that is, an Earth-sized planet in the habitable zone of its star, including both red dwarf and more Sun-like stars — is finally within our reach. (Credit: National Academies / Astro2020 decadal survey)
Great observatories and technology maturation
There are three big differences between how this current flagship is being approached as compared with all other previous astronomy and astrophysics flagship missions.
The first is arguably the most important: the maturation of concept is going to be done first. This has never been done at an institutional level like this before. From Hubble to James Webb, a telescope's design and architecture have traditionally been the starting point, and then the various components — optics, instruments, the assembly, and other necessary parts — are all designed and built. If there was something new that needed to be developed, like the sunshield or the unfolding of the segmented mirrors for James Webb, those technologies were developed and built in parallel to the rest of the observatory.
We are not doing it that way any longer. While the architecture of James Webb, by necessity, had to change as it was developing and evolving, the focus on the maturation of the technology first will mean that all of the necessary components will be designed, specified, and accurately cost-and-labor-estimated before construction begins. The current architecture is only outlined and won't be finalized until this technological development component is complete. By focusing on the technology maturation first, it eliminates the possibility of the very cost overruns that plagued James Webb throughout its construction.
The timeline of future medium and large missions, as set forth in the Astro2020 decadal. For the first time, multiple true flagship missions (B to B and up) have been slated for construction and development simultaneously. (Credit: National Academies / Astro2020 decadal survey)
No waiting for the next step
Only once the technological maturation necessary for this flagship has been established will the architecture be finalized, and only once the architecture is finalized will the rapid construction of the observatory commence.
Traditionally, it isn't until the construction phase is complete that major steps are taken to address the next flagship. That is where the second major difference from all previous decadal surveys comes in: once the architecture for this next flagship mission is finalized, then not only will construction of this observatory begin, but concurrently, the technological maturation of concept for the next flagship mission will commence.
The Astro2020 decadal issued recommendations for that as well: either a next-generation far-infrared observatory (like the proposed Origins) or a next-generation X-ray observatory (like the proposed Lynx) should proceed. By having the technological development of the next flagship's components proceed concurrently with the previous flagship's construction, this should prevent the large gap in time between flagships that we experienced prior to James Webb, and that will, by necessity, exist between Nancy Roman and this next flagship mission. But there is a third piece to the puzzle that is also new to the Astro2020 decadal and that is worth just as much as the other two pieces.
This chart, lifted directly from the Astro2020 decadal report, shows the medium and large expenditures that are planned for the decade from 2023-2033. Note how many different initiatives and stakeholders are served by this comprehensive plan. (Credit: National Academies / Astro2020 decadal survey)
Every scientific stakeholder is well-served by these recommendations
Thirdly, nobody on the cutting edge of astronomy winds up with their scientific dreams on the cutting room floor. The long-term vision put forth by the Astro2020 decadal recognized the need for observatories and facilities that addressed the full suite of needs of astronomers and astrophysicists. As written in the Astro2020 report itself:
"Establishing a panchromatic suite of observatories over the next 30 years is essential to address key questions in all three of the survey's priority science themes. The large strategic mission implementations presented to the survey cannot all be built and launched in an optimal timeframe given the current designs, available budgets, and approaches to mission development."
This means that all wavelengths, from X-ray to ultraviolet to optical to infrared all the way down to radio light, are slated for both maintenance of current observatories and developments of new facilities and instruments. It means that both ground-based and space-based enterprises are going to be invested in. It means that hunts for signals from the Universe that go beyond light, such as gravitational wave, cosmic ray, and neutrino sources are all being advocated for, along with the requisite funding. In the opinion of a great many professionals, the combination of these three factors makes this the most optimistic, pragmatic decadal ever put forth by the community.
In order to achieve our ambitious goals of understanding the Universe, a panchromatic suite of astronomical observatories, both space-based and ground-based, are required. The Astro2020 decadal plan addresses all of these needs. (Credit: National Academies / Astro2020 decadal survey)
Other major recommendations
Although the flagship missions are by far the most high-profile and most valuable part of these decadal surveys, maximizing how we efficiently reach for our science goals requires a variety of approaches.
For ground-based astronomy, the most exciting development is the construction of new, extremely large optical and infrared telescopes. While the current generation of large observatories is between 8-12 meters, the next generation will grow to 25-40 meters. One of them is a purely European endeavor: the European Extremely Large Telescope. However, two others, the Giant Magellan Telescope (in the southern hemisphere) and the Thirty Meter Telescope (in the northern hemisphere) are also under construction. The Astro2020 recommendation is that the National Science Foundation funds these in exchange for dedicated telescope time.
They recommend at least a 25 percent stake in each, assuming that consent issues with indigenous populations concerning TMT can be resolved; if not, they recommend at least a 50 percent stake in GMT. As stated in the decadal report:
"Because of their transformative scientific potential, as well as readiness, the success of at least one U.S. ELT is a critical priority for investment for ground-based astronomy in the coming decade."
The Giant Magellan Telescope is already well under construction and should reach completion by the decade's end. At 25 meters in diameter, it will be either the largest or second largest telescope in the world upon completion, depending only on whether the 39-meter diameter E-ELT is completed first. (Credit: Giant Magellan Telescope / GMTO Corporation)
Many other sweeping recommendations are also a part of this plan.
In the microwave portion of the sky, next generation observatories to measure the fluctuations and polarization in the cosmic microwave background should be constructed. These will measure small-scale fluctuations in multiple microwave wavelengths, will include polarization, and will teach us about everything from the gas in our own galaxy to whether large-amplitude gravitational waves from inflation are present in the Universe.
Radio astronomy should receive a new, unprecedentedly powerful array of telescopes: a next-generation Very Large Array (ngVLA). According to the Astro2020 decadal:
"The ngVLA would resolve protoplanetary disks on scales more than 20 times finer than ALMA, potentially capturing images of planet formation in action. The ngVLA facility would be absolutely unique worldwide in both sensitivity and frequency coverage."
They recommend the construction of a next-generation neutrino astronomy facility, a more powerful successor to IceCube.
They recommend the construction of a next-generation ground-based gravitational wave facility, a more powerful successor to LIGO.
They recommend time-domain and multi-messenger missions as part of a balanced portfolio that includes cost-effective, sub-flagship medium-scale missions, and small, explorer class missions. And they recommend foundational activities like substantial and sustained investments in early career researchers, supporting the cost of operating existing facilities and upgrading their instruments, fully funding the support staff, increasing available fellowships and scholarships, developing real and virtual tools like simulations, machine learning initiatives, and data archives and pipelines, as well as stopping harassment and discrimination, while simultaneously broadening access to opportunities.
Gravitational waves span a wide variety of wavelengths and frequencies and require a set of vastly different observatories to probe them. The Astro2020 decadal offers a plan to support science in every one of these regimes, furthering our knowledge of the Universe as never before. (Credit: National Academies / Astro2020 decadal survey)
If this were set up incorrectly, it could lead to tremendous budgetary problems. Many have been worried that a cost overrun on a flagship could wind up eating the budget of the other necessary science activities. But if the Astro2020 recommendations are followed — by NASA, the NSF, and the DOE — then the problems we have seen with funding in the past will not continue downstream.
We have seen budgetary and political whims sabotage long-term scientific projects in the past. Given that each of these proposed new flagships will take 15-20 years from the start of development until the start of science operations, slashed funding is an existential threat to the entire enterprise each and every year.
The benefit of this new approach, in summary, is this: budgetary resilience is built into the technology maturation process. If the process tells you, "This isn't the best way to build this," we will build it the smarter way, and we will not have lost anything because the technology maturation studies come before the architecture is finalized. If we phase money appropriately for technology development early, previously disastrous snags would not be mission-blockers any longer. This process targets co-maturation of all mission technologies, so that every aspect of construction is understood before the architecture is even finalized. For the next coming flagship, flexibility is a feature, not a bug. Right now, we don't know if this will be:
a single dish telescope or a set of segmented mirrors,
an on-axis or off-axis configuration,
more LUVOIR-like or HabEx-like,
or whether the initial guesses of a ~6 meter mirror and an ~$11 billion pricetag will wind up being part of the final architecture. There is tremendous opportunity for growth, and the budgetary projections only rely on a ~2% annual increase moving forward.
Over and above the existing commitments to astronomy and astrophysics, the blue line represents the remaining projected budget for all other expenses. This chart shows projected new expenditures to follow the Astro2020 recommendations. Note that, over time, there is sufficient funding to do everything that is planned. (Credit: National Academies / Astro2020 decadal survey)
The science rewards that will come from implementing the recommendations of Astro2020 are remarkable. The new X-ray missions will teach us about stars across all star types and inform habitability studies. Next-generation ground-based and space-based optical and infrared telescopes will measure early light from colliding neutron stars that cannot be seen today and will probe exoplanet atmospheres of Earth-sized planets in their star's habitable zones. Small-and medium-scale missions across all wavelengths will show us yet-unmeasured properties of transient sources, while the next-generation radio array will teach us about jets of gas produced by neutron stars and black holes, including in mergers and cataclysms. Throughout it all, no wavelength range is neglected.
The large strategic missions, or flagships, are truly transformative, as Hubble has been and James Webb surely will be. But, the plan is not to put all of our eggs in one basket, but rather to spend our money as wisely as possible over both the short- and long-term, leading to a future where both human knowledge and the health of the field grows as never before. As the Astro2020 decadal report summarizes it:
"21st century astrophysics owes much of its richness to NASA's panchromatic suite of Great Observatories that spanned the spectrum from gamma rays to infrared, and which were accomplished with a wide range of scales, from what today is referred to as 'Probe scale' up to the very ambitious HST and JWST missions. Diverse missions of all scales, national and international, designed to view the universe in a multiplicity of complementary ways are now essential to progress in modern astrophysics."
Travel the Universe with astrophysicist Ethan Siegel. Subscribers will get the newsletter every Saturday. All aboard!
For perhaps the first time in history, the future of astronomy and astrophysics is bright and clear, and the biggest remaining uncertainty is just what this fantastic suite of new missions, observatories, instruments, facilities, and the next generation of scientists all will help to uncover in our never-ending quest to understand this Universe.
Space & Astrophysics
Ask Ethan: Will dark energy cause the Big Bang to disappear?
If we were born trillions of years in the future, could we even figure out our cosmic history?
Can just one equation describe the entire history of the universe?
As the first Friedmann equation celebrates its 99th anniversary, it remains the one equation to describe our entire universe.
Could there be a parallel universe identical to our own?
If there really is another version of you out there in a parallel universe, what can that teach us about reality?
Dark energy might be neither particle nor field
Everything else in the universe is either a particle or field. Dark energy behaves as neither, and it may be a property inherent to space itself.
A new window to the early universe (and aliens?)
The upcoming launch of the James Webb Space Telescope is the event of a lifetime.
Is "Dune" worth the hype?
Tighten your 'thopter seatbelts and get those worm-hooks ready: we're going to unpack the hype surrounding Dune, both the book and the movie.


