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Background

In my hard sci-fi settings, there is an advanced Type II Civilization, that are progressing toward Type III Civilization (yes, I am referring to Kardashev Scale). They are sprawling across the galaxy with relativistic ships, generational ships, and perhaps counless unmanned Von Neumann probes (but there is no faster than light travel). Most are under the rule of benevolent (or even malevolent) AI Gods, that also contributes in development of large scale projects and scientific advancements. Along with it are plenty of megastructures on astronomical scales, including Dyson Swarms and many esoteric megastructures we might not be able to comprehend their functions at all.

One of the advancement they achieve is the introduction of economically feasible stargates. Apparently the AI Gods managed to solve most technical issues with wormholes, including development of stargates with sufficiently small negative mass requirements, and of significant size that most starships can pass through. They are connecting major star systems and clusters, and so many developed systems possess stargates of enormous size, up to several kilometers wide, but typically stargates are about less than kilometer in size.

The stargate is supposedly based on Thorne-Morris wormhole, modified slightly by Kuhfittig to allow arbitrarily small requirements of exotic matter. However (call me lazy but...) I have no idea what the equations inside those papers mean.

Question

For my worldbuilding purpose, I want to know how to calculate the gate's mass, given its throat radius, so we comes with this question: how to calculate the mass of a stargate, given desired radius is known?

Considerations:

  • The question seeks answers that could devise a wormhole given certain radius, that requires arbitrarily small amount of exotic matter. Hopefully, for a gate of some kilometers big, its mass is around the same mass as Earth's moon, and ideally it should be less than Earth-mass.

  • Answers that describe the behavior and structure of said gate is weighed up. Especially, on stability (for example, mass limit of objects that can safely traverse a gate of certain radius, or when the gate collapsed, what would happen to the exotic matter, and would it create a black hole of what size?), and at what radius did the exotic mass distribution ceases (see Miscellaneous section).

  • Answers that are based on the cited papers and sources are preferred, but if there is any other paper that could produce gates with more desirable properties (smaller mass for a given radius is an example, or perhaps without the need of exotic matter!), they are too, appreciated.

  • Regarding traversable wormhole, I consider this to be the standard. So answers must produce wormholes that are symmetrical and static, containing a throat that connects two asymptotically flat regions of spacetime, no event horizon, bearable tidal forces, reasonable transit time, stable against perturbations, and feasible mass requirements.

  • For the purpose of this question, assume that wormhole is not impossible, and exotic matter of some form can be obtained. Also, to solve problems with the existence of wormholes implies possibility of time travel, assume that it is impossible to arrange wormholes into time machine, and any wormholes that could lead to violation of causality would be unstable and collapse (in another words, chronology protection conjecture is assumed true).

  • The answer must give a mention on whether or not it is using geometrized unit system or in SI Units. It is preferable if the answer is to provide conversion to SI Units for its conclusion.

Miscellaneous

Probably worth mentioning, but I consider that the stargate in question is Medium-Exotic-Region Wormhole, or even Small-Exotic-Region Wormhole according to this answer. I don't know whether or not it will change the property of said wormhole, but I read somewhere (I can't find the link) that extending the exotic region into larger volume of space equals increased mass. By that logic I suspected that smaller region of exotic matter means smaller mass. And because the answer stated that if the exotic matter is restricted so closely to the throat, it is "absurdly benign", I think, it is a safer bet that the exotic matter would be loosely restricted to the throat (which, is also how Kuhfittig approached the feasibility of a wormhole with arbitrarily small exotic requirements of exotic matter, in the cited paper).

About the mass of a wormhole, as first mentioned by Dubukay on sandbox, I discovered that the mass of a wormhole is its ADM Mass. Perhaps this is also unrelated, but I discovered this answer, that explained that a wormhole gains mass of the incoming object in the entrance mouth, and lost equivalent mass of the out-coming object from the exit mouth.

