text:
All-In-One Washers and Dryers
Down Draft Range Hoods
Electric Smokers
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Baby Moses Baskets
Baby Movement Monitors
3-in-1 Shampoos
Antioxidant Creams
Blotting Papers for Men
Blow-Dry Sprays
Body Glitters
Bonnet Hair Dryers
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GRE Books
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Firefighter Exam Prep Books
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Flutes For Beginners
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Braun Electric Shavers
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Invisible Fences
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Blue Buffalo Dog Food
Car Pet Barriers
Cat Flea and Tick Prevention
Moving Service Companies
Extended Auto Warranty Services
Anti-Virus Protection Services
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Mold Remediation Services
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Fitbits
28.5-inch Basketballs
Adidas Basketball Shoes for Women
Adidas Running Shoes for Men
Adidas Running Shoes for Women
Adidas Soccer Cleats for Women
Adjustable Kettlebells
Adult Bicycle Helmets
Adults' Swim Caps
Auto Darkening Welding Helmets
BLACK+DECKER Chainsaws
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Trip Essentials
Why trust BestReviews?
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BestReviews spends thousands of hours researching, analyzing, and testing products to recommend the best picks for most consumers. We buy all products with our own funds, and we never accept free products from manufacturers.Read more
Feature-packed
The king of the tablets, it has a beautiful screen and top-notch features.
Large screen. A very advanced iOS device. Apple Pencil turns it into a drawing tablet.
Pricey. Apple Pencil is an additional purchase. Heavier than most tablets.
Best Bang for the Buck
Best for Amazon Fans
An especially good choice if you're invested in Amazon's content ecosystem.
Great value. Has a High Definition screen. Good speakers. Dual-band Wi-Fi. Decent screen that can display up to 1080p content.
Displays Amazon ads unless you pay extra. Tied into the Amazon ecosystem.
iPad Air 10.5-inch
Most Versatile
The 10.5-inch screen is an ideal compromise between the giant iPad Pro and the smaller standard iPad. If you're looking for an iPad with high-end specs but a low-end price, look no further.
It's got a high-resolution "retina" display. It's affordable compared to Apple's other tablets. The A12 processor is surprisingly fast, and it's one of the last devices to support TouchID, Apple's built-in fingerprint reader.
It's only compatible with the first-generation Apple Pencil. It relies on lightning cables for connecting to a computer instead of the newer USB-C standard. It doesn't support FaceID, Apple's face recognition service.
Samsung's top-of-the-line Galaxy is truly an incredible tablet. With a super-AMOLED display and Cinematic Dolby Atmos Surround sound, this is one of the few tablets around that can compete with a laptop.
The proprietary stylus, the S Pen, is included. It's compatible with DeX, Samsung's software for creating a laptop experience on a tablet. It's even got voice recognition.
It's running Android Oreo, a slightly outdated version of the mobile OS (although it's slated to get a free upgrade to Android Pie). The DeX feature works best with a keyboard, which you'll need to buy separately.
ZPad 10-inch Tablet
Bargain Pick
ZTE's ZPad is one of the most straightforward Android tablets around. It's got a 10" HD screen, LTE capability, and it's dirt cheap.
The built in 9,070mAh battery helps it last all day without a problem. The 1900x1200 screen is gorgeous, and the best screen we've seen at this price point. The microSD slot allows you to add up to 128GB.
It comes pre-configured for an LTE plan with US Cellular, although you can use it with any CDMA-based wireless carrier like Sprint or Verizon. ZTE has not indicated if they will publish future versions of Android for the ZPad.
We purchase every product we review with our own funds — we never accept anything from product manufacturers.
61 Models Considered
9 Hours Researched
1 Experts Interviewed
711 Consumers Consulted
Zero products received from manufacturers.
Buying guide for the best tablets
By Jaime Vazquez
When Apple first introduced the iPad, they changed the world — and turned tablets from fictional gadgets from the future into essential everyday companions. Tablets are everywhere nowadays, ready to run any app we want, and they come in just about every shape, size, and color you can imagine.
That's great news for affordability, but the tablet market has grown so crowded that it can sometimes be hard to tell the differences between the cream of the crop and the latest no-name tablet. Tablets have also evolved to become incredibly powerful; in some cases, they're more powerful than an average laptop.
Whether you're looking for a tablet for casual use or one that can keep up with you and the work you do, we've got you covered. We have everything you need to know to find the tablet that's perfect for you.
Jaime Vazquez has been writing about technology and geeking out with gadgets since 2000. He loves trying the latest electronics so that his readers don't have to, and using his inner cheapskate to find the best bargains.
Jaime Vazquez | Writer
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Some tablets come with lock-screen widgets that allow you to access your most common activities — like checking the weather, looking for your next appointment, or tracking your screen time — without having to unlock your tablet.
Before you start shopping, take a moment to consider how you'll be using your tablet. These questions will help you get started.
What's your ideal screen size?
The most important decision to make when you're shopping for a tablet is what size screen you want. Tablets come in a variety of screen sizes ranging from seven to 14 inches, so you've got a lot of options. Your choice should take both readability and portability into account. If you're looking for a tablet you can hold in one hand and you're comfortable with a smaller screen, a seven-inch tablet may be perfect for you. On the other hand, if you find yourself squinting at your phone, or if you want a screen that's roughly the same size as a piece of paper, a 9.7-inch or 12.9-inch tablet may be more appropriate.
Are you an Android smartphone user or an iPhone user?
While there's certainly no rule against owning devices from different manufacturers, there are definitely advantages to buying a tablet that's built on the same platform as your phone. The biggest benefit is familiarity. If you're already familiar with Android or iOS as operating systems, you'll feel right at home the first time you power up your tablet if you buy one with the same OS. In addition, some tablets and phones from the same brand offer extended functionality when used together. For example, if you're an iPhone owner, you can set up an iPad so you can answer phone calls from it.
Product in depth
Apple iPad Pro 12.9-Inch
Apple has always led the tablet industry in terms of innovation and design, and they've done it again with the iPad Pro. The 12.9-inch version is their most powerful tablet yet, and it pulls out all the stops: it's got a nearly edge-to-edge screen, facial recognition, 4K video recording, and professional-level apps that help blur the line between tablet and laptop. If you simply want the best tablet there is, full stop, this is the one.
Will you be using your tablet for work?
If you plan to use your tablet for tasks that you might normally complete on a laptop, you'll want to look for a tablet that can keep up — or a 2-in-1 laptop. If you need a tablet for presentations, writing, or number-crunching, get one with a larger screen and a faster processor.
EXPERT TIP
If you want a keyboard that's got the same aesthetics as your tablet, look into the options offered by the manufacturer. If that doesn't matter to you, or if you need a full-size keyboard, any Bluetooth keyboard will do.
Staff | BestReviews
TESTING TABLETS
We spent 9 hours researching tablets, then tested our top picks from Apple, Amazon, and Samsung. We used the testing results to make our picks.
Standard tablet features
Storage: Tablets are basically computers under the hood, and every computer needs file storage for the operating system, apps, and personal files. Storage amounts vary between 16GB and 256GB, with the associated cost increase you'd expect. The "right" amount of storage will vary depending on the user; if you keep most of your stuff in the cloud, you don't need a ton of space, but if you like to keep a lot of movies or TV shows with you — or you have a large photo collection — you'll want to invest in a tablet with enough room.
Cameras: You can use a tablet camera to take pictures, but holding one up for just the right angle can get pretty awkward and unwieldy, so most people use them for video conferencing services like Google Hangouts, Skype, or FaceTime.
WiFi: WiFi is a standard feature for all tablets; they're not that useful without an internet connection. That said, it's important to get a tablet that supports the fastest WiFi speeds available, so make sure the one you buy supports the 802.11ac WiFi standard. (If you have an older router, that's OK, too; tablets are backward-compatible with older WiFi standards like 802.11n.)
Processor: Every tablet needs a central processing unit (CPU) to run. Most tablet manufacturers make their own processors, so it's often difficult to compare, say, the iPad's A12X chip with the Qualcomm processors found in many Samsung tablets. To get a sense of how different tablets actually perform, watch video reviews and see them in action.
Speakers: While tablet speakers can't hold a candle to headphones or a pair of proper speakers, built-in speakers still matter. Most tablets have two speakers for achieving a stereo effect, but some still rely on a single speaker for mono sound. If you plan on playing music through your tablet's speakers, get one with speakers on either side for optimal sound separation.
Both Google and Apple offer built-in security software, so you can track the location of your tablet in the event that you get separated from it. Just keep in mind that it's not automatic, so you'll need to set it up in the Settings app to activate it.
Premium features and upgrades
Once you've got a solid handle on the basics, consider the features you might be willing to pay more for. These are our favorites.
A premium stylus
Styluses have been available for tablets from the beginning, but new innovations are giving the available options a boost. Now, it's possible to get a stylus that's custom-designed for your tablet. Proprietary styluses include advanced features. For example, many of them let you use the top end as an on-screen eraser, while others support multiple pen types and allow you to switch between thick pen strokes and thin ones with the click of a button.
Kid-friendly designs and apps
If you've got little ones, it's important to keep an eye on their device usage and put controls in place to keep them from going places online that they shouldn't. While you can use parental control software for this on any tablet, we recommend buying a kid-friendly tablet instead. Tablets made for kids often include ultra-durable construction so they can be dropped, and they focus on making it easy for parents to keep kids safe.
Much like laptops, some tablets include Secure Digital (SD) card slots for adding more SD and microSD memory cards. Expandable storage is an incredibly convenient feature because it gives you an option if you ever find yourself running out of storage space. You can even keep extra cards and swap them in as needed.
LTE connectivity
If you'll primarily be using your tablet at home, WiFi connectivity will be enough. But if you want to access the web from your tablet when you don't have access to WiFi, you'll need one with an LTE radio. With LTE connectivity, tablets can get online anywhere. Just keep in mind that you'll need to add your tablet to your mobile data subscription plan from a wireless provider like Verizon, AT&T, T-Mobile, or Sprint.
Your tablet's screen is responsible for the majority of its battery drain. If you're not satisfied with your tablet's battery life, consider lowering your screen's brightness, or better yet, set your screen to auto-dim.
Make sure your tablet is always protected and powered up with these peripherals.
Tablet case: ProCase Universal Case for 9-10 inch Tablet
We always recommend getting a case for any tablet you own to keep it safe from damage and to protect its resale value. If you've got a nine- or ten-inch tablet, the ProCase is one of the best options available: it's affordable, has a kickstand, and looks close enough to leather for our tastes. Best of all: it comes in multiple colors.
Screen protector: SPARIN Tempered Glass Screen Protector New iPad 9.7" (2018 & 2017) / iPad Pro 9.7
It's also important to protect your tablet's screen! Screen protectors keep tablet screens scratch-free, and they make it a lot easier to wipe off dirt and fingerprints, too. If you've got a 9.7" tablet, we recommend SPARIN's screen protectors. They're made from tempered glass, so they're a lot stronger than typical screen protectors made from plastic. But despite being so strong, they don't get in the way at all; you'll still be able to use features like stylus input or a fingerprint reader without a problem. Prevent screen cracks before they happen, and be sure to pick up the right size screen protector for your tablet.
Power bank: Anker PowerCore 10000
If you take your tablet with you to places where there isn't always power, it's important to get a power bank so you can recharge on the go as needed. Anker makes some of the best power banks in the business, and their PowerCore line is their flagship brand. The PowerCore 10000 holds a whopping 10,000 milliamp hours (mAh), which is enough to recharge the average tablet three times. It's also got PowerIQ, so you know that whatever device you've got plugged in is charging as fast as it can handle.
Tablet prices
Budget and refurbished tablets start around $50 to $100. Tablets in this price range have smaller screens and less powerful processors. They skimp on features like file storage and camera quality. If you're buying a kids' tablet or a tablet for short-term use, you can find a good deal for less than $100. In most other cases, we recommend spending more.
The best values in the tablet market are between $100 and $400. Tablets in this price range offer good compromises. For example, it's not hard to find a nine- or ten-inch tablet with 32GB of storage for less than $400. If you need high-end features like a giant screen, a fast processor, or facial recognition, you won't find good options in this price range. However, if you're in the market for a tablet that will last more than a few years and is powerful enough to keep up with everyday tasks, this is the price range to focus on.
High-end tablets start at $400 and can get as expensive as $1,400. Despite the wide range, there aren't that many tablets in this price range; it's just that individual add-ons (like more built-in file storage) can easily ratchet up the price by a few hundred dollars. If you want to use your tablet for school or work or for creating content, plan on spending a lot to get a tablet that's powerful enough.
If you're buying a tablet that has expandable storage, consider buying one with the minimal amount of built-in storage. Storage space can really drive up the price of a tablet. Sometimes, to get the next size up, you'll have to pay more than $100. Conversely, microSD cards and other external storage are all pretty cheap. If you need a lot of file storage space and don't want to spend a ton, consider buying a tablet with expandable storage along with an extra microSD card or two. By taking advantage of removable storage, you can get a lot more space for your money.
If you're taking your tablet to an area in which you may not be able to charge it, bring a portable battery with you. Tablet batteries last a while, but they don't last forever. It's important to have a backup plan for instances where recharging isn't an option. Our favorite fix for that is a portable battery — a gadget that's easy to carry around and always good for an extra charge (or three).
Avoid refurbished tablets. While buying a refurbished tablet can sound like a great way to save a little money, it's usually not worth the risk. Because there are no recognized standards for refurbishing, you never know what you're getting — or if anything was done at all. It's especially risky with tablets because batteries can degrade over time, so it's possible that a refurbished tablet could come with a battery that's not holding a full charge anymore. Save yourself the headache: buy new.
Amazon Fire HD 10
Put it on my tab(let)
Amazon's Fire HD 10 is easily one of the best values on the market. It's got a full-HD screen, above-average speakers, and best of all, it's got Amazon's best-in-class digital assistant, Alexa, on board. If you need a speedy, durable tablet but don't want to pay a fortune, get a Fire HD 10 — you won't regret it.
Other products we considered
If you're in the market for a kids' tablet, start by checking out the Fire HD 8 Kids' Edition. It's got the technical chops of a grown-up tablet, but it's built with a thick blue exterior that Amazon dubs a "kid-proof case." Plus, it includes a year's subscription to Amazon's FreeTime Unlimited service (which offers shows, movies, educational apps, and audiobooks, all geared for the younger crowd). You'll still need to use parental controls, but if you're looking for one of the best ways to introduce your kids to tech, this is a great place to start.
If you're not ready to buy your kids a full-blown tablet — or you just want an affordable option for your young ones — Leapfrog's My First Learning Tablet is a great place to start. No Android or iOS here: just big, easy-to-press buttons and apps for introducing animals, shapes, and colors to your toddler. Our favorite part: it's strong enough to withstand drops and falls, and there's no glass screen to crack, so you don't have to worry about expensive damage or anyone getting hurt.
If you've got a favorite or must-have app, check to make sure the tablet you buy supports it. Most mainstream apps are available for both iOS and Android — but not all of them.
Q. How long do tablet batteries usually last?
A. It depends on what you're using the tablet for. In most cases, tablets will last for anywhere from three to six hours on a battery charge. More intense tasks, like streaming video, can reduce that to between two and three hours. In standby mode, most tablets can last a few days without needing to be recharged.
Q. Should I buy a screen protector for my tablet?
A.In the early days of tablets, screens were prone to scratches, which made screen protectors vital. Since then, the glass used in tablet screens has gotten stronger and more scratch-resistant (although definitely not scratch-proof). The bottom line: if you prefer to go without a screen protector, we're not going to judge. With some basic precautions and the right case, your screen will likely only face minimal scratches. On the other hand, if you plan to sell your tablet down the line, keeping your screen scratch-free is crucial, so a screen protector may be your safest bet.
Q. Can I send text messages from a tablet?
A. Yes — sort of. Text messages come in two flavors: SMS and internet-based messages. SMS messages require a cellular data (LTE) connection, while web-based messages only need a connection to the internet. For example, third-party messaging apps like Facebook's WhatsApp or Apple's Messages send data over the web and work well on tablets. In contrast, tablets aren't usually able to send SMS messages to other devices.
Other Products We Liked
Fire HD 8
Fire HD 8 Kids' Edition
Surface Tablets
The team that worked on this review
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text:
\section{Introduction}