Travis Cochran is serving as pastor of North River Community Church several miles outside of Cosmopolis, WA. He married Jennifer in 2009 and has four children–Russell, Michael, Ellie and Meri.
After graduating from Moody Bible Institute in 2008, Travis joined Village Missions, a mission organization that is focused on revitalizing rural churches in the United States and Canada. He is passionate about seeing small, country churches become vital gospel fellowships for the glory of Jesus.
Travis loves spending time with his family, bicycling, going to the beach, reading, guitar and piano, painting, and poetry.


Britney Spears is the latest star to suffer in the property market slump - she's set to lose millions of dollars in the sale of her California home.
The singer put the six-bedroom mansion on the market for $7.9 million (£5.3 million) back in September, 2008 but the property failed to attract any buyers.
And now Spears has dropped the price - she relisted the home on Monday (15Mar10) for $5.5 million (£3.7 million) - a loss of $2.4 million (£1,6 million), according to TMZ.com.
The home, which features three garages, a library, a spa and a media room, is where the Toxic hitmaker suffered a series of meltdowns in 2007 and 2008, most notably being removed from the house and transferred to a psychiatric hospital ward after refusing to hand her sons back to her ex-husband Kevin Federline.


Hey all, things have been crazy.
My twitch streams have been offline for a long time. But I've been doing a lot of work over at www.rotu.com! To keep track of where I've been and am, join the discord channel.
My name is Charles Logan, but my buddies call me Charlie! I'm an illustrator, a performer and a musician. I live in Boston, Massachusetts!
I have dreams of making it big. If you dream big as well, we'll get along really well! (: You can follow my Facebook page from down below or track down my Tumblr!


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\section{introduction} \label{sec:1}

Einstein and Rosen introduced the concept of wormholes, as early as in 1935 \cite{Einstein:1935tc}, which generate `short-cuts' in spacetime and allow `apparently faster than light' travel \citep{Visser:1995cc, Hawking:1973uf, Wald:1984rg, lobo2017wormholes, Witten:2019qhl} between two distinct spacetime points, dubbed as the Einstein-Rosen bridge. Wheeler, later, referred these solutions as `wormholes' \cite{Wheeler:1957mu}. It was soon realised that the wormhole solutions do not form a stable structure-- it's `throat' closes up too quickly when subjected to even tiny perturbation \citep{Kruskal:1959vx, Fuller:1962zza, Eardley:1974zz, Wald:1980nm}. Morris and Thorne showed that one can get a `traversable' wormhole solution by threading exotic form of matter (matter with negative energy density that violates null energy condition (NEC)) at or near the throat \citep{Morris:1988cz, Visser:1995cc}. Remarkably, the concept of exotic matter (such as dark energy, dark matter and phantom energy) is found to be useful in explaining several cosmological observations such as: the late time accelerated expansion of the universe, behaviour of the galactic rotation curves and the mass discrepancy in clusters of galaxies \cite{Lobo:2008sg}. It seems that the natural habitat for such matter would be of quantum origin. However, it has been noted that approaches using quantum features of standard model matter are insufficient for creating macroscopic wormholes \cite{Witten:2019qhl}.

Despite the facts stated earlier, certain classical ways are devised to circumvent the necessity for matter with negative energy density that violates energy conditions \cite{Hochberg:1990is, Bhawal:1992sz, Agnese:1995kd, Samanta:2018hbw, Fukutaka:1989zb, Ghoroku:1992tz, Furey:2004rq, Bronnikov:2009az}. Alternative theories of gravity, often known as the modified gravity theories, also offer new ways to avoid energy condition violations. Even though the convergence condition of null geodesics is violated in some cases, the literature has a considerable number of non-exotic matter models under modified gravity \citep{Lobo:2008zu, Kanti:2011jz, Kanti:2011yv, Zubair:2017oir, Shaikh:2016dpl, Ovgun:2018xys, Canate:2019spb}. So far, the $f(R)$, $f(R, T)$, $f(Q)$, and higher order gravity theories have garnered sufficient attention. It has been demonstrated that cosmic acceleration can be explained using $f(r)$ gravity \cite{Nojiri:2003ft} and that these models can provide wormhole solutions with viable matter sources \citep{Lobo:2009ip, Garcia:2010xb, MontelongoGarcia:2010xd, Sajadi:2011oei, Moraes:2017dbs, Sahoo:2017ual, Moraes:2019pao, Sahoo:2020sva, Hassan:2021egb, Mustafa:2021ykn}.

Although wormholes are still regarded speculative, new advancements in precision black hole measurements have enhanced the need of evaluating plausible wormhole models (as black hole mimickers). Studies of events such as wormhole merger \citep{Krishnendu:2017shb, Cardoso:2016oxy} and their quasinormal modes \citep{Aneesh:2018hlp, DuttaRoy:2019hij} can aid in the detection of wormhole signature in the cosmos. Similarly. their lensing effects, shadows, Einstein ring and other features may be analysed in detail for possible detection \citep{Abe:2010ap, Toki:2011zu, Takahashi:2013jqa, Cramer:1994qj, Perlick:2003vg, Tsukamoto:2012xs, Bambi:2013nla, Nedkova:2013msa, Zhou:2016koy,  Dzhunushaliev:2012ke, Dzhunushaliev:2013lna,  Aringazin:2014rva, Dzhunushaliev:2016ylj}. Any of these signatures, if detected, may also favour modified gravity theories over general relativity.

The Ellis-Bronnikov wormhole (4D-EB) \citep{Ellis:1973yv, Bronnikov:1973fh} was constructed separately by Ellis and Bronnikov employing phantom scalar field (a field with negative kinetic term) and is one of the most researched wormhole geometries. Several studies on this spacetime can be found in the literature, including geometry of spinning 4D-EB spacetime \cite{Chew:2016epf}, generalized spinning of 4D-EB wormhole in scalar-tensor theory \cite{Chew:2018vjp}, hairy Ellis wormholes solutions \cite{Chew:2020svi}, Ellis wormholes in anti-de Sitter space \cite{Blazquez-Salcedo:2020nsa}, stability analysis of 4D-EB solution in higher dimensional spacetime \cite{Torii:2013xba} as such. Kar et al. presented a generalised version of 4D-EB spacetime (4D-GEB) \cite{Kar:1995jz}, where the necessity for exotic matter is {\em partially} evaded by introducing a new wormhole parameter, $m \geq 2$ ($m = 2$ corresponds to 4D-EB geometry). Quasinormal modes, echoes and some other aspects of 4D-GEB spacetimes is reported in \cite{DuttaRoy:2019hij}. We recently suggested a further generalisation \cite{Sharma:2021kqb} where the 4D-GEB geometry is embedded in a five-dimensional warped spacetime and showed that energy conditions are satisfied even for $m=2$ i.e. a novel 5D-EB geometry.
Such embedding is also considered in the context of Schwarzschild-like spacetime \cite{Culetu:2021zun} as well. Recently Kar has proposed another 5D warped wormhole model where the warping chosen is largely inspired by the non-static Witten bubble \cite{Kar:2022omn}.

The theories of extra spatial dimensions started with the work of Kaluza (1921) and Klein (1926) \cite{Kaluza:1921tu, Klein:1926tv}, who attempted to combine electromagnetism and gravity in a 5D-gravity framework. In fundamental physics, be it the string theory \cite{Green:1987sp}, which is a work in progress in unifying the quantum theory and gravity or in the context of symmetries of particle physics  \cite{Furey:2015yxg, Baez:2001dm, Baez:2010ye, Furey:2018yyy, Furey:2018drh, Gillard:2019ygk}, the extra  dimensions seems to appear {\em naturally}. Development of string theory also motivated the so-called brane-world scenarios-- where our 4D Universe (3-brane) is embedded in a higher dimensional bulk. The Dvali-Gabadadze-Porrati (DGP) models produce infra-red modification with extra dimensional gravity dominating at low energy scale \cite{Dvali:2000hr}. Further generalization to cosmology of DGP model lead to late-accelerated-expansion \cite{Deffayet:2000uy}.
Perhaps, the most popular of these models are the `warped braneworld' models \cite{Rubakov:1983bb, Gogberashvili:1998iu, Gogberashvili:1998vx,Randall:1999ee, Randall:1999vf} that generate ultra-violet modification to general relativity with extra dimensional gravity dominating at high energy scale. This model proposes a non-factorizable geometry- a curved five-dimensional spacetime in which the 4D-metric is warped by the extra dimension. Though some recent research on wormholes embedded in higher-dimensional spacetime has been published \citep{Lobo:2007qi, deLeon:2009pu, Wong:2011pt, Kar:2015lma, Banerjee:2019ssy, Wang:2017cnd}, warped braneworld models have not been examined.