Edit 1 (21 January 2019)

The title is edited so that now it focus to the way to calculate its mass. And to clarify comments about whether nor not the hard science tag is eligible for this question, I use this tag under the need for a question backed up with equations, and relevant theories, preferably from peer-reviewed scientific papers. The basis of my argument is on this quote from tag info of "hard-science" (here):

Ideally, answers should be backed up by equations, relevant theories, and citations where possible - arXiv can be quite good for citations, though Wikipedia is usually OK too.

On comment about whether I could understand the math behind this question, I have it outlined above that the answer I seek must explain how to calculate the mass of said wormhole by knowing its radius, a qualification of model accepted (the less mass for a given radius, the better), and a requirement to state which units used (geometrized units or in SI, where SI units or conversion to SI units is preferable). All of those requirements ensures that all answers boils down to how to calculate the gate mass given only its throat radius.

I believe with all of those requirements, the answer is not impossible, based on known (albeit speculative) current theoretical physics. This is something I could not get from just science-based tag.

Edit 2 (28 January 2019)

I rephrase certain parts of the question and considerations to reflect corrections provided by the commenters.

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ If I'm not mistaken there is no general relationship between the radius and total ADM mass of each side of a wormhole; the mass can be as small as you want, positive, negative or even zero, see for example the first introductory paragraph here. Also, as the PhysicsForums comments mentioned in your last link say, the mass of each mouth would change with every object that passes through (and this has to hold as a generic statement for any reasonable wormhole), even though the sum of the two masses would stay constant. $\endgroup$ – pregunton Jan 20 at 20:54
  • $\begingroup$ However, if you have a fixed wormhole structure, just by dimensional analysis I'd expect the mass to increase roughly in direct proportion to the radius like in an ordinary black hole, since the throat length doesn't contribute to the mass AFAIK, and there don't seem to be any other relevant dimensional parameters at play (I could be wrong though, better wait for an expert physicist like HDE to confirm). $\endgroup$ – pregunton Jan 20 at 20:54
  • $\begingroup$ Your title asks for the mass of the wormhole, and the text asks for the mass in relation to a transit station. Can you elaborate on how you're picturing the gate structure/station looks? $\endgroup$ – Dan Jan 27 at 21:49
  • $\begingroup$ @Dan The paper by Kuhfittig implies there exist a space of certain radius from the throat where space is approaching asymptotic flatness, as far as I could digest it. Therefore I assume the transit station would orbit the gate slightly beyond (or on) that point where space is asymptotically flat. First and foremost, the question asks how to calculate mass of a gate of certain throat radius, supposedly separation from throat to station is optional, not mandatory. Thank you for pointing it out :), as the way I phrase it raises the confusion. I'll edit it accordingly after I finished my flight. $\endgroup$ – Hendrik Lie Jan 27 at 22:24
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I finally found how to calculate the mass in the general case. Here is an answer summarizing my comments and calculations (but I'm still not an expert in general relativity, so please take it with a grain of salt).


Mass in general relativity can be a tricky concept. In particular, the ADM mass of a system is only defined relative to far away observers, ideally at infinity (this is why we need the asymptotic flatness condition), and it has in general little to do with the quantity of matter needed to create it. It is a formal parameter which is conserved in time and roughly describes the "force" that these observers would feel: gravitational attraction if the ADM mass is positive, repulsion if it is negative, and nothing at all if it is zero.

ADM mass of an arbitrary (static, spherically symmetric) wormhole

Maybe it's just my inexperience in GR speaking, but the ADM mass seems quite tedious to calculate for an arbitrary spacetime. Luckily, we can make two simplifying assumptions here, which leads us to a formula for the mass of any static and spherically symmetric system:

  • For a static spacetime (where the metric coefficients do not depend on the time coordinate) it is known that the ADM mass coincides with the Komar mass, see this ref. This is another definition of mass which is somewhat easier to calculate with.