Nucleation and subsequent evolution of vacuum bubbles are very interesting issues in gravity, field theory, and cosmology. We know that our universe experiences accelerated expansion and this state may be described by a positive cosmological constant. In addition, we know that our universe had experienced a stage of inflation. The primordial inflation can be described by the existence of a false vacuum in a potential of a field and this necessarily requires a phase transition of the field from the false vacuum to the current true vacuum. In this context, the nucleation and evolution of vacuum bubbles are quite natural to introduce. In addition, such a study of vacuum bubbles can be interesting and important.

The primordial inflation of the current universe will probably be described by a potential with some fields that causes inflation \cite{Guth:1980zm}. One of the important points of consensus is that the phase transition that finishes the primordial inflation seems to be slow-rolling, not the first order phase transition \cite{Linde:1981mu}. In other words, it is difficult to explain the ending of inflation by the nucleation and percolation of true vacuum bubbles, since the inflating universe quickly expands, so that bubbles in general cannot percolate sufficiently \cite{Guth:1982pn}. Although we know this pessimistic information, the study of bubble nucleation and dynamics is still quite important. There are at least two main reasons why.
\begin{enumerate}
\item \textbf{Cosmic landscape and multiverse:} String theory requires moduli stabilization and this may be possible by flux compactification \cite{Kachru:2003aw}. The flux compactification can generate a huge number of different vacua \cite{Bousso:2000xa}, the so-called cosmic landscape \cite{Susskind:2003kw}. If eternal inflation happens, then all vacua can be populated by first order phase transitions (quantum tunneling) and each bubble will form a pocket universe. The totality of pocket universes in the eternally inflating background is called the multiverse. In the multiverse and cosmic landscape, study of the nucleation and subsequent dynamics of vacuum bubbles is illuminated again.
\item \textbf{In the context of thermal inflation:} After the primordial inflation ended, some unnecessary massive particles can be overproduced \cite{Lyth:1995ka}. Such massive particles should disappear, and this may be possible by introducing thermal inflation. In many phenomenological models of thermal inflation, it will be ended by the first order phase transition, not the second order phase transition \cite{Easther:2008sx}. Therefore, our universe may have some signs of the first order phase transition, bubble nucleation, and bubble percolation.
\end{enumerate}

The bubble nucleating process including gravitation was first studied by Coleman and DeLuccia \cite{Coleman:1980aw}. If we include non-minimal coupling effects that can be motivated from string theory, then possible vacuum transitions can be complicated \cite{Lee:2006vka}. The subsequent evolution of the bubble including gravitation can be approximated by the thin-wall approximation \cite{Blau:1986cw}. However, the thin-wall approximation itself is an approximation and hence it may lose dynamics at the transition region. The related study beyond the thin-wall approximation was studied by numerical method \cite{Hansen:2009kn}\cite{Hwang:2010gc}.