We showed in \cite{Sharma:2021kqb} that a GEB (and EB) spacetime embedded in 5D warped bulk (5D-WWEB) model meets the energy conditions. As a follow-up, we analysed the timelike trajectories in these spacetimes in detail in \cite{Sharma:2022tiv}. In this work, we investigate the congruence of geodesics in the original 4D-GEB geometry as well as the 5D-WEB spacetime and compare them to see how the wormhole parameter and warped extra dimension effect the congruence evolution. Note that the 5D line element we utilised is the well-known thick brane model \cite{Dzhunushaliev:2009va, Ghosh:2008vc}, in which the warp factor is a smooth function of the extra dimension (thus there are no derivative jump or delta functions in the curvature and connections).

The following is a breakdown of the structure of this article. Section (\ref{sec:2}) introduces the wormhole spacetimes that correspond to the novel 5D-WGEB model. It includes a summary of the geometric properties and geodesic equations. In Section (\ref{sec:3}), we used timelike velocity vector field to obtain analytic expressions for the expansion and shear (with zero rotation) variables corresponding to the geodesic congruences and solve the Raychaudhury equations for 4D-GEB geometry.  In section (\ref{sec:4}), we numerically computed the ESR variables for two cases-- without rotation and with rotation for both 4D-GEB and 5D-WGEB geometries. In Section (\ref{sec:5}), to get further insight about the congruences in 4D and 5D, the evolution of cross-sectional area of a geodesic congruence of timelike geodesics is numerically determined and presented graphically. Finally, in Section (\ref{sec:6}) we summarise the key results and discuss future direction of research.

\section{The 5D-WGEB geometry and Geodesics} \label{sec:2}

The specific line element of 5D-WGEB geometry introduced in \cite{Sharma:2021kqb} is as follows,
\begin{equation}
ds^{2} =  e^{2f(y)} \Big[ - dt^{2} +  dl^{2} + r^{2}(l)~\big(  d\theta^{2} + \sin^{2}\theta~d\phi^{2} \big) \Big] + dy^{2}, \label{eq:5d-line-element}
\end{equation}
where, the extra spatial dimension is represented by $y$ ($ - \infty \leq y \leq \infty$). $f(y)$ is the warping factor that we chose as, $f(y) = \pm \log[\cosh(y/y_{0})]$, corresponding to a thick brane scenario. The four dimensional part of the above metric (\ref{eq:5d-line-element}) is a spherically symmetric, ultra-static Generalized Ellis-Bronnikov wormhole spacetime, given by,
\begin{equation}
ds^{2}_{4D} = - dt^{2} +  dl^{2} + r^{2}(l)~\big(  d\theta^{2} + \sin^{2}\theta~d\phi^{2} \big) \label{eq:generalised-E&B-l}
\end{equation}
\begin{equation}
\mbox{where}~~~~~ r(l) = (b_{0}^{m} + l^{m})^{1/m}. \label{eq:r(l)}
\end{equation}
Here, $l$ is the `proper radial distance' or the `tortoise coordinate'. The `radius' of the wormhole throat is given by $b_{0}$, and $m$ is the wormhole parameter that essentially generalises the EB geometry ($m=2$) to GEB geometry($m > 2$). For smoothness of $r(l)$, it is necessary to have even valued $m$. In the usual radial coordinate $r$, the above metric (\ref{eq:generalised-E&B-l}) can be written as,
\begin{equation}
ds^{2} = - dt^{2} + \frac{dr^{2}}{\Big( 1 - \frac{b(r)}{r} \Big)} + r^{2} \big( d\theta^{2} + \sin^{2}\theta d\phi^{2} \big), \label{eq:generalised-E&B-r}
\end{equation}

where $r$ and $l$ are related through the {\em shape function} $b(r)$ as,
\begin{equation}
dl^{2} = \frac{dr^{2}}{ \Big( 1 - \frac{b(r)}{r} \Big)}~~~~~\implies ~~~~~\quad b(r) = r - r^{(3-2m)} (r^{m} - b_{0}^{m})^{ \Big( 2 - \frac{2}{m} \Big)}. \label{eq:r-l-relation}
\end{equation}

The geometric and curvature quantities for both the 5D-WGEB and 4D-GEB spacetimes have been discussed in detail in \cite{Sharma:2021kqb}. Interested reader is requested to go through that article. In \cite{Sharma:2022tiv}, we further discussed the possible timelike trajectories in detail. In the following we shall focus on the congruence of geodesics.
For completeness, we rewrite the geodesic equations corresponding to metrics (\ref{eq:5d-line-element}) and (\ref{eq:generalised-E&B-l}) as these equations are to be solved along with the geodesic deviation equations to determine the properties of the congruences
The geodesic equations for 4D-GEB are,
\begin{equation}
\frac{d^{2}t}{d\lambda^{2}} = 0 \label{eq:geodesic-1}
\end{equation}
\begin{equation}
\frac{d^{2}l}{d\lambda^{2}} - l^{-1+m}  ~\big( b_{0}^{m} + l^{m} \big)^{-1 + \frac{2}{m}} ~\Big[ \Big( \frac{d\theta}{d\lambda} \Big)^{2} + \sin^{2}\theta  ~\Big( \frac{d\phi}{d\lambda} \Big)^{2} \Big] = 0 \label{eq:geodesic-2}
\end{equation}
\begin{equation}
\frac{d^{2}\theta}{d\lambda^{2}} + \frac{2l^{-1+m}}{\big( b_{0}^{m} + l^{m} \big)} \frac{dl}{d\lambda} \frac{d\theta}{d\lambda} - \sin\theta \cos\theta \Big( \frac{d\phi}{d\lambda} \Big)^{2} = 0 \label{eq:geodesic-3}
\end{equation}
\begin{equation}
\frac{d^{2}\phi}{d\lambda^{2}} + 2 \cot\theta \frac{d\theta}{d\lambda} \frac{d\phi}{d\lambda} + \frac{2l^{-1+m}}{ \big( b_{0}^{m} + l^{m} \big) } \frac{dl}{d\lambda} \frac{d\phi}{d\lambda} = 0 \label{eq:geodesic-4}
\end{equation}
and the geodesic equations for the 5D-WGEB model are as follows.
\begin{equation}
\frac{d^{2}t}{d\lambda^{2}} + 2 ~f'(y) ~\frac{dt}{d\lambda} ~\frac{dy}{d\lambda} = 0 \label{eq:geodesic-11}
\end{equation}
\begin{equation}
\frac{d^{2}l}{d\lambda^{2}} + 2~f'(y)~\frac{dl}{d\lambda}~\frac{dy}{d\lambda}  - l^{-1+m}  ~\big( b_{0}^{m} + l^{m} \big)^{-1 + \frac{2}{m}} ~\Big[ \Big( \frac{d\theta}{d\lambda} \Big)^{2} + \sin^{2}\theta  ~\Big( \frac{d\phi}{d\lambda} \Big)^{2} \Big] = 0 \label{eq:geodesic-22}
\end{equation}
\begin{equation}
\frac{d^{2}\theta}{d\lambda^{2}} + 2 ~f'(y) ~\frac{d\theta}{d\lambda} ~\frac{dy}{d\lambda} + ~\frac{2l^{-1+m}}{(b_{0}^{m} + l^{m})} ~\frac{d\theta}{d\lambda} ~\frac{dl}{d\lambda} - \sin\theta ~\cos\theta ~\Big( \frac{d\phi}{d\lambda} \Big)^{2} = 0 \label{eq:geodesic-33}
\end{equation}
\begin{equation}
\frac{d^{2}\phi}{d\lambda^{2}} + 2 ~f'(y) ~\frac{d\phi}{d\lambda} ~\frac{dy}{d\lambda} + 2 ~\cot\theta ~\frac{d\theta}{d\lambda} ~\frac{d\phi}{d\lambda} + \frac{2l^{-1+m}}{ \big( b_{0}^{m} + l^{m} \big) } ~\frac{dl}{d\lambda} ~\frac{d\phi}{d\lambda} = 0 \label{eq:geodesic-44}
\end{equation}
\begin{equation}
\frac{d^{2}y}{d\lambda^{2}} + f'(y)~e^{2f(y)}~\Big[ \Big( \frac{dt}{d\lambda} \Big)^{2} - \Big( \frac{dl}{d\lambda} \Big)^{2} - (b_{0}^{m} + l^{m})^{2/m} ~\Big[ \Big( \frac{d\theta}{d\lambda} \Big)^{2} + \sin^{2}\theta ~\Big( \frac{d\phi}{d\lambda} \Big)^{2} \Big] \Big] = 0 \label{eq:geodesic-55}
\end{equation}
 Here $\lambda$ is the affine parameter. The differences between 4D and 5D geodesic equations can be seen explicitly from the above expressions, which are $y$ and $\dot{y}$ dependent term in the right hand sides of the Eqs. (\ref{eq:geodesic-11}) to (\ref{eq:geodesic-55}), and an extra equation for motion along the extra dimension. Note that Eq. \ref{eq:geodesic-55}, implies that in presence of growing warp factor we have confined trajectories (whose  $y$-coordinate oscillate about $y=0$) and in presence of a decaying warp factor we have runaway trajectories ($y \rightarrow \pm \infty$ with evolving $\lambda$) \cite{Sharma:2022tiv,Ghosh:2009ig}.