  • The Komar mass for the particular case of a spherically symmetric metric, such as the ones in this problem, can be calculated using for example Eq. 17 here. That is, if we have any metric of the form

$$ds^2 = - e^{2A(r)} dt^2 + e^{2B(r)} dr^2 + r^2 d\Omega^2,$$

the Komar mass of each throat can be found using

$$M(r) = \frac{r^2}{2} e^{-(A(r)+B(r))} \left( e^{2A(r)} \right)' = r^2 e^{A(r)-B(r)} A'(r),$$

and taking the limit $r\to +\infty$ (first mouth) or $r \to -\infty$ (second mouth).

Ellis wormhole

In the case of the Thorne-Morris wormhole (perhaps more properly called Ellis wormhole, see ref. 14 here), the mass turns out to be exactly zero since $\alpha(r)=0$. This means that the spacetime around the wormhole would be approximately Minkowski, rather than Schwarzschild-like, on both sides far away from the throat, so the hole would produce no gravitational attraction (as is also said in the Wikipedia article).

Note that this does not mean that you can't put the wormhole in orbit around other bodies such as the Earth or the Sun: due to the principle of equivalence, everything feels gravity no matter how much mass it has (even massless light). It's just that you can't put things in orbit around the wormhole.

As a curiosity, there is a generalization of the Ellis wormhole where both sides have (different) mass; it is called the Ellis drainhole.

Kuhfittig wormhole

In the case of the Kuhfittig wormhole, replacing $A(r), B(r)$ by their definitions in the paper $\gamma_2(r), \alpha_2(r)$, and if the reasoning so far is correct, we obtain a mass of

$$M = \frac{c^2}{G}b(r_3-r_0)$$

for both sides, which depends on the three parameters $b$, $r_3$ and $r_0$. The factor of $c^2/G$ is just to convert from geometrized to SI units, as required in the question. For the example proposed in the paper ($b=0.5, r_0 \approx 0, r_3 = 0.00005$ light years $=4.73\cdot 10^{11}$ m), this gives a mass of $+3.2 \cdot 10^{38}$ kg, or roughly 160 million solar masses, meaning that far away observers would feel a attractive gravitational force similar to the one of a black hole of that mass.

The formula matches what we would have expected from dimensional analysis: the mass increases roughly in direct proportion to a radial parameter, in this case $r_3$.

Changes in mass

Of course, these are the masses only at the beginning, before anything passes through the throat. As you mention in the question, whenever an object traverses the wormhole the mass of each mouth changes as this Physics SE answer indicates, and this is because the metric itself must change to accomodate the object.

Since an object usually enters the wormhole from a given direction, the new metric would not be spherically symmetric anymore, so the above formula would not necessarily apply, but we can approximate the final mass of a mouth with

$$M_{\mathrm{final}} = M_{\mathrm{initial}} + \sum_i m_{i, \:\mathrm{in}} - \sum_j m_{j, \:\mathrm{out}},$$

where $m_{i, \:\mathrm{in}}$ is the mass of the $i$th incoming object and $m_{j, \:\mathrm{out}}$ the mass of the $j$th outcoming object. In the case of humans or spaceships the small amount of mass gained/lost by the mouths is quite small by general relativistic scales, so would have no important effects gravitationally speaking.

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  • $\begingroup$ Out of curiosity, on Kuhfittig wormhole, what do the parameters a, r0, and r3 represent? I get that r3 is the outermost boundary, r0 is throat, but what is a? $\endgroup$ – Hendrik Lie Feb 12 at 8:13
  • $\begingroup$ @HendrikLie The parameters $a$ and $b$ are used as exponents in the paper. They seem to be related to some sort of "slope" of the wormhole's shape near the throat, but I don't know if there is a more immediate physical interpretation like there is for the radial parameters. Btw, I fixed a mistake in the calculations, the Kuhfittig wormhole's mass is actually positive (which makes more sense since it's supposed to minimize exotic matter). $\endgroup$ – pregunton Feb 12 at 12:52
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The Thorne-Morris black hole is new to me. However - reading the paper, unless I've missed it, this is a modification to a Schwarzchild black hole to provide a traversable opening (they start with the same equations).