In this paper, we focus on bubble collision issues. This issue has a long history. First, without considering gravitation, there were some studies using analytic methods or numerical methods \cite{Hawking:1982ga}. One of the interesting points of colliding bubbles is that the colliding bubble may induce a vacuum transition \cite{Johnson:2010bn} or a production of particles \cite{Zhang:2010qg} using the colliding energy. Second, including gravitation, it is highly non-trivial to study the dynamics. We may use the thin-wall approximation \cite{Wu:1984eda}\cite{Moss:1994iq}. In the study of Freivogel, Horowitz, and Shenker \cite{Freivogel:2007fx} (for more advanced review, see \cite{Chang:2007eq}), they discussed two colliding true vacuum bubbles in the de Sitter background: one is flat and the other is anti de Sitter. The analysis in itself is very concise and important, but this may not be able to describe the dynamics of fields on the colliding walls. For example, this cannot describe the vacuum transition behaviors. Hence, we may need further numerical studies.

There were some numerical studies of bubble collisions with gravitation. Very recently, Johnson, Peiris, and Lehner \cite{Johnson:2011wt} succeeded in studying bubble collisions with gravitation beyond the thin-wall approximation. They assumed hyperbolic symmetry from the Birkhoff-like theorem of colliding bubbles and assumed initial data from the Coleman-DeLuccia type solutions. They could solve Einstein and field equations numerically and observe and report on symmetric/asymmetric bubble collisions and vacuum transitions.

In this paper, our study can be an alternative approach to the problem using the double-null formalism \cite{Hamade:1995ce}\cite{Hong:2008mw} (note that, \cite{BlancoPillado:2003hq} studied $5$-dimensional anti de Sitter bubble collisions for different purposes). We can list possible differences with the previous studies:
\begin{enumerate}
\item \textbf{Symmetry}: When two bubbles are colliding and there are no other perturbations that disturb bubble collisions, then we can assume the hyperbolic symmetry. However, if many bubbles collide and the background is very complicated, then the Birkhoff-like theorem can be highly non-trivial. For the multi-bubble colliding case \cite{Moss:1994iq}, the spherical symmetry can be a simplified model; also, it may be useful to try to extend the symmetry to be not only hyperbolic, but also planar.

\item \textbf{Beyond the Coleman-DeLuccia conditions}: At once we fix a certain symmetry (spherical, hyperbolic, or planar), we can assign various initial conditions for the field within the symmetry. These initial conditions can come from the Coleman-DeLuccia solution, but it is not necessary in general\footnote{For a general spherical bubble, it may be difficult to give an exact relation between a spherical slice and a hyperbolic slice. However, we can justify the hyperbolic `approximation' at least for the vicinity of the colliding walls when each walls become sufficiently large. Since the lapse function is approximately $\sim 1 - 2M/r + \mathcal{O}(r^{2})$, around the large wall of a spherical bubble, one can approximate de Sitter, flat, or anti de Sitter. In these limits, we can justify hyperbolic slices from spherical slices. The planar or hyperbolic symmetry can be the best trial for general spherical bubble collisions beyond the Coleman-DeLuccia conditions: we can try to see qualitative results and we can choose a certain symmetry by a practical way. The authors thank to a referee to point this out.}. The Coleman-DeLuccia solution assumes $O(4)$ symmetry. Therefore, their solutions are de Sitter, Minkowski, or anti de Sitter. Then the initial tension and the initial size of the bubble are already fixed. However, in general, bubbles can be massive and hence the tension and size can be free parameters \cite{Blau:1986cw}, although the probability of the bubble can be reduced. It is reasonable that if a massive bubble is nucleated in the de Sitter background, then the causal future of the massive bubble can see the effect of the mass, while the causal past may not notice such behavior \cite{Hansen:2009kn}\cite{Hwang:2010gc}. In the double-null coordinates, it is very easy to implement this kind of massive thick-wall conditions.

\item \textbf{Causal structure}: The double-null formalism does not have coordinate singularities in general, since it does not use the coordinate time. In addition, the double-null coordinate preserves causality. Therefore, it is easy to draw causal structures for complicated cases.
\end{enumerate}

This paper is organized as follows. In Section~\ref{sec:mod}, we discuss our model for bubble collisions in the double-null formalism with spherical, planar, and hyperbolic symmetry. In Section~\ref{sec:var}, we show various phenomena of bubble collisions: bubble percolation, dynamics of bubble walls, and vacuum destabilization. Finally, in Section~\ref{sec:dis}, we summarize our results.

\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{bubbles}
\caption{\label{fig:bubbles}Left: General bubble percolation. Gray regions are true vacuum regions and white regions are false vacuum regions. Middle: Approximately, this can be described by an out-going true vacuum bubble and an in-going false vacuum bubble. Right: When two bubbles are colliding, we can magnify around the colliding shells, so that the region can be approximated by left-moving and right-moving walls.}
\end{center}
\end{figure}

\section{\label{sec:mod}Model for bubble collisions}

In this paper, we study a gravitational system in Einstein gravity with a scalar field that is governed by a potential. Then the action $S$ becomes:
\begin{eqnarray}
S = \int dx^{4} \sqrt{-g} \left[ \frac{1}{16\pi} R - \frac{1}{2}\nabla_{\mu} \phi \nabla^{\mu} \phi - V(\phi) \right],
\end{eqnarray}
where $R$ is the Ricci scalar, $\phi$ is the scalar field, and $V(\phi)$ is the potential of the scalar field.

Einstein equations and scalar field equations are as follows:
\begin{eqnarray}\label{eq:Einstein}
G_{\mu\nu} &=& 8 \pi T^{\phi}_{\mu\nu} \\
\label{eq:T_Phi}
T^{\phi}_{\mu\nu} &=& \phi_{;\mu}{\phi}_{;\nu} - \frac{1}{2}\phi_{;\rho}\phi_{;\sigma}g^{\rho\sigma}g_{\mu\nu} -V(\phi) g_{\mu\nu} \\
\label{eq:Phi}0 &=& \phi_{;\mu\nu}g^{\mu\nu}- \frac{dV}{d\phi}
\end{eqnarray}

In this paper, we will choose one of the following metric ansatz of double-null coordinate:
\begin{enumerate}
\item Spherical symmetry: $dS^{2} = d\theta^{2} + \sin^{2} \theta d\varphi^{2}$
\begin{eqnarray}
ds^{2} = -\alpha_{\mathrm{s}}^{2}(u,v) du dv + r_{\mathrm{s}}^{2}(u,v) dS^{2}
\end{eqnarray}
\item Planar symmetry: $dR^{2} = dx^{2} + dy^{2}$
\begin{eqnarray}
ds^{2} = -\alpha_{\mathrm{p}}^{2}(u,v) du dv + r_{\mathrm{p}}^{2}(u,v) dR^{2}
\end{eqnarray}
\item Hyperbolic symmetry: $dH^{2} = d\chi^{2} + \sinh^{2} \chi d\varphi^{2}$
\begin{eqnarray}
ds^{2} = -\alpha_{\mathrm{h}}^{2}(u,v) du dv + r_{\mathrm{h}}^{2}(u,v) dH^{2}
\end{eqnarray}
\end{enumerate}
Note that $0 \leq \theta \leq \pi$, $0 \leq \varphi < 2\pi$, $-\infty < x,y < \infty$, and $0 \leq \chi < \infty$.

Figure~\ref{fig:bubbles} shows general motivations of each symmetries. When vacuum bubbles are colliding and percolated, there can be a certain true vacuum bubble that is surrounded by a false vacuum region, where the outside of the false vacuum region is already transited to the true vacuum (left). We may be able to approximate such a situation by an in-going false vacuum bubble and an out-going true vacuum bubble with spherical symmetry (middle). In addition, if we focus on the collision of two vacuum bubbles (red square in left), the colliding region locally looks like two walls colliding (right). Such walls can be approximated by the planar symmetry or the hyperbolic symmetry.

\subsection{Comments on mass functions}

To give the proper initial conditions, first we have to understand the value of metric ansatz in the static limit. For spherical and hyperbolic cases, we can impose the Birkhoff theorem and obtain the concepts of the local mass functions. We know that the general solutions for $V(\phi)=V_{0}$ will look like
\begin{eqnarray}
ds^{2} = - N(r)^{2} dt^{2} + \frac{{dr}^{2}}{N(r)^{2}} + r^{2} d\Omega_{\kappa}^{2},
\end{eqnarray}
where $\kappa = +1$, $0$, or $-1$,
\begin{eqnarray}
N^{2} = \kappa - \frac{2M}{r} - \frac{8\pi V_{0} r^{2}}{3},
\end{eqnarray}
and $d\Omega_{+1}=dS$, $d\Omega_{0}=dR$, or $d\Omega_{-1}=dH$, respectively \cite{Freivogel:2007fx}.

\begin{enumerate}
\item The solution in the spherical symmetry is
\begin{eqnarray}
ds^{2} = - N_{\mathrm{s}}(r_{\mathrm{s}})^{2} dt_{\mathrm{s}}^{2} + \frac{dr_{\mathrm{s}}^{2}}{N_{\mathrm{s}}(r_{\mathrm{s}})^{2}} + r_{\mathrm{s}}^{2} dS^{2},
\end{eqnarray}
where
\begin{eqnarray}
N_{\mathrm{s}}^{2} = 1 - \frac{2M}{r_{\mathrm{s}}} - \frac{8\pi V r_{\mathrm{s}}^{2}}{3}.
\end{eqnarray}
Here, $r_{\mathrm{s}}$ is the space-like parameter and $t_{\mathrm{s}}$ is the time-like parameter.
We define coordinate transformation:
\begin{eqnarray}
dr_{\mathrm{s}} &=& {r_{\mathrm{s}}}_{,u}du + {r_{\mathrm{s}}}_{,v}dv,\\
dt_{\mathrm{s}} &=& \frac{\alpha_{\mathrm{s}}^{2}}{4} \left( -\frac{dv}{{r_{\mathrm{s}}}_{,u}} + \frac{du}{{r_{\mathrm{s}}}_{,v}} \right),
\end{eqnarray}
and obtain the double-null metric $ds^{2} = -\alpha_{\mathrm{s}}^{2}(u,v) du dv + r_{\mathrm{s}}^{2}(u,v) dS^{2}$. Note that we can choose the in-going direction ${r_{\mathrm{s}}}_{,u}<0$ and the out-going direction ${r_{\mathrm{s}}}_{,v}>0$.
Thus, we can show
\begin{eqnarray}
N_{\mathrm{s}}(r_{\mathrm{s}})^{2} = - \frac{4{r_{\mathrm{s}}}_{,u}{r_{\mathrm{s}}}_{,v}}{\alpha^{2}}.
\end{eqnarray}
Therefore, we can identify that the Misner-Sharp mass function is
\begin{eqnarray}
m_{\mathrm{s}}(u,v) = \frac{r_{\mathrm{s}}}{2} \left( 1 + \frac{4 {r_{\mathrm{s}}}_{,u} {r_{\mathrm{s}}}_{,v}}{\alpha_{\mathrm{s}}^{2}} - \frac{8\pi V}{3} {r_{\mathrm{s}}}^{2} \right).
\end{eqnarray}