\section{Derivation of ESR From the Velocity Field} \label{sec:3}

In the following we solve the the Raychaudhuri equations to analyse the flow of geodesic congruences through the kinematic variables expansion $\Theta$, shear $\Sigma_{AB}$ and rotation $\Omega_{AB}$ (ESR). Note that we shall consider those trajectories that cross the throat (for details of {\em crossing trajectories} see \cite{Sharma:2022tiv}).
The Raychaudhuri equation for expansion \cite{Kar:2006ms,Poisson:2009pwt}, given by
\begin{equation}
\frac{d\Theta}{d\lambda} + \frac{1}{3}\Theta^{2} + R_{AB}u^{A}u^{B} + \Sigma^{2} - \Omega^{2} = 0, \label{eq:raychaudhuri-eq}
\end{equation}
is dependent on other kinematic variables such as the shear and rotation, as well as the curvature term, $R_{AB}u^{A}u^{B}$. The above equation is a non-linear first order differential equation that can be transformed into a second order {\em linear} form via redefining the $\Theta$ as $\Theta = 3 \frac{\dot{F}}{F} $. Thus, Eq. (\ref{eq:raychaudhuri-eq}) turn out to be,
\begin{equation}
\frac{d^{2}F}{d\lambda^{2}} + \frac{1}{3} \left( R_{AB}u^{A}u^{B} + \Sigma^{2} - \Omega^{2} \right)F = 0. \label{eq:raychaudhuri-eq-2}
\end{equation}
The congruence converges at a finite $\lambda$ where $F = 0$, $\dot{F} < 0$. Using the Sturm comparison theorems in differential equations, one can show that convergence happens when
\begin{equation}
R_{AB}u^{A}u^{B} + \Sigma^{2} - \Omega^{2} \geq 0 .\label{eq:convergence-condition}
\end{equation}
The role of the terms that appear in the evolution equation for expansion, is clearly shown in the above convergence condition. While rotation works against convergence, shear works in its favour. When $\Omega_{AB} = 0$ then $R_{AB}u^{A}u^{B} \geq 0$ leads to focusing of the congruences.
In the case of vanishing rotation, we derived analytic expressions for ESR variables using the first integral of the metric and plotted them below.
ESR variables can be derived directly from their definitions where the velocity vector field is $u^{A}$. The following are the formal definitions for the expansion $\Theta$, shear $\Sigma_{AB}$, and rotation $\Omega_{AB}$ \citep{Poisson:2009pwt}:
\begin{equation}
\Theta = \nabla_{A}u^{A} , \label{eq:theta-definition}
\end{equation}
\begin{equation}
\Sigma_{AB} = \frac{1}{2}~\left( \nabla_{B}u_{A} + \nabla_{A}u_{B}  \right) - \frac{1}{n - 1}~h_{AB}~\Theta , \label{eq:sigma-definition}
\end{equation}
\begin{equation}
\Omega_{AB} = \frac{1}{2}~\left( \nabla_{B}u_{A} - \nabla_{A}u_{B}  \right). \label{eq:omega-definition}
\end{equation}
Here, $n$ is the dimension of spacetime, $h_{AB} = g_{AB} \pm u_{A}u_{B} $ is the projection tensor and $u_{A}u^{A} = \mp 1$ (The plus and minus sign stands for the timelike and spacelike geodesics, respectively). In case of the 4D-GEB geometry the calculation of the first integral $u^{A}$ (whose expressions are independent of the affine parameter $\lambda$) corresponding to each coordinate could be found. These are listed bellow in the case of timelike trajectories (for sake of simplicity, we choose $\theta = \pi/2 $ without loosing any generality).
\begin{equation}
\dot{t} = k ,\label{eq:t-dot}
\end{equation}
\begin{equation}
\dot{\phi} = \frac{h}{\left( b_{0}^{m} + l^{m} \right)^{\frac{2}{m}}} ,\label{eq:phi-dot}
\end{equation}
\begin{equation}
\dot{l} = \sqrt{k^{2} - \frac{h^{2}}{\left( b_{0}^{m} + l^{m} \right)^{\frac{2}{m}}} - 1} .\label{eq:l-dot}
\end{equation}
Eq. (\ref{eq:l-dot}) obtained by using Eqs, (\ref{eq:t-dot}), (\ref{eq:phi-dot}) and timelike constraint on geodesics, $ g_{AB}u^{A}u^{B} =  -1  $. Here, $k$ and $h$ are the conserved energy and angular momentum per unit mass of the timelike particle.
For given velocity vector field (Eqs. \ref{eq:t-dot} to \ref{eq:l-dot}) of 4D-GEB wormhole geometry, the expansion scalar and amplitude squared of shear is,
\begin{equation}
\theta = l^{-1+m} \left[ \frac{h^{2} + \left(b_{0}^{m} + l^{m} \right)^{\frac{2}{m}} \left(k^{2} - \frac{h^{2}}{\left(b_{0}^{m} + l^{m} \right)^{\frac{2}{m}}} - 1 \right)}{\left(b_{0}^{m} + l^{m} \right)^{1 + \frac{2}{m}} \sqrt{k^{2} - \frac{h^{2}}{\left(b_{0}^{m} + l^{m} \right)^{\frac{2}{m}}} - 1}} \right] , \label{eq:GEB-theta}
\end{equation}
\begin{equation}
\resizebox{\textwidth}{!}
{%
$ \Sigma^{2} = \frac{l^{ -2 + 2m } \left(  b_{0}^{m} + l^{m} \right)^{\frac{-2 \left( 1 + m \right)}{m}} \left[ h^{4} \left( 13 + 4k^{2} + k^{4} \right) - h^{2} \left( - 16 + 12k^{2} + 3k^{4} + k^{6} \right) \left( b_{0}^{m} + l^{m} \right)^{2/m} + \left( - 1 + k^{2} \right)^{2} \left( 9 - k^{2} + k^{4} \right) \left( b_{0}^{m} + l^{m} \right)^{\frac{4}{m}} \right]}{9 \left[ - h^{2} + \left( - 1 + k^{2} \right) \left(  b_{0}^{m} + l^{m} \right)^{\frac{2}{m}} \right]} $%
} \label{eq:GEB-sigsq}
\end{equation}
One can easily check that if $l \rightarrow 0$ or $l \rightarrow \pm \infty$ the expansion scalar $\Theta \rightarrow 0$ and the shear scalar $\Sigma^{2}\rightarrow$ $0$. This is true irrespective of the value of $h$. Thus there will be no expansion/contraction and distortion of timelike geodesic congruences at the wormhole throat and at asymptotic flat regions which is expected. The profile of $\Sigma^2$ is symmetric about $l=0$, on the other hand, $\theta$ profile is exactly asymmetric about the throat. Fig.(\ref{fig:GEB-ESR-analytic}) clarifies the effect of the wormhole parameter $m$ on expansion and shear. The same $\Theta$-variation has also been reported in \cite{DuttaRoy:2019hij}. While plotting the figures the numerical values are chosen to be consistent with the corresponding `crossing trajectory conditions' \cite{Sharma:2022tiv}.
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{GEB-crossing-theta-analytic.pdf}
\hspace{1cm}
\includegraphics[scale=.5]{GEB-crossing-sigma-analytic.pdf}
\caption{Evolution of $\Theta$ and $\Sigma^{2}$ in case of crossing geodesic congruences with different choice of wormhole parameter $m = 2, 4, 6$ (blue , black and red curves). Where, $b_{0} = 1$, $k = \sqrt{3}$ and $h = 1$. }
\label{fig:GEB-ESR-analytic}
\end{figure}
Variation in  $\Theta$ (or $\Sigma^{2}$) is qualitatively similar in the asymptotic regions for $m = 2$ and $m > 2$ geometries. However, they are very different at about the throat which is a reminiscent of the fact that upto $m^{th}$ order derivative of the geodetic potential vanishes at the throat \cite{Sharma:2022tiv}. Thus a congruence of geodesics coming from $l=\infty$, with zero initial ESR, expands and then contract before crossing the throat with zero ESR. However, the location of the extrema moves farther away from the throat with increasing $m$.
In case of 5D-WGEB model, it is difficult build any intuition from the equations because of their complexity. Therefore, in the following we numerically analyse the 4D and 5D scenario that is difficult to solve analytically.

\section{Numerical Analysis of ESR variables} \label{sec:5}

To describe the behaviour of evolution of expansion $\Theta$ in both the 4D-GEB and 5D-WGEB models, we numerically solved Eq. (\ref{eq:evolution-of-B_AB}) along with the geodesic equations. For this, we have chosen two types of boundary conditions along with $\Theta$ and $\Sigma_{AB}$ being zero at the throat: (a) without rotation ($\Omega_{AB}=0$) and (b) with rotation ($\Omega_{AB}\neq 0$). The velocity components for timelike geodesics in the 4D-GEB background satisfying the timelike constraint are chosen as $\{ \dot{t}(0) = \sqrt{3} $, $\dot{l}(0) = 1.41421$, $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = 0 \} $  and the same for the 5D-WGEB models with growing and decaying warp factor are  $\{ \dot{t}(0) = 1.71485 $, $\dot{l}(0) = 1.39665 $, $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = 0 $, $ \dot{y(0)} = 0 \}$ and $ \{ \dot{t}(0) = 1.74943 $, $\dot{l}(0) = 1.43195 $, $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = 0 $, $ \dot{y(0) = 0} \}$  respectively (see Appendix and \cite{Sharma:2022tiv}). Fig. (\ref{fig:GEB-ESR-wor2}) to (\ref{fig:5d-decay-ESR-wr2}) show the evolution of ESR variables with (continuous curves) and without (dashed curves) rotation for three different values of $m$ for 4D-GEB and 5D-WGEB (with growing and decaying warp factor) models, respectively.