More simplistically, the construction appears to be a black hole, then sufficient exotic mass-energy to create another black hole of about the same size.

The mass requirement to achieve a Schwarzchild radius of $R$ comes from Schwarzchild's solution to General Relativity :

$R = {{GM} \over {c^2}} \rightarrow M = {{Rc^2} \over G}$

Where: $c^2$ is the speed of light (~3E+8 m/s) squared, $G$ is the gravitational constant (6.67E-11), $R$ is the desired wormhole outer radius, and $M$ is the mass of your desired wormhole.

For the inner radius, the same equation applies. Also, the outer radius must be larger than the inner one. $R_{outer} > R_{inner}$

In an instance where the two radii are close enough to equal ($R_{outer} \approx R_{inner}$), the mass for your traversable wormhole is $M = 2 {{Rc^2}\over G}$

Let's try this out for a 1 kilometer wormhole :

$R = 1,000 \therefore M = 2.69E+30$ kg. For some perspective, the mass of the sun is $1.988E+30$ kg, so a little over 1 solar mass would be required to generate a 1 kilometer-radius (2 km diameter) wormhole of this type. Assuming there aren't some technologies that draw down this requirement.

Evaporation

I'm a little curious what evaporation would be in this condition. As black holes become smaller, their lifetimes become shorter and they become hotter.

Assuming my notes are right, the lifetime of a singularity of any size is :

t = $M^3 ((5120 \pi G) / (h c^4)$

Where the only new term is $h$, Plank's constant (6.60E-34) Js

Given a 2 kilometer diameter (1km radius) traversable wormhole of this type, weighing in at around 2 solar masses, how long until it evaporates into a fine explosion of energy? I get $3.9E+84$ seconds, which is longer than the universe is old. So, it won't be evaporating any time soon.

Effect on Surrounding Environment

On another recent thread people from this forum calculated the closest approach two solar systems could make to one another without disrupting all of the planets and other bodies. Depending on the technology your alien race has to dampen out far range gravitational effects, your traversable wormhole of (1km rad/2km diam @ ~1 solar mass) would be disrupting bodies out to a distance of 30 light-days (5,000 astronomical units (AU)), or out to the outer edge of the Oort cloud.

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    $\begingroup$ Unfortunately this answer is wrong. The metric for the Thorne-Morris wormhole (perhaps more properly called Ellis wormhole, see ref. 14 here) is given by Eq. (B2a) in the article you linked, and it is not a modification of the Schwarzschild metric but of the flat Minkowski metric. In fact the Ellis wormhole's ADM mass seems to be exactly zero (see e.g. section III here), so it would cause no important disruptions on surrounding environment (at least far from the throat). $\endgroup$ – pregunton Feb 11 at 12:57
  • $\begingroup$ What would happen if the inner and outer radius is significantly different? And for @pregunton according to this answer, when we enter a throat, it gained mass, and on the exit throat, a wormhole loses equivalent mass. That, and assuming that mass of each throat can't be zero, wouldn't it be ideal for the throat mass to be non-zero? Is it even possible to make the throats to have mass? $\endgroup$ – Hendrik Lie Feb 11 at 14:54
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    $\begingroup$ @HendrikLie Yes, the masses would be zero only at the beginning. Whenever an object traverses the wormhole the mass of each mouth changes as your link indicates, and this is because the metric itself changes to accomodate the object. However, in the case of humans or spaceships the small amount of mass gained/lost by the mouths would have no important effects gravitationally speaking. I don't know if it would be ideal to increase the mass, I guess it depends on the story. (This applies only to the Ellis wormhole, I'm not yet sure whether Kuhfittig's wormhole mouths have mass and of what sign). $\endgroup$ – pregunton Feb 11 at 19:05

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