\item The solution in the hyperbolic symmetry will look like
\begin{eqnarray}
ds^{2} = - N_{\mathrm{h}}(r_{\mathrm{h}})^{2} dt_{\mathrm{h}}^{2} + \frac{dr_{\mathrm{h}}^{2}}{N_{\mathrm{h}}(r_{\mathrm{h}})^{2}} + r_{\mathrm{h}}^{2} dH^{2},
\end{eqnarray}
where
\begin{eqnarray}
N_{\mathrm{h}}^{2} = - \left( 1 + \frac{2M}{r_{\mathrm{h}}} + \frac{8\pi V r_{\mathrm{h}}^{2}}{3} \right)
\end{eqnarray}
from the Birkhoff theorem and $N_{\mathrm{h}}^{2}$ is chosen to be negative. Therefore, now $t_{\mathrm{h}}$ is the space-like parameter and $r_{\mathrm{h}}$ is the time-like parameter.
We define coordinate transformation:
\begin{eqnarray}
dr_{\mathrm{h}} &=& {r_{\mathrm{h}}}_{,u}du + {r_{\mathrm{h}}}_{,v}dv,\\
dt_{\mathrm{h}} &=& \frac{\alpha_{\mathrm{h}}^{2}}{4} \left( - \frac{dv}{{r_{\mathrm{h}}}_{,u}} + \frac{du}{{r_{\mathrm{h}}}_{,v}} \right),
\end{eqnarray}
and obtain the double-null metric $ds^{2} = -\alpha_{\mathrm{h}}^{2}(u,v) du dv + r_{\mathrm{h}}^{2}(u,v) dH^{2}$. Note that we should choose both left-moving and right-moving null directions ${r_{\mathrm{h}}}_{,u}>0$ and ${r_{\mathrm{h}}}_{,v}>0$.
Thus, we can show
\begin{eqnarray}
N_{\mathrm{h}}^{2} = -\frac{4{r_{\mathrm{h}}}_{,u}{r_{\mathrm{h}}}_{,v}}{\alpha_{\mathrm{h}}^{2}}.
\end{eqnarray}
Therefore, we can identify that the mass function in the double-null coordinate by
\begin{eqnarray}
m_{\mathrm{h}}(u,v) = -\frac{r_{\mathrm{h}}}{2} \left( 1 - \frac{4 {r_{\mathrm{h}}}_{,u} {r_{\mathrm{h}}}_{,v}}{\alpha_{\mathrm{h}}^{2}} + \frac{8\pi V}{3} r_{\mathrm{h}}^{2} \right).
\end{eqnarray}
Note that usual black hole type solutions in Minkowski vacuum will happen for $m_{\mathrm{h}} < 0$ limit, and hence, in this paper, we are interested in $m_{\mathrm{h}} \leq 0$.
\end{enumerate}

\subsection{\label{sec:imp}Implementation of double-null formalism}

From now, if there is no confusion, then we will omit the subscripts $\mathrm{s}$, $\mathrm{p}$, or $\mathrm{h}$.

Define
\begin{eqnarray}
\sqrt{4\pi}\phi \equiv S
\end{eqnarray}
and use conventions \cite{Hong:2008mw}
\begin{eqnarray}\label{eq:conventions}
h \equiv \frac{\alpha_{,u}}{\alpha},\quad d \equiv \frac{\alpha_{,v}}{\alpha},\quad f \equiv r_{,u},\quad g \equiv r_{,v},\quad W \equiv S_{,u},\quad Z \equiv S_{,v}.
\end{eqnarray}

Einstein tensor components are
\begin{eqnarray}
\label{eq:Guu}G_{uu} &=& -\frac{2}{r} \left(f_{,u}-2fh \right),\\
\label{eq:Guv}G_{uv} &=& \frac{1}{2r^{2}} \left( 4 rf_{,v} + \kappa \alpha^{2} + 4fg \right),\\
\label{eq:Gvv}G_{vv} &=& -\frac{2}{r} \left(g_{,v}-2gd \right),\\
\label{eq:Gthth}G_{aa} &=& -4\frac{r^{2}}{\alpha^{2}} \left(d_{,u}+\frac{f_{,v}}{r}\right),
\end{eqnarray}
where $\kappa=+1, 0, -1$ and $a=\theta, x, \chi$ for spherical, planar, hyperbolic cases, respectively.
Energy-momentum tensor components are
\begin{eqnarray}
\label{eq:TPhiuu}T^{\phi}_{uu} &=& \frac{1}{4 \pi} W^{2},\\
\label{eq:TPhiuv}T^{\phi}_{uv} &=& \frac{\alpha^{2}}{2} V(S),\\
\label{eq:TPhivv}T^{\phi}_{vv} &=& \frac{1}{4 \pi} Z^{2},\\
\label{eq:TPhithth}T^{\phi}_{aa} &=& \frac{r^{2}}{2 \pi \alpha^{2}} WZ -r^{2}V(S).
\end{eqnarray}

Therefore, simulation equations are as follows:
\begin{eqnarray}
\label{eq:E1}f_{,u} &=& 2 f h - 4 \pi r T^{\phi}_{uu},\\
\label{eq:E2}g_{,v} &=& 2 g d - 4 \pi r T^{\phi}_{vv},\\
\label{eq:E3}f_{,v}=g_{,u} &=& - \kappa \frac{\alpha^{2}}{4r} - \frac{fg}{r} + 4\pi r T^{\phi}_{uv},\\
\label{eq:E4}h_{,v}=d_{,u} &=& -\frac{2\pi \alpha^{2}}{r^{2}}T^{\phi}_{aa} - \frac{f_{,v}}{r}.
\end{eqnarray}
In addition, we include the scalar field equation:
\begin{eqnarray}
\label{eq:S}Z_{,u} = W_{,v} = - \frac{fZ}{r} - \frac{gW}{r} -\pi\alpha^{2} V'(S).
\end{eqnarray}

\subsection{\label{sec:ini}Initial conditions}

We need initial conditions for all functions ($\alpha, h, d, r, f, g, S, W, Z$) on the initial $u=u_{\mathrm{i}}$ and $v=v_{\mathrm{i}}$ surfaces, where we set $u_{\mathrm{i}}=v_{\mathrm{i}}=0$.

\paragraph{Spherical symmetry}

We have gauge freedom to choose the initial $r$ function. Although all constant $u$ and $v$ lines are null, there remains freedom to choose the distances between these null lines. Here, we choose $r(0,0)=r_{0}$, $f(u,0)=r_{u0}$, and $g(0,v)=r_{v0}$, where $r_{u0}<0$ and $r_{v0}>0$ such that the radial function for an in-going observer decreases and that for an out-going observer increases.

It is convenient to choose $r_{u0}=-1/2$ and $r_{v0}=1/2$; we choose that the mass function on $u_{\mathrm{i}}=v_{\mathrm{i}}=0$ vanish, since we can think that the initial point is in the causal past of the bubbles. Hence, to specify a false vacuum background, for given $r(0,0)=r_{0}$ and $S(0,0)=S_{\mathrm{f}}$, and if the field is at the local minimum, then
\begin{eqnarray}
\alpha(0,0) = \left( 1 - \frac{8 \pi V(S_{\mathrm{f}}) r_{0}^{2}}{3} \right)^{-1/2}.
\end{eqnarray}

\paragraph{Planar symmetry}

We first choose $r(0,0)=r_{0}$, $f(0,0)=0$, $g(0,0)=0$, $\alpha(u,0)=1$, and $\alpha(0,v)=1$, for simplicity. Therefore, $h(u,0)=d(0,v)=0$.

\paragraph{Hyperbolic symmetry}

We have gauge freedom to choose the initial $r$ function. Although all constant $u$ and $v$ lines are null, there remains freedom to choose the distances between these null lines. Here, we choose $r(0,0)=r_{0}$, $f(u,0)=r_{u0}$, and $g(0,v)=r_{v0}$, where $r_{u0}>0$ and $r_{v0}>0$ such that the function $r$ increases for both of in-going and out-going observers.

It is convenient to choose $r_{u0}=1/2$ and $r_{v0}=1/2$; we choose that the mass function on $u_{\mathrm{i}}=v_{\mathrm{i}}=0$ vanish. Hence, to specify a pure de Sitter background, for given $r(0,0)=r_{0}$, $S(0,0)=S_{\mathrm{f}}$, and if the field is at the local minimum, then
\begin{eqnarray}
\alpha(0,0) = \left( 1 + \frac{8 \pi V(S_{\mathrm{f}}) r_{0}^{2}}{3} \right)^{-1/2}.
\end{eqnarray}
To compare previous studies, we can choose the initial mass function nonzero. Then,
\begin{eqnarray}
\alpha(0,0) = \left( 1 + \frac{8 \pi V(S_{\mathrm{f}}) r_{0}^{2}}{3} + \frac{2 m_{0}}{r_{0}} \right)^{-1/2}
\end{eqnarray}
with a free parameter $m_{0}$.