\subsection{case-1: Congruences in 4D-GEB spacetime with and without rotation}

The evolution of numerically solved expansion $\Theta$ (without rotation) in the 4D-GEB  scenario  shows exact similarity with the analytic behaviour discussed earlier thus proving the accuracy of the numerical computation. The quantitative difference wrt Fig. \ref{fig:GEB-ESR-analytic} is because we have chosen angular momenta $h$ of the geodesics of the congruence to be zero for computational simplicity. For readers sake, we also showed the evolution of the curvature term $R_{AB}u^{A}u^{B}$ along with the expansion and shear.
The boundary values used in this subsection is shown in Appendix \ref{app-1}.

\begin{figure}[H]
\centering
\includegraphics[scale=.4]{GEB-crossing-theta-wor2.pdf}
\hspace{.3cm}
\includegraphics[scale=.4]{GEB-crossing-sigma-wor2.pdf}
\hspace{.2cm}
\includegraphics[scale=.45]{GEB-crossing-curvature-wor2.pdf}
\caption{Evolution of $\theta$, $\Sigma^{2}$ and the curvature term without rotation in case of crossing geodesic congruences for wormhole parameter, $m = 2, 4$, and $6$ (blue, black and red curves) with $b_{0} = 1$. }
\label{fig:GEB-ESR-wor2}
\end{figure}

Something interesting happens in presence of rotation in the congruence (see Fig. \ref{fig:GEB-ESR-wr2}), even for 4D-GEB that could not be derived through analytic approach.
Posing a boundary condition at the throat with non-zero rotation $\Omega_{AB}(\lambda=0) \neq 0$ seems to have an effect of nullifying the effect of $m$ the wormhole parameter. Thus the $\Theta$ profile (vs $l$) for different values of $m$ overlaps. This raises a possibility that a {\em rotating} Ellis-Bronnikov wormhole may not require exotic matter to be stable. However a detailed study of this interesting case is not possible within the scope of this article.
Fig. \ref{fig:GEB-ESR-wr2} further shows that the initial rotation also dies away in the asymptotic regions.
 \begin{figure}[H]
\centering
\includegraphics[scale=.4]{GEB-crossing-theta-wr2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.4]{GEB-crossing-sigma-wr2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.4]{GEB-crossing-omega-wr2.pdf}
\hspace{0.2cm}
\caption{Evolution of ESR in case of crossing geodesic congruences with rotation for wormhole parameter, $m = 2, 4$, and $6$ (blue, black and red curves) with $b_{0} = 1$. }
\label{fig:GEB-ESR-wr2}
\end{figure}

\subsection{Case-2: Congruences in 5D-WGEB spacetime with growing warp factor}

In the presence of growing warp factor, the evolution of ESR variables with rotation and without rotation are shown in figures \ref{fig:5d-grow-ESR-wor2} and \ref{fig:5d-grow-ESR-wr2} respectively.
The boundary values used in this subsection is shown in Appendix \ref{app-2}.
The general observation on evolution of $\Theta$ implies that a congruence of dense geodesics (focussed) coming from positive $l$ will defocus on the other side and vice versa. The presence of geodesic singularity at finite $l$ is essentially because of bounded trajectories along $y$ \cite{Ghosh:2010gq}. For $m=2$ geometry, there is only one extrema on either side of $l=0$, but for $m > 2$ we have two extrema on either side.
Shear on the other hand increases monotonically at $l \rightarrow \pm \infty$. The curvature term particularly become positive in the asymptotic region and around the throat which is different from the 4D case where this term was negative everywhere.

\begin{figure}[H]
\centering
\includegraphics[scale=.4]{5d-grow-crossing-theta-wor2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.4]{5d-grow-crossing-sigma-wor2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.45]{5d-grow-crossing-curvature-wor2.pdf}
\caption{Evolution of $\theta$, $\Sigma^{2}$ and the curvature term without rotation in case of crossing geodesic congruences with growing warp factor for $m = 2, 4$, and $6$ (blue, black and red curves) with $b_{0} = 1$. }
\label{fig:5d-grow-ESR-wor2}
\end{figure}

Introduction of non-zero rotation, $\Omega_{AB}$, at the throat again compensate for the differences in different $m$ geometries.
The evolution of ESR variables are similar as the 4D case with rotation to some extent about. Congruence singularity sustains the presence of rotation. Though expansion diverges, shear on the other hand vanishes (unlike the 4D case and without rotation case) in the asymptotic region like rotation.
\begin{figure}[H]
\centering
\includegraphics[scale=.4]{5d-grow-crossing-theta-wr2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.4]{5d-grow-crossing-sigma-wr2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.4]{5d-grow-crossing-omega-wr2.pdf}
\hspace{0.2cm}
\caption{Showing evolution of ESR  in case of crossing geodesic congruences  for growing warp factor wormhole parameter, $m = 2, 4$, and $6$ (blue, black and red curves) with as $b_{0} = 1$. }
\label{fig:5d-grow-ESR-wr2}
\end{figure}

\subsection{case-3: Congruences in 5D-WGEB spacetime with decaying warp factor}

Fig. \ref{fig:5d-decay-ESR-wor2}, in the presence of a decaying warp factor, without rotation, the differences in the $\Theta$-profiles between $m=2$ and $m > 2$ geometries is decreased-- both have one of extrema on either side of the throat.
Further, the congruence singularities is moved towards $l \rightarrow \pm \infty$.
Shear has local minima at finite $\pm l$ (except the global minima at $l=0$) and increases monotonically beyond them.
The curvature term, though positive in the asymptotic region, around the throat it contributes negatively (even for $m > 2$) which is similar to the 4D case and opposite to the case with a growing warp factor.

\begin{figure}[H]
\centering
\includegraphics[scale=.4]{5d-decay-crossing-theta-wor2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.4]{5d-decay-crossing-sigma-wor2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.45]{5d-decay-crossing-curvature-wor2.pdf}
\caption{Evolution of $\theta$, $\Sigma^{2}$ and the curvature term without rotation in case of crossing geodesic congruences with decaying warp factor for $m = 2, 4$, and $6$ (blue, black and red curves) with $b_{0} = 1$.}
\label{fig:5d-decay-ESR-wor2}
\end{figure}

Introduction of non-zero rotation, $\Omega_{AB}$, at the throat, again compensate for the differences in different $m$ geometries. Remarkably, it slows the evolution down and avoid the divergences at large $l$ as it happened in case with growing warp factor.
The evolution of shear and rotation are similar to the case with a growing warp factor.
The boundary values used in this subsection is shown in Appendix \ref{app-3}.
\begin{figure}[H]
\centering
\includegraphics[scale=.4]{5d-decay-crossing-theta-wr2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.4]{5d-decay-crossing-sigma-wr2.pdf}
\hspace{0.2cm}
\includegraphics[scale=.4]{5d-decay-crossing-omega-wr2.pdf}
\hspace{0.2cm}
\caption{Showing evolution of ESR  in case of crossing geodesic congruences  for decaying warp factor wormhole parameter, $m = 2, 4$, and $6$ (blue, black and red curves) with as $b_{0} = 1$. }
\label{fig:5d-decay-ESR-wr2}
\end{figure}

We can make some general conclusion for the evolution of ESR variables from the figures given above. In case with zero rotation, at $l = 0$ (this is by choice) and at $l = \pm b_{0} = \pm 1$, expansion and shear have same value for all $m$ for both 4D and 5D models.
The effect of wormhole parameter $m$ neutralises when we introduce rotation ($\Omega_{AB}\neq 0$) in the congruence. Occurrence of divergences in the $\Theta$ and $\Sigma^2$ profiles of the geodesic congruences or congruence singularity is due to the extra dimension or the warping factor to be specific. However, the presence of rotation can avoid these divergences in the case of decaying warp factor.

\section{Evolution of cross-sectional area of Congruences} \label{sec:4}

In this section we present a different perspective on the evolution of ESR variables.
Let us consider a congruence of four timelike geodesics coming from one side ($l = - \infty$) of the throat, crosses the throat, and reaches the other side to infinity ($l = \infty$). One may ask how the cross sectional area of these geodesic congruences evolve as the affine parameter or time evolves? To answer this question, we numerically solve the following set of equations (which are essentially the Raychaudhury and the geodesic deviation equations),
\begin{equation}
u^{C}\nabla_{C}B_{AB} = - B_{AC}B^{C}_{B} - R_{ACBD}u^{C}u^{D} \label{eq:evolution-of-B_AB}
\end{equation}
\begin{equation}
\xi^{A}_{;B}u^{B} = B^{A}_{B} \xi^{B} \label{eq:evolution-deviation-vector}
\end{equation}
along with the geodesic equations (\ref{eq:geodesic-1} - \ref{eq:geodesic-4}). Here, $B_{AB} = \nabla_{A}u_{B}$ is gradient of velocity vector field, $\xi^{A}$ representing the deviation vector between two geodesics and Eq. (\ref{eq:evolution-deviation-vector}) is the evolution equation for the deviation vector. To see the evolution from the perspective of a local observer, we express the tensorial quantities in the frame basis. In coordinate and frame basis, the metric tensors and components of deviation vector are related via the vierbein field $e^{A}_{a}$ as
\begin{equation}
g^{AB} = e^{A}_{a}e^{B}_{b} \eta_{ab} ,\label{eq:relation-g-eta}
\end{equation}
\begin{equation}
\xi^{A} = e^{A}_{a} \xi^{a} .\label{eq:relation-deviation-A-a}
\end{equation}
Here capital and small latin indices stand for general spacetime coordinate (w.r.t coordinate basis) and local laboratory coordinate or local Lorentz frame (w.r.t. frame basis) respectively.
As a first case, let us choose the boundary conditions (on $B_{AB}$) such that all the rotation components vanish (i.e. $\Omega_{AB}=0$). We choose a congruence of four geodesics (along with a reference geodesic at the {\em origin}) such that the cross-sectional area projected on $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{33}} \xi^{3}$ or $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{44}} \xi^{4}$ planes is of square shape at $\lambda = 0$ (or at the throat as $ l(\lambda=0) = 0$ for crossing geodesics). Then Eq. (\ref{eq:evolution-deviation-vector}) is solved for the four deviation vectors which represent the distances/deviations of the four geodesics (at the four corners of the square) from a central geodesic (co-moving observer) which is at the centre of the imaginary square (in the figures below).
The following figures show the evolution of the projected cross sectional area for both 4D-GEB and 5D-WGEB geometries with respect to $\lambda$ (as geodesic congruence crosses the throat at $\lambda=0$). Various subcases correspond to different values of the wormhole parameter $m = 2, 4, 6$ and growing/decaying warp factor is analysed. In case where $m=4$ and $m=6$ geometries show similar behaviour we have presented evolution for $m=4$. Note that we choose the angular momentum `$h$' to be non-zero (and consistent with all the constraints), in this section, which plays an interesting role in 5D scenario.