\paragraph{Assignments of initial conditions}

Now it is possible to assign all of initial conditions along initial $u=u_{\mathrm{i}}$ and $v=v_{\mathrm{i}}$ surfaces.

\begin{description}
\item[Initial $v=v_{\mathrm{i}}$ surface:]
We choose
\begin{eqnarray}
S(u,0) = \left\{ \begin{array}{ll}
S_{\mathrm{f}} & u < u_{\mathrm{shell}},\\
\left| S_{\mathrm{t}}-S_{\mathrm{f}} \right| G(u) + S_{\mathrm{t}} & u_{\mathrm{shell}} \leq u < u_{\mathrm{shell}}+\Delta u,\\
S_{\mathrm{t}} & u_{\mathrm{shell}}+\Delta u \leq u,
\end{array} \right.
\end{eqnarray}
where $G(u)$ is a pasting function which goes from $1$ to $0$ by a smooth way. We choose $G(u)$ by
\begin{eqnarray}
G(u) = 1 - \sin^{2} \left[\frac{\pi(u-u_{\mathrm{shell}})}{2\Delta u}\right].
\end{eqnarray}
Then, we know $W(u,0)=S_{,u}(u,0)$. If $\kappa=\pm1$, then $h(u,0)$ is given from Equation~(\ref{eq:E1}), since $f_{,u}=0$ along the in-going null surface. Then, using $h(u,0)$, we obtain $\alpha(u,0)$. If $\kappa=0$, then we obtain $f(u,0)$ from Equation~(\ref{eq:E1}).

We need more information to determine $d, g$, and $Z$ on the $v=0$ surface. We obtain $d$ from Equation~(\ref{eq:E4}), $g$ from Equation~(\ref{eq:E3}), and $Z$ from Equation~(\ref{eq:S}).

\item[Initial $u=u_{\mathrm{i}}$ surface:]
We choose
\begin{eqnarray}
S(0,v) = \left\{ \begin{array}{ll}
S_{\mathrm{f}} & v < v_{\mathrm{shell}},\\
\left| S_{\mathrm{f}} - S_{\mathrm{t}} \right| G(v) - S_{\mathrm{t}} & v_{\mathrm{shell}} \leq v < v_{\mathrm{shell}}+\Delta v,\\
S_{\mathrm{t}} & v_{\mathrm{shell}}+\Delta v \leq v,
\end{array} \right.
\end{eqnarray}
where
\begin{eqnarray}
G(v) = 1 - \sin^{2} \left[\frac{\pi(v-v_{\mathrm{shell}})}{2\Delta v}\right].
\end{eqnarray}

If $\kappa=\pm1$, then we obtain $d(0,v)$ from Equation~(\ref{eq:E2}), since $g_{,v}(0,v)=0$. If $\kappa=0$, then we obtain $g(0,v)$ from Equation~(\ref{eq:E2}). By integrating $d$ along $v$, we have $\alpha(0,v)$.

We need more information for $h, f,$ and $W$ on the $u=0$ surface. We obtain $h$ from Equation~(\ref{eq:E4}), $f$ from Equation~(\ref{eq:E3}), and $W$ from Equation~(\ref{eq:S}). This finishes the assignments of the initial conditions.
\end{description}

We used the second order Runge-Kutta method. The details of the numerical scheme and convergence/consistency checks are included in Appendix A and B.

\subsection{Simulation parameters}

Finally, we can specify all simulation parameters. In addition, we need information on potentials for detailed purposes.

We illustrate all simulation parameters.
\begin{enumerate}
\item Initial $r$ and Misner-Sharp mass functions: $r_{0}$, $m_{0}$
\item Thickness of the shells: $\Delta u$, $\Delta v$
\item Size of the shells: $u_{\mathrm{shell}}$, $v_{\mathrm{shell}}$
\item Field values: $S_{\mathrm{t}}$, $S_{\mathrm{f}}$
\item Potentials
\end{enumerate}

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{pot}
\caption{\label{fig:pot}Potentials we used. $V_{\mathrm{f}}=0.0001$ and $\phi_{\mathrm{f}}$ or $\phi_{\mathrm{t}}$ are chosen to be $0.1$.}
\end{center}
\end{figure}

Especially, we need further comments on the potentials (Figure~\ref{fig:pot}).
\begin{enumerate}
\item To study bubble percolation, we use the potential
\begin{eqnarray}
V_{1} (\phi) = \frac{V_{\mathrm{f}}}{12 \beta \phi_{\mathrm{f}}} \int_{0}^{\phi} \frac{\bar{\phi}}{\phi_{\mathrm{f}}} \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} - \frac{\beta+1}{2} \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} -1 \right) d \bar{\phi},
\end{eqnarray}
where $\beta = 0.1$ is a free parameter, $V_{\mathrm{f}}$ is the vacuum energy of the false vacuum, and $\phi_{\mathrm{f}}$ is the field value of the false vacuum.

\item To study bubble collision with different true vacua, we use the potential
\begin{eqnarray}
V_{2} (\phi) = \frac{V_{\mathrm{f}}}{C_{2}} \int_{\phi_{\mathrm{t}}}^{\phi} \left[ \frac{\bar{\phi}}{\phi_{\mathrm{t}}} \left( \frac{\bar{\phi}}{\phi_{\mathrm{t}}} - 1 \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{t}}} + 1 \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{t}}} - \frac{1}{2} \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{t}}} + \frac{1}{2} \right) \right] d \bar{\phi},
\end{eqnarray}
where $V_{\mathrm{f}}$ is the vacuum energy of the false vacuum and $C_{2}$ is a normalization constant with
\begin{eqnarray}
C_{2} = \int_{\phi_{\mathrm{t}}}^{0} \left[ \frac{\bar{\phi}}{\phi_{\mathrm{t}}} \left( \frac{\bar{\phi}}{\phi_{\mathrm{t}}} - 1 \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{t}}} + 1 \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{t}}} - \frac{1}{2} \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{t}}} + \frac{1}{2} \right) \right] d \bar{\phi},
\end{eqnarray}
and $\phi_{\mathrm{t}}$ is the field difference between the false vacuum and the true vacuum.

\item If we tilt the potential $V_{2}$ as follows:
\begin{eqnarray}
V_{3} (\phi) = V_{2}(\phi) - D_{3} \frac{\phi}{\phi_{\mathrm{t}}},
\end{eqnarray}
where $D_{3}/V_{\mathrm{f}} = 0.5$ is fixed, we then observe the asymmetric bubble collisions.

\item To study vacuum destabilization by bubble collisions, we use the potential:
\begin{eqnarray}
V_{4} (\phi) = \frac{V_{\mathrm{f}}}{C_{4}} \int_{\phi_{\mathrm{f}}}^{\phi} \left[ \frac{\bar{\phi}}{\phi_{\mathrm{f}}} \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} - 1 \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} + 0.75 \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} - \frac{1}{2} \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} + \frac{0.75}{2} \right) \right] d \bar{\phi} + D_{4} \left(\frac{\phi}{\phi_{\mathrm{f}}}\right)^{3},
\end{eqnarray}
where $V_{\mathrm{f}}$ is a constant, $D_{4}/V_{\mathrm{f}}=1.5$ is fixed, and $C_{4}$ is a normalization constant with
\begin{eqnarray}
C_{4} = \int_{\phi_{\mathrm{f}}}^{0} \left[ \frac{\bar{\phi}}{\phi_{\mathrm{f}}} \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} - 1 \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} + 0.75 \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} - \frac{1}{2} \right) \left( \frac{\bar{\phi}}{\phi_{\mathrm{f}}} + \frac{0.75}{2} \right) \right] d \bar{\phi}.
\end{eqnarray}
Note that in this case, if we choose a certain $S_{\mathrm{f}}$, the local maximum is shifted to $\simeq 0.9502 \times S_{\mathrm{f}}$, and we considered this effect.
\end{enumerate}
Therefore, for all cases, $V_{\mathrm{f}}$ and $S_{\mathrm{f}} = \sqrt{4\pi} \phi_{\mathrm{f}}$ (or $S_{\mathrm{t}} = \sqrt{4\pi} \phi_{\mathrm{t}}$) are free parameters of each potential.

\section{\label{sec:var}Various phenomena of bubble collisions}

\subsection{Bubble percolation}

First, we study bubble percolation. By percolation, we mean that two true vacuum bubbles are collided and have eventually emerged to one true vacuum region. The question is whether this process can continuously happen, and what is the new phenomena that could not be observed by the thin-wall approximation?

\subsubsection{Spherical symmetry}

Figure~\ref{fig:per_sphe} shows the collision of an in-going false vacuum bubble and an out-going true vacuum bubble. In this case, we choose the same field value for the true vacuum. Therefore, after the collision, bubbles should be emersed and be percolated. In these calculations, we used the potential $V_{1}$ with $V_{\mathrm{f}}=0.00001$. We also used $r_{0}=50$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and varying $S_{\mathrm{f}}=0.1, 0.2, 0.3, 0.4$.

\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{per_sphe}
\caption{\label{fig:per_sphe}Bubble percolation with spherical symmetry: $V_{1}$ with $V_{\mathrm{f}}=0.00001$, $r_{0}=50$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and varying $S_{\mathrm{f}}=0.1, 0.2, 0.3, 0.4$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{spherical1}
\caption{\label{fig:spherical1}A schematic diagram of bubble percolation with the spherical symmetry.}
\end{center}
\end{figure}

As $S_{\mathrm{f}}$ increases, the tension and the total energy increase. Therefore, the location of the apparent horizon and singularity becomes lower and lower as $S_{\mathrm{f}}$ increases; this implies that the apparent horizon increases, and a singularity relatively quickly appears.

In Figure~\ref{fig:per_sphe}, the yellow region is the false vacuum region, while the skyblue region is the true vacuum region. The black colored region is around the local maximum of the potential $V_{1}$ and hence it shows the dynamics of the thick wall.

The behavior of the out-going true vacuum wall before colliding (for $v<30$) is changed as $S_{\mathrm{f}}$ varies. If $S_{\mathrm{f}}$ is sufficiently small, then initially it shrinks (upper left). However, as $S_{\mathrm{f}}$ increases, it tends to expand (lower right). Unless the shell has sufficient energy, it tends to collapse. In addition, if the shell has sufficient energy, it can expand, although the inner region becomes destabilized; the inside of the out-going shell ($v<30$ and $u>50$) is not exactly zero and the fields are slightly perturbed (we can compare the upper left and upper right diagrams). These results are consistent with that of the false vacuum bubble cases \cite{Hansen:2009kn}.

The upper left of Figure~\ref{fig:per_sphe} shows interesting behaviors: after the collision, some fields can roll up to the false vacuum region ($50<v<60$ and $u>80$). This is due to the fact that the colliding energy perturbs and pushes the field to the other local minimum. This is related to the vacuum destabilization by bubble collisions, and it will be discussed further in Section~\ref{sec:vacdeep}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{causal1}
\caption{\label{fig:causal1}Causal structure of percolating bubbles with spherical symmetry.}
\end{center}
\end{figure}

We can intuitively draw as in Figure~\ref{fig:spherical1}. Initially, there is an in-going false vacuum bubble and an out-going true vacuum bubble (left). The false vacuum region becomes narrower (middle). Eventually, the collided region collapses and shrinks beyond the apparent horizon (right). This can be compared to the Figure~\ref{fig:spherical2} in Section~\ref{sec:vacdeep}. This mimics the situation that bubble collisions can generate a black hole: Figure~\ref{fig:causal1}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{per_plan}
\caption{\label{fig:per_plan}Bubble percolation with planar symmetry: $V_{1}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=1$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and varying $S_{\mathrm{f}}=0.1, 0.2, 0.3, 0.4$. Black curves are contour curves of $r$ and the differences of each of the curves are $1$ and drawn from $r=0$ to $r=100$.}
\end{center}
\end{figure}

\subsubsection{Planar symmetry}

Figure~\ref{fig:per_plan} shows the bubble percolation of two planar bubbles. We assume these conditions: $V_{1}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=1$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and varying $S_{\mathrm{f}}=0.1, 0.2, 0.3, 0.4$.

The first clear observation is that after the collision, walls oscillates back-and-forth (upper left). As a wall oscillates, oscillating field amplitudes decreases (upper right) and eventually the wall disappears (lower left).

When the tension of the shell is sufficiently small, the shell initially expands to the true vacuum region, although it eventually and quickly shrinks to the center (upper left). As the tension of the shell increases, the initial expansion of the shell becomes slower and slower (upper right and lower left). Eventually, if the shell energy is sufficiently large, it slowly approaches and it may not be able to collide, since the false vacuum region inflates too quickly (lower right).

We need more comments on the lower right of Figure~\ref{fig:per_plan}. For all diagrams in Figure~\ref{fig:per_plan}, we can see the details of the $r$ contours. In this case, $r$ is not a real radius; rather, this is just a metric function that relates the planar direction and the orthogonal direction. For the inside of the false vacuum region, it is space-like, while it is time-like for the true vacuum region. The $r$ function increases from lower left to the upper right; and it diverges in the lower right of Figure~\ref{fig:per_plan}. If we only see this diagram, it is not easy to interpret the space-like piece in this diagram, where the numerical simulation cannot work. It seems to not be a singularity, since two bubbles are not collided yet. To interpret correctly, we have to compare Figure~\ref{fig:per_hype_massless} in Section~\ref{sec:hypsym}; in the hyperbolic symmetry, we can interpret that the region is future infinity, where the time-like parameter diverges. Then we can finally interpret the figure that two bubbles cannot collide since the false vacuum region is rapidly expanded.

\subsubsection{\label{sec:hypsym}Hyperbolic symmetry}

\paragraph{Massless case} In Figure~\ref{fig:per_hype_massless}, we observe bubble percolation with the hyperbolic symmetry. The results are similar to the planar symmetric case. We used $V_{1}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=1$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and varying $S_{\mathrm{f}}=0.1, 0.2, 0.3, 0.4$.

When two bubbles collide, the collided region oscillates back-and-forth (upper left). The collided region eventually decays and disappears (upper right). Similar to the planar symmetric case, as the tension increases, the wall slowly moves. For all cases, $r$ contours are always space-like. This is not strange, since our metric ansatz assumed $r$ as a time-like parameter. Therefore, $r$ works as a global time function. For figures in the lower left and lower right in Figure~\ref{fig:per_hype_massless}, the function $r$ diverges as it approaches the upper right region of the figure. Therefore, we can interpret that two bubbles may not be able to collide and the space-like boundary is the future infinity of the false vacuum region, where the in-going and out-going null boundaries are future infinity of the Minkowski regions.

To summarize, we obtain two interesting causal structures in Figure~\ref{fig:causal2}: left is normal bubble percolation when the tension is sufficiently small; right is the large tension case and due to the repulsiveness of large tension walls, bubbles cannot collide and bubble walls touch the future infinity.

\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{per_hype_massless}
\caption{\label{fig:per_hype_massless}Bubble percolation with hyperbolic symmetry: $V_{1}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=1$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and varying $S_{\mathrm{f}}=0.1, 0.2, 0.3, 0.4$. Black curves are contour curves of $r$ and the differences of each of the curves are $10$ and drawn from $r=0$ to $r=1000$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{causal2}
\caption{\label{fig:causal2}Causal structure of percolating bubbles with planar/hyperbolic symmetry. Left: bubbles are percolated. Right: when the tension increases, bubbles may not be able to collide.}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{per_hype}
\caption{\label{fig:per_hype}Bubble percolation with hyperbolic symmetry: $V_{1}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and varying $S_{\mathrm{f}}=0.1, 0.2, 0.3, 0.4$. Black curves are contour curves of $r$ and the differences of each of the curves are $10$ and drawn from $r=100$ to $r=1100$.}
\end{center}
\end{figure}

\paragraph{Massive case} We can also choose the $m_{0} < 0$ case. In Figure~\ref{fig:per_hype}, we especially choose $-m_{0} = r_{0} = 100$, since $m_{0}$ and $r_{0}$ should be the same order to see the effect of the mass term. We used $V_{1}$ with $V_{\mathrm{f}}=0.0001$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and varying $S_{\mathrm{f}}=0.1, 0.2, 0.3, 0.4$.

As $S_{\mathrm{f}}$ increases, it is qualitatively similar to the massless case; however, the wall collides relatively rapidly than the massless case. Therefore, the existence of mass allows a wide range of tensions that allows collision. Therefore, for convenience, we choose this kind of initial conditions for the latter sections, since we want to build colliding bubbles.

\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{fie_hype}
\caption{\label{fig:fie_hype}Bubble collisions of symmetric different true vacua with hyperbolic symmetry: $V_{2}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{f}}=0$, and varying $S_{\mathrm{t}}=0.1, 0.2, 0.3, 0.4$. Black curves are contour curves of $r$ and the differences of each of the curves are $10$ and drawn from $r=100$ to $r=1100$.}
\includegraphics[scale=0.6]{causal3}
\caption{\label{fig:causal3}Causal structure of colliding symmetric bubbles with different field values.}
\end{center}
\end{figure}

\subsection{Dynamics of bubble walls}

Second, we study dynamics of bubble walls. In this case, we especially focus on the case that the left true vacuum region and the right true vacuum region have different field values. Therefore, the bubble wall region will not disappear. In this section, we will observe their back-reaction to the causal structure. Also, we see when the initial conditions of the left and the right are asymmetric.

\subsubsection{Symmetric bubbles}

In Figure~\ref{fig:fie_hype}, we studied bubble collisions of symmetric bubbles with different field values of true vacuum region: $V_{2}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{f}}=0$, and varying $S_{\mathrm{t}}=0.1, 0.2, 0.3, 0.4$. Note that we only used hyperbolic symmetry, since the planar case is qualitatively similar to the hyperbolic case.

As $S_{\mathrm{t}}$ increases, the oscillating frequency of the thick wall increases and is quickly stabilized for sufficiently large $S_{\mathrm{t}}$. One interesting thing to notice is that after a sufficient time, two regions cannot be emersed, since two regions have different field values. Therefore, between two true vacuum regions, there should be always a false vacuum region (blue colored region). In a finite advanced and retarded time, the function $r$ diverges. Therefore, we can interpret that the coordinate time diverges and hence there is the future infinity. For the true vacuum part, the future infinities are null. Between two future null infinities, the false vacuum region can roll a crossing point between two null boundaries. This is a quite different behavior than the bubble percolation case (Figure~\ref{fig:causal3}).

\subsubsection{Asymmetric bubbles}

In Figure~\ref{fig:asy_hype}, we studied bubble collisions of asymmetric bubbles with different field values of true vacuum region. First we change tensions of two walls: $V_{2}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=20$, $\Delta v=40$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{f}}=0$, and varying $S_{\mathrm{t}}=0.1, 0.3$. Second we change vacuum energy between two true vacuum regions: $V_{3}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{f}}=0$, and varying $S_{\mathrm{t}}=0.1, 0.3$.

\paragraph{Asymmetry of tension}

If one side has larger tension (in our case, the skyblue side has larger tension, since the thickness of the shell is narrower than the yellow side), it can push the smaller tension region, since it is related to the energy of the shell. The upper two figures show this behavior.