\subsection{Case-1: 4D-GEB spacetime}

Congruences in the 4D-GEB spacetime is shown in Fig. (\ref{fig:4d-deviation-evolution}).
The boundary values used in this subsection is shown in Appendix \ref{app-4}.
Four snapshots of the area with increasing and decreasing $\lambda$ is depicted in left and right plots respectively. These plots show features of expansion and shear with no rotation as expected. The cross sectional area focuses as geodesic congruence approaches towards the throat $l=0$, while distortion in projected area decreases until it hits the throat and increases after passing through the throat.
\begin{figure}[h]
\centering
\includegraphics[scale=.60]{4d-deviation-2-negative-l.pdf}
\hspace{1cm}
\includegraphics[scale=.60]{4d-deviation-2-positive-l.pdf}
\hspace{1cm}
\includegraphics[scale=.60]{4d-deviation-4-negative-l.pdf}
\hspace{1cm}
\includegraphics[scale=.60]{4d-deviation-4-positive-l.pdf}
\caption{Showing evolution of projected geodesic congruence plane (without rotation) in case of crossing congruences with different choice of wormhole parameter $m = 2, 4$ and $b_{0} = 1$. }
\label{fig:4d-deviation-evolution}
\end{figure}

The evolution of cross sectional area is symmetric about the throat which is opposite to the overall expansion profile as shown in the the previous section.
However, there are no qualitative difference in the evolution of cross-section between $m = 2$ and $m > 2$ geometries. Apparently, a congruence which is widespread and largely distorted coming from one asymptotic region crossing the throat with minimum size and distortion and regenerates its original shape and size on the other side.

\subsection{Case-2: 5D-WGEB spacetime with growing warp factor}

The evolution profiles of congruence cross-sections projected on the $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{33}} \xi^{3}$ and $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{44}} \xi^{4}$ plane for 5D-WGEB model with growing warp factor is shown in the Fig. (\ref{fig:5d-grow-deviation-evolution}) for two different values of $m = 2, 4$.
The boundary values used in this subsection is shown in Appendix \ref{app-5}.
Since the evolution is symmetric about $\lambda=0$ (or $l=0$) we only present here plots for $\lambda \geq 0$ (or $l \geq 0$).

\begin{figure}[H]
\centering
\includegraphics[scale=.6]{5d-grow-deviation-2-positive-l.pdf}
\hspace{1cm}
\includegraphics[scale=.6]{5d-grow-deviation-2-positive-y.pdf}
\includegraphics[scale=.60]{5d-grow-deviation-4-positive-l.pdf}
\hspace{1cm}
\includegraphics[scale=.60]{5d-grow-deviation-4-positive-y.pdf}
\caption{Showing evolution of projected geodesic congruence plane (without rotation) in case of crossing congruences with growing warp factor for wormhole parameter $m =2, 4$ and $b_{0} = 1$. }
\label{fig:5d-grow-deviation-evolution}
\end{figure}

The evolution of the area element projected on $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{33}} \xi^{3}$ plane has almost similar profile as the 4D case with a crucial difference-- the area has rotation (that slowly increases with increasing $m$) too along with expansion and shear even though the rotation was zero at the throat as in the 4D case. Thus the warping factor triggers a rotation in the congruence if individual geodesics has angular momenta. In other words, congruence of geodesics with angular momenta that has initial rotation at asymptotic regions may become rotation-less while passing through the throat.
The other effect of the growing warp factor is visible through the projection of the evolution on the $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{44}} \xi^{4}$ plane which shows that the geodesics whose $y$-coordinate are some distant apart at $\lambda=0$, have the same $y$-coordinate in the asymptotic regions, i.e. separation along $y$-axis vanishes among the geodesics as $\lambda \rightarrow \pm \infty$.
However, there is no rotation of the congruence on this plane.

\subsection{Case-3: 5D-WGEB spacetime with decaying warp factor}

The figures (\ref{fig:5d-decay-deviation-evolution-2}) and (\ref{fig:5d-decay-deviation-evolution-4}) show the evolution of projected area element with different $m$ values ($m = 2, 4$) in $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{33}} \xi^{3}$ and $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{44}} \xi^{4}$ plane for the 5D-WGEB model with a decaying warp factor.
The boundary values used in this subsection is shown in Appendix \ref{app-6}.
There are few factors remarkably different from the case with a growing warp factor.
First, evolution is not similar before and after crossing the throat. So we keep plots for both positive (on right) and negative (on left) $\lambda$.
Second, the presence of the decaying warp factor has amplified the magnitude of rotation of the area element considerably when $\lambda \rightarrow \infty$ whereas rotation is almost negligible when $\lambda \rightarrow -\infty$. This means a congruence with low rotation become highly rotating after crossing the throat and vice versa.

\begin{figure}[H]
\centering
\includegraphics[scale=.6]{5d-decay-deviation-2-negative-l.pdf}
\hspace{0.2cm}
\includegraphics[scale=.6]{5d-decay-deviation-2-positive-l.pdf}
\includegraphics[scale=.6]{5d-decay-deviation-2-negative-y.pdf}
\hspace{0.2cm}
\includegraphics[scale=.6]{5d-decay-deviation-2-positive-y.pdf}
\caption{Showing evolution of projected geodesic congruence plane (without rotation) in case of crossing congruences with decaying warp factor for wormhole parameter $m = 2$ and $b_{0} = 1$. }
\label{fig:5d-decay-deviation-evolution-2}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.60]{5d-decay-deviation-4-negative-l.pdf}
\hspace{0.2cm}
\includegraphics[scale=.60]{5d-decay-deviation-4-positive-l.pdf}
\includegraphics[scale=.60]{5d-decay-deviation-4-negative-y.pdf}
\hspace{0.2cm}
\includegraphics[scale=.60]{5d-decay-deviation-4-positive-y.pdf}
\caption{Showing evolution of projected geodesic congruence plane (without rotation) in case of crossing congruences with decaying warp factor for wormhole parameter $m = 4$ and $b_{0} = 1$. }
\label{fig:5d-decay-deviation-evolution-4}
\end{figure}
In contrast to the model with a growing warp factor, the evolution of the area element projected on the $\sqrt{g_{11}} \xi^{1}$-$\sqrt{g_{44}} \xi^{4}$ plane in the case of decaying warp factor has the opposite profile.
Further, these figures reveal that the wormhole parameter, $m$, affects the area element in the same way for all the models, i.e. the magnitude of the expansion and distortion decreases as $m$ grows.
Perhaps the most significant aspect of this section is the unique perspective that let us look into the details of the congruence rotation or {\em accretion}.

\section{Discussion} \label{sec:6}

Earlier we showed that the generalised 4D-Ellis-Bronnikov spacetime  embedded in warped 5D bulk satisfies the energy conditions thus is a viable wormhole model within the framework of general relativity. We further discussed the particle trajectories in 4D-GEB and 5D-GEB models and compared them. These investigations show that even the simple 5D-EB model ($m=2$ case) has nice properties as a wormhole.
This work can be considered as the third article in this series where we have analysed congruence of timelike crossing geodesics in detail which is essential to understand the viability of travelling through these class of wormholes.
We derived the evolution of the so-called ESR variables for both the 4D-GEB and 5D-WGEB (with growing or decreasing warp factor) spacetimes by solving geodesic equations, geodesic deviation equations and the Raychaudhuri equation simultaneously and compared the results. The key findings of this analysis are as follows.

\begin{itemize}

\item For 4D-GEB spacetime, analytic expressions of ESR variables of the congruence were first determined without rotation ($\Omega_{AB} = 0$). The resulting plots reveal that $m = 2$ and $m > 2$ geometries can be easily distinguished based on the evolution of expansion scalar $\Theta$. ESR variables all vanish in a similar way in the asymptotic regions at $l \rightarrow \pm \infty$. This essentially happens because the geometries are asymptotically flat.

\item In the 4D-GEB geometry, for congruence with rotation, numerical results are consistent with the analytic calculation of ESR variables. Numerical analysis further show that the differences between ESR profiles in $m = 2$ and $m > 2$ geometries, that appear in absence of rotation, in fact disappear if rotation is present while the congruence is crossing the throat.

\item The 5D-WGEB models are studied with growing and decaying warp factors. In absence of any rotation in the congruence, expansion and shear becomes very large in the asymptotic regions i.e. the congruence fails to retain its shape and size while crossing the throat. Remarkably, presence of rotation improves the situation for expansion scalar in case of growing warp factor and completely removes the divergences in case of decaying warp factor model. In fact the divergence in shear at asymptotic region is removed as well.
An universal feature, in absence of rotation, in both the 4D-GEB and 5D-WGEB models is that the expansion scalar corresponding to all values of $m$ has exactly same value at length scale $l = \pm b_{0}$.

\item In the snapshots of cross-sectional area of a congruence of timelike geodesics projected on 2D surfaces ($l-\phi$ and $l-y$ planes) provide another interesting perspective on the evolution of the ESR variables. This analysis clearly shows that, a congruence with zero rotation near the throat may have rotation in the asymptotic regions if the individual geodesic possesses angular momenta. This effect is considerably large in presence of decaying warp factor compared to the case of a growing warp factor. One may say, the warping factor essentially leads to {\em accretion}. Further an effect of increasing $m$ (which essentially means a steeper throat) has similar effect on the evolutions for all the 4D and 5D models.

\item It is important to note that when the warp factor is growing, geodesics have a turning point along the extra dimension \cite{Ghosh:2010gq}, which forces congruence singularity to occur, whereas when the warp factor is decaying, the occurrence of congruence singularity is completely dependent on boundary conditions.
Our visualization analyses do show how the square elements change shape and get rotated because of variations in the metric functions and boundary conditions. We may contemplate such accretion effects for flows around brane-world wormholes (similar to braneworld black-hole scenario \cite{Pun:2008ua}). In such situations, it will become useful to pursue an approach very similar to what we have used in this article.

\end{itemize}

It still remains to analyse the congruences for {\em all} possible boundary conditions which should be imposed in the asymptotic regions. To avoid computational difficulty, in most cases, we imposed those conditions at the throat. Even then we could understand various effects the congruences will be subjected to while traversing through the hole. There are various other features yet to be studied for these class of wormholes regarding lensing, stability under perturbation as such. One crucial pointer is towards rotating Ellis-Bronnikov wormholes which seems to be possessing useful properties and yet not been addressed in the literature as much. We shall report on these issues in future communications.