\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{asy_hype}
\caption{\label{fig:asy_hype}Upper: Bubble collisions of asymmetric different true vacua with hyperbolic symmetry: $V_{2}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=20$, $\Delta v=40$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{f}}=0$, and varying $S_{\mathrm{t}}=0.1, 0.3$. Lower: Bubble collisions of asymmetric different true vacua with hyperbolic symmetry: $V_{3}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{f}}=0$, and varying $S_{\mathrm{t}}=0.1, 0.3$.}
\end{center}
\end{figure}

\paragraph{Asymmetry of vacuum energy}

If one true vacuum side has smaller vacuum energy (in our case, the yellow region is anti de Sitter and hence it is smaller than the skyblue region), it can push the larger vacuum energy region, since the difference of the vacuum energy between the false vacuum and the true vacuum is changed to the pressure of the wall. Therefore, in the lower figures, the yellow region pushes the skyblue region.

The lower right in Figure~\ref{fig:asy_hype} is important to emphasize. This shows the collision between a de Sitter vacuum and an anti de Sitter vacuum. This problem was systematically discussed by \cite{Freivogel:2007fx}. However, here we can see details from numerical calculations.

\begin{figure}
\begin{center}
\includegraphics[scale=0.3]{asy_hype_wider}
\caption{\label{fig:asy_hype_wider}Bubble collisions of asymmetric different true vacua with hyperbolic symmetry: $V_{3}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{f}}=0$, and $S_{\mathrm{t}}=0.3$. Black contours are $r$ and the differences of each of the contours are $10$ and drawn from $r=0$ to $r=1000$. Red and blue curves are $r_{,v}=0$ and $r_{,u}=0$ contours. Black arrows are directions where $r$ increases.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=1]{asy}
\caption{\label{fig:asy}Causal structure of bubble collisions between de Sitter and anti de Sitter bubbles. We denote local horizons and Bousso wedges.}
\end{center}
\end{figure}

Figure~\ref{fig:asy_hype_wider} is the detailed version of the lower right of Figure~\ref{fig:asy_hype}. Black contours are $r$ and the distances of each of the contours are $10$. Red and blue curves are $r_{,v}=0$ and $r_{,u}=0$ contours. Black arrows are directions where $r$ increases. Therefore, for the de Sitter region, $r$ contours are always space-like. However, inside of the anti de Sitter region, the signature of $r$ changed. After $r$ changes the signature two times ($r_{,u}<0$ and $r_{,v}<0$), eventually  $r$ decreases to zero. This whole smooth evolution of the bubble collision and the formation of big crunch singularity are possible due to the numerical calculations.

We summarized in Figure~\ref{fig:asy}. This is the advanced version of the previous work \cite{Freivogel:2007fx}. The left bubble is de Sitter (in principle, Minkowski is possible). The left bubble is collided into the right anti de Sitter bubble. The wall moves in a time-like direction, is biased to the left direction, and eventually becomes parallel to the null direction; the tension via the difference of vacuum energy is larger for the anti de Sitter side and the speed of the wall will eventually approach the speed of light. The future infinity of the left side is space-like (for Minkowski, it should be null). There is a future infinity on the wall, and the wall will touch the future infinity of the left side. When the wall touches the future infinity of the left side, the wall may or may not generate a Cauchy horizon; this depends on initial conditions. However, at least in the data of Figure~\ref{fig:asy_hype_wider}, we can see that \textit{there should be a Cauchy horizon} in the anti de Sitter side (green dotted line) because of two reasons: (1) the wall almost moves in the left-going null direction and (2) the right side of the wall is approximately pure anti de Sitter as time goes on, and therefore, the blue curve ($r_{,u}=0$ horizon) should be parallel to the null direction. Therefore, the wall and the $r_{,u}=0$ horizon will not meet each other, and hence, there should be a Cauchy horizon for the anti de Sitter side. We also notice the Bousso wedges, that implies the direction where $r$ increases. Due to the horizons, the Bousso wedges are changed.

There are two main differences between the thin-wall approximation \cite{Freivogel:2007fx} and our numerical studies:
\begin{enumerate}
\item The crossing point of two horizons ($r_{,v}=0$ and $r_{,u}=0$) can happen before the causal future of the bubble collision, while this was not allowed by the thin-wall approximation. This is not strange, because the horizons are dynamically deformed \textit{on the wall} and connected between de Sitter and anti de Sitter regions.
\item The future infinity can have the \textit{Cauchy horizon} as we have commented. However, the time parameter $r$ will diverge on the wall when the wall touches the future boundary. Therefore, in terms of the analytic coordinates of the anti de Sitter side, they should be identified to the time-like $r=\infty$ boundary (gray diagrams beyond the Cauchy horizon in Figure~\ref{fig:asy}), although we cannot determine beyond the Cauchy horizon from the past data.
\end{enumerate}

\subsection{\label{sec:vacdeep}Vacuum destabilization to deeper vacuum}

Finally, we focus on the vacuum destabilization to deeper vacuum by bubble collisions. We tested two cases. First, for spherical symmetry, $V_{4}$ with $V_{\mathrm{f}}=0.00001$, $r_{0}=50$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and $S_{\mathrm{f}}=0.1$. Second, for hyperbolic symmetry, $V_{4}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and $S_{\mathrm{f}}=0.1$.

\begin{figure}
\begin{center}
\includegraphics[scale=0.2]{dee}
\caption{\label{fig:dee}Transition to deeper vacuum. Upper: Spherical symmetry: $V_{4}$ with $V_{\mathrm{f}}=0.00001$, $r_{0}=50$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and $S_{\mathrm{f}}=0.1$. Lower: Bubble collisions of asymmetric different true vacua with hyperbolic symmetry: $V_{4}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and $S_{\mathrm{f}}=0.1$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{spherical2}
\caption{\label{fig:spherical2}Deeper vacuum transition by bubble collisions, for spherical symmetric case.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{causal4}
\caption{\label{fig:causal4}Causal structure of deeper vacuum transitions via bubble collisions with spherical (left) and hyperbolic (right) symmetry.}
\end{center}
\end{figure}

Figure~\ref{fig:dee} shows the spherical and hyperbolic cases. The skyblue region is the deeper vacuum.

For the spherical case (left of Figure~\ref{fig:dee}), after two walls are collided, a deeper region is generated. The boundary between the deeper vacuum and the true vacuum moves along the out-going null direction. Hence, it can be outside of the event horizon, if we properly choose the tension. This can be interpreted by Figure~\ref{fig:spherical2}. After the bubble collision, the deeper vacuum region expanded along the out-going direction, and hence it can be outside of the apparent horizon.

For the hyperbolic case (right), the deeper vacuum region and the big crunch singularity of the anti de Sitter region are symmetrically generated, as one expected. To summarize, we draw the causal structures for vacuum destabilization for spherical and hyperbolic cases in Figure~\ref{fig:causal4}.

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

In this paper, we studied bubble collisions with gravitation by numerical calculations. We used the double-null formalism to implement numerical simulations. We used three types of symmetries: spherical, planar, and hyperbolic. Three kinds of symmetries can be qualitatively useful in various situations during the first order phase transition (Figure~\ref{fig:bubbles}). In this discussion, we will summarize our results.

First, we tested bubble percolation by bubble collisions. After bubbles collided, the false vacuum region disappeared and two true vacuum regions are emersed. For the spherical symmetric case, it will eventually induce a black hole in a true vacuum background (Figure~\ref{fig:causal1}). For the planar or hyperbolic symmetric case, two bubbles are emersed (left of Figure~\ref{fig:causal2}). However, as tension increases, if the wall is not sufficient to push the false vacuum region, then two walls may not be able to meet (right of Figure~\ref{fig:causal2}).

Second, we tested bubble collisions with different field values to see the dynamics of bubble walls. For symmetric cases, the collided wall moves symmetrically and the wall will touch the future infinity between two true vacuum regions (Figure~\ref{fig:causal3}). For asymmetric cases, the wall is biased due to the difference of pressures. The interesting example is the collision of the de Sitter vacuum and the anti de Sitter vacuum, and the causal structure is analyzed in Figure~\ref{fig:asy}.

Finally, we tested vacuum destabilization due to the bubble collisions. For the spherical symmetric case, the deeper vacuum region can be outside of the event horizon (left of Figure~\ref{fig:causal4}). When we compare this to Figure~\ref{fig:causal1}, we can notice that this can give a significant difference for outside observers. For the hyperbolic symmetric case, the deeper vacuum region is expanded over the null directions and there will be a space-like big crunch singularity (right of Figure~\ref{fig:causal4}).

There were a number of previous studies on bubble collisions, and this paper can confirm these studies. In addition, there are mainly new contributions provided by this paper.
\begin{itemize}
\item The spherical symmetric bubble collision (including black hole formation) was studied numerically with gravitation (Figure~\ref{fig:per_sphe} and left of Figure~\ref{fig:dee}). This can give a \textit{general intuition for multiple bubble collisions} (Figures~\ref{fig:spherical1} and \ref{fig:spherical2}).\footnote{Of course, we do not claim that all multiple bubble collisions generate a black hole; rather, a formation of a black hole requires a concentration of energy and our study shows that a black hole can be formed in these special cases. See the references in \cite{Khlopov}.}
\item We observed more general initial conditions than Coleman-DeLuccia type bubbles. This was easily done, because we used the double-null formalism. One interesting observation is that as tension increases \textit{walls slowly moves} and they may not be collided (lower right in Figure~\ref{fig:per_plan} and lower left and right in Figure~\ref{fig:per_hype_massless}).
\item The smooth transitions of metric functions (especially $r$) are realized. This cannot be done by the thin-wall approximation. We obtained future boundary formation from smooth initial data without assuming thin-wall and classified various causal structures. Moreover, Figure~\ref{fig:asy} shows more complicated behavior of horizons of $r$ during bubble collisions.
\end{itemize}

There can be further applications and there will be many other models and issues for bubble collisions. Also, there can be many problems that can be solved by double-null formalism with hyperbolic or planar symmetry. We leave these for future work.