\section{Appendix}

\subsection{Boundary conditions to determine ESR profiles in 4D-GEB spacetimes} \label{app-1}

\noindent
$k = \sqrt{3}$, $h = 0 $;
$t(0) = 0$, $l(0) = 0$, $\theta(0) = \pi/2$, $\phi(0) = 0$ ;\\
$\dot{t}(0) = k $, $\dot{l}(0) = 1.41421$, $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = \frac{h}{(1 + l(0)^{m})^{2/m}} = 0 $ ;\\
$B_{AB}(0) = 0$ for without rotation and (initially $\theta$, $\Sigma_{AB}$ and $\Omega_{AB}$ is zero) ;\\
For with rotation case: $\theta$ and $\Sigma_{AB}$ are zero at $\lambda=0$ but non-zero $\Omega_{AB}$ are chosen such that $\Omega_{AB}u^{B} = 0 = u^{A}\Omega_{AB}$ at $\lambda = 0$

\subsection{Boundary conditions to determine ESR profiles in 5D-WGEB  spacetimes with growing warp factor}\label{app-2}

\noindent
$T = \sqrt{3}$, $H = 0 $;
$t(0) = 0$, $l(0) = 0$, $\theta(0) = \pi/2$, $\phi(0) = 0$, $y(0) = 0.1 $ ;\\
$\dot{t}(0) = \frac{T}{e^{2f(y)}} = 1.71485 $, $\dot{l}(0) = 1.39665 $, $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = \frac{H}{(1 + l(0)^{m})^{2/m}} = 0 $,$ \dot{y}(0) = 0 $ ;\\
$B_{AB}(0) = 0$ for without rotation that implies $\theta(0)$, $\Sigma_{AB}(0)$ and $\Omega_{AB}(0)$ is zero ;\\
For with rotation case: $\theta$ and $\Sigma_{AB}$ are zero at $\lambda=0$ but non-zero $\Omega_{AB}$ are chosen such that $\Omega_{AB}u^{B} = 0 = u^{A}\Omega_{AB}$ at $\lambda = 0$

\subsection{Boundary conditions to determine ESR profiles in 5D-WGEB  spacetimes with decaying warp factor}\label{app-3}
\noindent
$T = \sqrt{3}$, $H = 0 $ ;
$t(0) = 0$, $l(0) = 0$, $\theta(0) = \pi/2$, $\phi(0) = 0$, $y(0) = 0.1 $ ;\\
$\dot{t}(0) = \frac{T}{e^{2f(y)}} = 1.74943 $, $\dot{l}(0) = 1.43195 $, $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = \frac{H}{(1 + l(0)^{m})^{2/m}} = 0 $,$ \dot{y}(0) = 0 $ ;\\
$B_{AB}(0) = 0$ for without rotation that implies $\theta(0)$, $\Sigma_{AB}(0)$ and $\Omega_{AB}(0)$ is zero ;\\
For with rotation case: initially $\theta$ and $\Sigma_{AB}$ are still zero but non-zero $\Omega_{AB}$ are chosen such that $\Omega_{AB}u^{B} = 0 = u^{A}\Omega_{AB}$ at $\lambda = 0$

\subsection{Boundary conditions for evolution of cross-sectional area in 4D-GEB spacetimes}\label{app-4}
\noindent
$k = \sqrt{3}$, $h = \sqrt{\frac{k^{2} - 1 }{2}} = 1$ ;
$t(0) = 0$, $l(0) = 0 $, $\theta(0) = \pi/2$, $\phi(0) = 0$ ;\\
$\dot{t}(0) = k$, $ \dot{l}(0) = 1$, $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = \frac{h}{(1 + l(0)^{m})^{2/m}} = h = 1 $ ;\\
$\dot{l}(0)$ is calculated from the timelike geodesic constraint ;\\
$B_{AB}(0) = 0$ ;
$\xi^{1}(0) = \pm 1$, $\xi^{2}(0) = 0$, $\xi^{3}(0) = \pm 1$ ;\\
$\xi^{0}(0) = \pm \frac{2}{\sqrt{3}}, 0$ is calculated from $u_{A}\xi^{A} = 0$.

\subsection{Boundary conditions to determine evolution of cross-sectional area in 5D-WGEB spacetimes with growing warp factor}\label{app-5}

\noindent
$t(0) = 0$, $l(0) = 0$, $\theta(0) = \pi/2$, $\phi(0) = 0$, $y(0) = a = 0.1$ ;\\
$\dot{t}(0) = T$, $ \dot{l}(0) = 0.98758 $ , $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = \frac{H}{\cosh(a)^{2}(1 + l(0)^{m})^{2/m}} = 0.98758$, $\dot{y}(0) = b = 0$ ;\\
$T = \sqrt{3}$, $H = \sqrt{\frac{k^{2} - \cosh(a)^{2}(b^{2} + 1)}{2}} = 0.997489$ ;\\
$B_{\mu\nu}(0) = 0$ , where $\mu$ and $\nu$ runs from $0$ to $4$ ;\\
$\xi^{1}(0) = \pm \frac{1}{\cosh(a)} = \pm 0.995021$, $\xi^{2}(0) = 0$, $\xi^{3}(0) = \pm \frac{1}{\cosh(a)} = \pm 0.995021$, $\xi^{4}(0) = \pm \frac{1}{\cosh(a)} = \pm 0.995021$ ;\\
$\xi^{0}(0) = \pm 1.19213, \pm 0.0574475$ is calculated from $u_{A}\xi^{A} = 0$.

\subsection{Boundary conditions to determine evolution of cross-sectional area in 5D-WGEB spacetimes with decaying warp factor}\label{app-6}

$t(0) = 0$, $l(0) = 0$, $\theta(0) = \pi/2$, $\phi(0) = 0$, $y(0) = a = 0.1$ ;\\
$\dot{t}(0) = T$, $\dot{l}(0) = 1.01254 $ , $\dot{\theta}(0) = 0$, $\dot{\phi}(0) = \frac{H}{sech(a)^{2} (1 + l(0)^{m})^{2/m}} = 1.01254$, $\dot{y}(0) = b = 0$ ;\\
$T = \sqrt{3}$, $H = \sqrt{\frac{k^{2} - sech(a)^{2}(b^{2} + 1)}{2}} = 1.00248$ ;\\
$\xi^{1}(0) = \pm \frac{1}{sech(a)} = \pm 1.005$, $\xi^{2}(0) = 0$, $\xi^{3}(0) = \pm \frac{1}{sech(a)} = \pm 1.005 = \xi^{4}(0)$ ; \\
$\xi^{0}(0) = \pm 1.23305, \pm 0.0580239 $ is calculated from $u_{A}\xi^{A} = 0$.

\section*{Bibliography}

redpajama_set_name: RedPajamaArXiv


Poor pitching has the Colorado Rockies on the verge of their longest losing streak in 15 years, and giving the ball to Jordan Lyles again doesn't seem to be a solution.
Lyles looks to avoid losing a third straight start Wednesday night when the Rockies try to split this two-game set with the Los Angeles Angels.
Colorado's 5.48 ERA is the worst in the majors, and the staff has an 8.63 mark while losing 10 games in a row for the first time since 2005. The Rockies (11-18) last dropped 11 straight from July 4-17, 2000, and that's two shy of the franchise record set as an expansion team in 1993.
Rafael Betancourt provided no relief on the mound Tuesday, giving up three runs while getting just two outs in the eighth of a 5-2 defeat at Los Angeles.
Lyles (2-3, 4.42 ERA) has lost both of his starts during Colorado's current losing streak, giving up 11 runs and 17 hits with six walks in 12 innings. The right-hander was victimized by a big inning last Wednesday, surrendering three of the four runs he allowed in the fifth before completing seven in a 5-1 loss to Arizona.
"When in doubt, throw the fastball," Lyles said of his approach. "I had a couple of quick innings to start the game, then we had some trouble in the fifth."
Lyles is 2-1 with a 5.82 ERA in four games - three starts - against the Angels, and that all came two years ago with Houston. He won both of his interleague starts last year, yielding three runs in 14 2-3 innings.
Nick Hundley is batting .409 during a career-high 13-game hitting streak after getting an infield single Tuesday. The catcher has a .342 season average, 99 points higher than last year when he played 83 games between San Diego and Baltimore.
The Rockies have lost six straight meetings with the Angels (16-17), who have pounded their pitchers to the tune of a .308 batting average and 7.8 runs per game.
Los Angeles collected 10 hits with four doubles Tuesday, and Kole Calhoun had two of them among his three hits. The right fielder is 10 for 28 (.357) with four doubles in seven games.
Albert Pujols is looking to build on his clutch performance from Tuesday. He snapped out of an 0-for-13 skid with a leadoff single in the eighth before unexpectedly stealing second and scoring the go-ahead run on Johnny Giavotella's single.
"I've got the green light all the time, so I go whenever I feel like it," said Pujols, who hadn't stole a base since Aug. 15. "If I know that I'm going to make it, I go. That's it. ... In that opportunity, (the way) we've been struggling, you try to do whatever it takes to get that extra base. I felt that I could score easier from second base than first base."
The Angels are turning to Hector Santiago (2-2, 2.57), who is 1-0 with a 1.02 ERA over his last three home starts.
The only hit the left-hander gave up Thursday was a leadoff single, and he overcame four walks and two balks to toss 6 1-3 scoreless innings. However, closer Huston Street wasted the effort by giving up three runs in the ninth of a 3-2 home loss to Houston.
Santiago is facing the Rockies for the first time, and he's 1-2 with a 3.29 ERA in seven interleague starts.


The Police have deployed more than 2,000 officers to provide security during the festive season in the Colombo city and suburbs.
They have also strengthened security and deployed more manpower to provide security at major cities during the season. A special programme has been organised to minimise traffic congestion during the festive season.Around 1,200 officers have been assigned for traffic control.
To provide enhanced security, the Police have deployed 2,800 uniformed . The Police also said they have enhanced efforts to nab drunk drivers and will take stern action against those found guilty.


Quantum computers are proposed devices that would exploit quantum mechanics to solve certain specific problems dramatically faster than we know how to solve them with today's computers. In the popular press, quantum computers are often presented not just as an exciting frontier of science and technology (which they are), but also as magic devices that would work by simply trying every possible solution in parallel.
However, research over the past 25 years has revealed that the truth is much more subtle and problem-dependent. For some types of problems, quantum computers would offer only modest speedups or none at all.
These limitations are entirely separate from the practical difficulties of building quantum computers (such as "decoherence") and apply even to the fully error-corrected quantum computers we hope will be built in the future.
In this talk, Scott Aaronson will give a crash course on what computer science has learned about the capabilities and limitations of quantum computers. Then he will describe a remarkable and unexpected connection, made just within the past five years, where the conjectured limitations of quantum computers have been applied to problems in fundamental physics.
These include Hawking's black-hole information puzzle (in its modern incarnation as the "firewall paradox"), as well as the growth of wormholes in the so-called gauge/gravity duality that emerged from string theory.
Scott Aaronson is the David J. Bruton Centennial Professor of Computer Science at the University of Texas (UT), Austin.
He received his bachelor's degree from Cornell University and his Ph.D. from the University of California, Berkeley. He did postdoctoral fellowships at the Institute for Advanced Study and the University of Waterloo.
Before joining UT Austin, Aaronson spent nine years as a professor of electrical engineering and computer science at Massachusetts Institute of Technology (MIT).
Aaronson's research in theoretical computer science has focused on the capabilities and limits of quantum computers. His first book, "Quantum Computing Since Democritus," was published in 2013 by Cambridge University Press.
He is the recipient of the National Science Foundation's Alan T. Waterman Award, the United States PECASE Award, the Vannevar Bush Fellowship, and MIT's Junior Bose Award for Excellence in Teaching.
Editor's Note: This event was first announced by the Georgia Tech Algorithms and Randomness Center (ARC). For updates, check the original posting.


text:
No verdict reached Wednesday in Segundo-Clark murder trial
Jurors weigh murder case against Northern NY man
By Kelly O'Brien |
Posted: Wed 11:50 AM, Aug 14, 2019 |
PLATTSBURGH, N.Y. (WCAX) Jurors did not reach a verdict on their first day of deliberations in the Gustavo Segundo-Clark murder case.
Gustavo Segundo-Clark
The 25-year-old West Chazy, New York, man is accused of killing his grandmother, Ginger Clark, 73, at her home last Thanksgiving with a kitchen knife.
But after deliberating all afternoon, the jury did not yet reach a verdict and they were sent home for the night.
Our Kelly O'Brien was first on the scene the day of the murder and has been following the trial since it started. She was in court Wednesday for closing arguments and has an update.
In the opening statements, the prosecution compared their case to a puzzle. They outlined how they would show a timeline of the days leading up to and following the murder.
We didn't hear much from the defense during the trial, but Wednesday morning, they made their closing arguments and said there are holes in the prosecution's case.
"To my belief, there are missing pieces of the puzzle," defense lawyer Joseph Mucia said. "You all have no choice but to render a verdict of not guilty."
In his closing arguments, Mucia hit on missing pieces from the puzzle, like the lack of DNA linked to Segundo-Clark and that there was no DNA found on the knife. He also pointed out inconsistencies between witness testimonies and pointed to a lack of blood on the landline phone Segundo-Clark is accused of using after the murder.
"I think it could be inferred that there should be some fingerprints of blood on that landline," Mucia argued.
And while Segundo-Clark's story changed repeatedly during interrogation, one part stayed the same.
"I think one thing that remains consistent-- he never stated that he killed his grandmother. He never said that," Mucia said.
"You have to ask yourself, why would a person simply come out and admit to killing somebody?" argued Jeffrey Stitt, the Clinton County assistant district attorney.
The prosecution said the defense and prosecution are clearly looking at two different puzzles. Stitt reminded the jury of all of the physical evidence linking Segundo-Clark to the murder.
"It gives us one answer, did the defendant intentionally murder Ginger Clark? That's the only option we have, that's the only option the evidence gives us," Stitt said.
The prosecution recounted the events of Nov. 20, reciting points from the trial like Segundo-Clark's visits to counseling, wanting to leave his aunt's house and fighting with his grandmother about his drinking.
"That is where we start to see the defendant's motive build," Stitt said.
Stitt focused on how the case presented Segundo-Clark's behavior changing because those around him were nagging him. He played a voicemail left by Ginger Clark to Segundo-Clark's counselor the day of the murder. She is heard saying, "I can't take care of him anymore. Please call."
The prosecution says the motive was festered inside Segundo-Clark.
In a recording of an interrogation, Segundo-Clark is heard saying, "I'm tired of her telling me I can't use drugs."
"Ginger Clark was a problem for the defendant even when she was trying to help," Stitt told jurors.
The jury wrapped up deliberations Wednesday without a verdict. They will be back Thursday morning at 9:30 a.m.
Defense calls no witnesses in Segundo-Clark murder trial
Testimony: Murder suspect claimed not to know grandmother
Troopers testify about evidence in NY grandmother's murder
Witnesses testify murdered grandmother would do anything for suspect
Trial begins for New York man accused of murdering grandmother
Jury seated in North Country murder case
Lawyers for NY murder suspect seek mental fitness evaluation
Beekmantown man indicted for grandmother's murder
Beekmantown death ruled homicide
NY police: Grandson a person of interest in suspicious death

redpajama_set_name: RedPajamaCommonCrawl


John Bruegger represents clients in asbestos litigation throughout the U.S., handling high-value premises and product liability cases in state and federal courts. He is experienced in toxic tort, environmental, energy, insurance and other commercial litigation.
John has experience in mass torts, class actions, and complex, multi-district and whistleblower litigation, and he has successfully represented numerous Fortune 500 companies. He also has represented small and large businesses as plaintiffs and defendants in business tort, class action and contract cases.
Before joining Greensfelder, John was a senior associate at a commercial litigation firm, where he acted as national trial counsel and local counsel for businesses in numerous industries. He also previously was the trial team leader for the pharmaceutical litigation department at an Illinois mass tort and complex litigation firm.
"Freedom and the Rule of Law"
"Freedom, Legality, and the Rule of Law"
"'Depositions' and Discovering Evidence for Trial Through Requests for Admission"
"Anatomy of a Class Action Killer: Picking Off Named Plaintiffs in Illinois Class Actions After Barber"


Simply put, EMI is nothing but equated monthly instalments, that you pay per month as repayment for your home loan. The EMI will consist of both parts of the repayments which is the principal amount and the payment of interest. Try out our home loan EMI calculator to understand how much you will pay can help you analyse interest and outstanding principal repayments.


Hang on with a journey of astounding adventurous moments that will ring upon your mind-blowing anticipations as this tour is indeed for the adventure seekers. Gain a wondrous experience of thrill as you white water raft, mountain bike and witness the gigantic cetaceans: the whales whereas gather your breathtaking moments around the safari destinations. Feel the wonder of Sri Lanka as you'll visit a number of sites that will shine up your senses. So come on, take your step to this unique experience, an adventure around this loving isle.


Okay, Macs Cue 2 is out; the future is here. Time to upgrade this bad boy to a more modern platform. Who knows what that'll do to things. I've already lost a couple comments due to a bug in WP. Hopefully it all transfers to the new version.
Then I guess I have to come up with some stuff to say.
You are currently browsing the I Think I Need a Shoehorn weblog archives for the month February, 2011.


Oakhill's 12 month, cost effective Equine Worm Control Plan plays an integral part in reducing worm burden in your horses and ponies.
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If you are familiar with nootropics and how to consume them, you might already know what stacking means. However, if you don't, stacking is a method of combining different nootropics and consuming them to make them work together.
More often than not, the purpose of any nootropic stack is to gain nootropic synergy – a combination of two or more nootropic supplements with similar effects to improve each other's benefits. There are other stacks that people combine to achieve balance so that one nootropic alleviates the consequences of the other.
One of the most beneficial nootropic stacks that you would find today is the L theanine and caffeine stack that use both the elements to create one of the most powerful and effective nootropic stacks that you can find today. Let's dig deep into what this L-theanine and caffeine stack is, its dosage, and how it is beneficial for you.
What is L-theanine and caffeine stack?
As the name suggests, L-theanine and caffeine stack is a stack containing two nootropic substances named L-theanine and caffeine.
Theanine or L-theanine is a chemical that is abundantly found in green and black tea leaves. This compound is thought to be one of the best mind relaxers. One of the best things about this compound is, although it has calming properties, it does not have any sedative properties. It works by increasing the levels of GABA in the brain in addition to dopamine and serotonin.
Caffeine is a stimulant drug that is well known for its psychoactive abilities. It is naturally available in a variety of food and herbal sources, especially in coffee. It acts through various neurotransmitter systems by blocking the activation of adenosine receptors – the ones that are responsible for the sense of fatigue. It is one of the most commonly available nootropics and is used by most people in so e or the other way.
How does the L-theanine and caffeine stack work?
When both L-theanine and caffeine are stacked together, they create a great balance of benefits for both the compounds that are recommended for everyone. The best part of this stack is you get the best benefits of both the compounds while negating the side effects associated with them. Caffeine especially can be over-stimulating form many people. It can cause side effects, like insomnia, stomach problems, and others. However, when it's taken in a stack with L-theanine, you don't experience any such side effects.
On the other hand, L-theanine cannot be termed as a perfect nootropic on its own. This is because it doesn't cater to many parts of the brain. However, when it's taken with caffeine, the L-theanine tends to work more effectively by affecting most parts of the brain.
To make a balanced stack of L-theanine and caffeine, you must keep the ingredients in the ratio of 1:2, with one part of caffeine and two parts of L-theanine. The preferred dosage for the stack is 80-100 mg of caffeine combined with 160-200 mg of L-theanine. This ensures that you get the best results from both these compounds without being wary of the side effects associated with them.