\section*{Acknowledgment}

The authors would like to thank Ewan Stewart for discussions and encouragement. The authors also would like to thank Maxim Khlopov, Matthew Kleban and Dong-Hoon Kim for useful comments on this work and other references. DY, BHL, and WL were supported by the National Research Foundation of Korea grant funded by the Korean government (MEST) through the Center for Quantum Spacetime (CQUeST) of Sogang University with grant number 2005-0049409. DH was supported by Korea Research Foundation grants (KRF-313-2007-C00164, KRF-341-2007-C00010) funded by the Korean government (MOEHRD) and BK21. WL was supported by the National Research Foundation of Korea grant funded by the Korean government (MEST) [NRF-2010-355-C00017].

\begin{figure}
\begin{center}
\includegraphics[scale=1]{consistency}
\caption{\label{fig:consistency}Consistency test for Equation~(\ref{eq:constraint}) along $u=20, 40, 60, 80$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=1]{convergence}
\caption{\label{fig:convergence}Convergence test: $|r_{(1\times1)}-r_{(2\times2)}|/r_{(2\times2)}$ and $4|r_{(2\times2)}-r_{(4\times4)}|/r_{(4\times4)}$ along $u=20, 40, 60, 80$. This shows the second order convergence.}
\end{center}
\end{figure}

\section*{Appendix A: Numerical scheme}

The integration scheme that we used is the second order Runge-Kutta method. Here, we briefly discuss the integration scheme. As we see in Section~\ref{sec:imp}, we organized all equations by a set of first order differential equations. Let us assume that we know all data for initial surfaces $u=u_{\textit{i}}$ and $v=v_{\textit{i}}$ as we discussed in Section~\ref{sec:ini}, and observe how we can determine the point $(u=\Delta u + u_{\textit{i}}, v=\Delta v + v_{\textit{i}})$. For example, to obtain the function $r$, we have to solve
\begin{eqnarray}
r_{,u} &=& f,\label{eq:example1}\\
f_{,u} &=& \mathrm{Right\; hand\; side\; of\; Equation~(\ref{eq:E1})}\label{eq:example2}.
\end{eqnarray}
Of course, this is not the unique choice and we can use
\begin{eqnarray}
r_{,v} &=& g,\label{eq:example3}\\
g_{,v} &=& \mathrm{Right\; hand\; side\; of\; Equation~(\ref{eq:E2})},\label{eq:example4}
\end{eqnarray}
or
\begin{eqnarray}
r_{,v} &=& g, \label{eq:example5}\\
g_{,u} &=& \mathrm{Right\; hand\; side\; of\; Equation~(\ref{eq:E3})},\label{eq:example6}
\end{eqnarray}
and so on. In any case, all equations are first order differential equations. Therefore, for example, if we apply Equations~(\ref{eq:example1}) and (\ref{eq:example2}), we can successively solve
\begin{eqnarray}
f(\Delta u + u_{\textit{i}}, \Delta v + v_{\textit{i}}) &=& f(u_{\textit{i}}, \Delta v + v_{\textit{i}}) + \left( \mathrm{RHS\; of\; Equation~(\ref{eq:E1})\; at\;} (u_{\textit{i}}, \Delta v + v_{\textit{i}}) \right) \times \Delta u, \\
r(\Delta u + u_{\textit{i}}, \Delta v + v_{\textit{i}}) &=& r(u_{\textit{i}}, \Delta v + v_{\textit{i}}) + f(u_{\textit{i}}, \Delta v + v_{\textit{i}}) \times \Delta u.
\end{eqnarray}
Or, if we use Equations~(\ref{eq:example3}) and (\ref{eq:example4}), then we can successively solve
\begin{eqnarray}
g(\Delta u + u_{\textit{i}}, \Delta v + v_{\textit{i}}) &=& g(\Delta u + u_{\textit{i}}, v_{\textit{i}}) + \left( \mathrm{RHS\; of\; Equation~(\ref{eq:E2})\; at\;} (\Delta u + u_{\textit{i}}, v_{\textit{i}}) \right) \times \Delta v, \\
r(\Delta u + u_{\textit{i}}, \Delta v + v_{\textit{i}}) &=& r(\Delta u + u_{\textit{i}}, v_{\textit{i}}) + g(\Delta u + u_{\textit{i}}, v_{\textit{i}}) \times \Delta v.
\end{eqnarray}
If we use Equations~(\ref{eq:example5}) and (\ref{eq:example6}), then we can successively solve
\begin{eqnarray}
g(\Delta u + u_{\textit{i}}, \Delta v + v_{\textit{i}}) &=& g(u_{\textit{i}}, \Delta v + v_{\textit{i}}) + \left( \mathrm{RHS\; of\; Equation~(\ref{eq:E3})\; at\;} (u_{\textit{i}}, \Delta v + v_{\textit{i}}) \right) \times \Delta u, \\
r(\Delta u + u_{\textit{i}}, \Delta v + v_{\textit{i}}) &=& r(\Delta u + u_{\textit{i}}, v_{\textit{i}}) + g(\Delta u + u_{\textit{i}}, v_{\textit{i}}) \times \Delta v.
\end{eqnarray}
Although the actual calculations are more difficult than this since many functions are involved, the principle is the same. Note that these are based on the first order method (the Euler method). We can improve the integration scheme to the second order \cite{nr}: to solve the equation
\begin{eqnarray}
\frac{dy}{dx} = F(x,y),
\end{eqnarray}
we successively solve
\begin{eqnarray}
k_{1} &=& \Delta x \times F(x_{n},y_{n}),\\
k_{2} &=& \Delta x \times F(x_{n}+\Delta x / 2,y_{n}+k_{1}/2),\\
y_{n+1} &=& y_{n} + k_{2} + \mathcal{O}((\Delta x)^{3}),
\end{eqnarray}
where $n$ is the step and $\Delta x$ is the step size. It is not difficult to implement the second order Runge-Kutta method to the series of equations of this paper.

\section*{Appendix B: Consistency and convergence tests}

In this appendix, we report on the convergence and consistency tests for our simulations. As a demonstration, we check the case of hyperbolic symmetry: $V_{1}$ with $V_{\mathrm{f}}=0.0001$, $r_{0}=-m_{0}=100$, $\Delta u=\Delta v=20$, $u_{\mathrm{shell}}=v_{\mathrm{shell}}=30$, $S_{\mathrm{t}}=0$, and $S_{\mathrm{f}}=0.1$.

For consistency, we test one of the constraint functions:
\begin{equation}\label{eq:constraint}
\frac{\left|g_{,v}-2gd+4\pi r T^{\phi}_{vv}\right|}{\left(\left|g_{,v}\right|+\left|2gd - 4\pi r T^{\phi}_{vv}\right|\right)/2}
\end{equation}
around $u=20, 40, 60, 80$. Figure~\ref{fig:consistency} shows that it is less than $1$~\% except some points, where the denominator oscillatory vanishes ($g_{,v}\approx 0$); this will not be accumulated as one integrates along $v$. Therefore, this shows good consistency.

For convergence, we compared finer simulations: $1\times1$, $2\times2$, and $4\times4$ times finer for around $u=20, 40, 60, 80$.
In Figure~\ref{fig:convergence}, we see that the difference between the $1\times1$ and $2\times2$ times finer cases is $4$ times the difference between the $2\times2$ and $4\times4$ times finer cases,
and thus our simulation converges to second order. The numerical error is $\lesssim 10^{-6}\%$.

redpajama_set_name: RedPajamaArXiv


text:
Querrey gives US 1st US Open men's quarterfinalist since '11
Sam Querrey, of the United States, goes for a shot in the first set of his fourth round match against Mischa Zverev, of Germany, at the U.S. Open tennis tournament in New York, Sunday, Sept. 3, 2017. (AP Photo/Kathy Willens)
NEW YORK (AP) Sam Querrey was about as perfect as can be en route to becoming the first American man in the U.S. Open quarterfinals since 2011.
Not that Querrey feels any sort of burden when it comes to the carrying-the-flag aspect of his career-best run at Flushing Meadows.
''I never once thought about that this whole tournament or that match out there tonight,'' he said. ''I don't feel any extra pressure.''
Querrey never faced a break point and wound up with 55 winners and only eight unforced errors while easily beating 23rd-seeded Mischa Zverev of Germany 6-2, 6-2, 6-1 on Sunday night.
''I thought I played really well. I kind of couldn't have done anything better out there,'' the 17th-seeded Querrey told the crowd in his on-court interview. ''I kind of was in the zone.''
Sure was.
He delivered 18 aces and needed only 77 minutes to finish off Zverev, a left-hander who had that shoulder treated by a trainer during the match.
''Felt good out there from the beginning,'' said Querrey, who had never played a night match in Arthur Ashe Stadium. ''I felt like even when I was warming up, the ball was coming off clean. I felt like I was going to play well before the first point started.''
Querrey, a 29-year-old from California, reached the first Grand Slam semifinal of his career at Wimbledon in July, knocking off 2016 champion Andy Murray in the process.
Querrey has now won 16 of his past 19 matches overall.
If he can beat 28th-seeded Kevin Anderson of South Africa next, Querrey would give the host country its first male semifinalist at Flushing Meadows since Andy Roddick was the runner-up in 2006.
As it is, Querrey is the first U.S. man to get to the quarterfinals at the U.S. Open since Roddick and John Isner did it in 2011.
Roddick was the last American man to win any Grand Slam singles title, winning the championship in New York in 2003.
More AP tennis coverage: https://apnews.com/tag/apf-Tennis
More Tennis

redpajama_set_name: RedPajamaCommonCrawl


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> those other redirect codes...".
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