K^2

There is no "out there". So asking what's there is pointless. That's the hole point. The boundary or direction in which you think something else might be does not exist. It's not a feature of the universe. So what's out? Where is it? It's not just a matter of us not being to probe beyond our universe. That's one thing. But you keep insisting on something being beyond... what? Point me to some line that you have to cross to be outside. Or direction in which you need to go to get there. We have excluded all of these. What you see is what you get. It's that simple. From perspective of mathematics and science. Again, this isn't a limitation of science. It's science specifically telling us that what you're looking for doesn't exist. Which is far more than I can say about unicorns. I'm not being facetious on that matter. Unicorns are a real possibility in comparison.

Eh, these links expire after single view... I've taken a look at the first circuit board one, so maybe someone else would still be able to look at others. (Unless this is per IP?) These joints do look oxidized to hell, though. I guess Nuke had a good call on this. With practice, you should be able to make joints using flux core solder without additional flux, but maybe you did burn off all of your flux while making these. Good practice when working with simple wire joints is to use the soldering iron to heat up the wire, then touch it with the solder wire. That tends to produce very neat joints. These look like you've transferred solder to iron, then applied it to the joint. Anyways, unless you feel like having good practice in de-soldering and then re-soldering everything, next step would be to try and find a bad joint. You might be able to eliminate some possibilities with a multimeter.

Simple example. Surface of a sphere is a 2d manifold that can be embedded in 3d space. Locally, it behaves like a 2d surface, but if you want to picture it as a complete shape in flat space, you need at least one extra dimension. GR space-time is a 4d manifold. Some special cases can be embedded. You might have seen a picture of a funnel that represents curvature of space around a black hole. Of course, these representations are somewhat limited, since it is hard for us to picture more than 3 dimensions at the same time. But they exist, nonetheless. Well, an interesting result in GR is that a general solution to Einstein's Field Equations cannot be embedded. In other words, for any finite number of dimensions, there exists a mass arrangement that results in a space-time shape that cannot fit into that number of dimensions. Therefore, our space-time is not embedded. If it was, there would be significant and measureable differences in how gravity works. Sort of can. As mentioned above, ebedding is not an option. Some flavors of String Theory work with a bulk, but they also make predictions that do not pan out based on it. So bulk seems unlikely. Locally, topology is fixed, so there cannot be a connection to an outside of any sort. That leaves edges, but we observe a very uniform universe. So unless we are dead center of a huge, much larger than observable, but still finite universe, it does not look good for edges either. If you have some other definition of outside, please, bring it forward. As it is, I can claim lack of outside as a scientific fact. If you need a greater absolute, then yes, I cannot help you, being a mere mortal and all. But you'd have better luck looking for unicorns than an "outside" of our universe.

It is hard to even buy solder these days that is not a 60/40 with a flux core, but worth making sure. Dominatus, could you post some pictures? Bad joints are usually pretty distinct to an experienced eye. Maybe something will jump at one of us about wiring as well.

There isn't a "beyond". There is a set of locations and a metric. Everything else is just your brain trying to make sense of it based on everyday experience. There's no reason to expect it to be successful in that.

Nothing contains the universe. It's not an embedded manifold. In fact, GR pretty much states that our universe cannot be embedded. Possible, but very arbitrary and somewhat boring. If you are going to go with finite matter in infinite space, a fractal universe is much more elegant. Non-uniform structure is a requirement for infinite universe, by the way. Otherwise, sky would be infinitely bright.

You can, actually, recompute all of the physics to act on orbital parameters directly. This is something that KSP doesn't do. KSP does it exactly like you say. It takes velocity and position and computes the 6 orbital elements. Semi-major axis, eccentricity, inclination, argument of periapsis, longitude of the ascending node, and mean anomaly at epoch. If you apply a force, it recomputes velocity, then recomputes the orbital elements. That's why sometimes you see the orbits shake a little for your current craft. While warping, or if physics isn't being simulated, you usually go "on rails". That is, you'll just follow along a fixed orbit. The other way to do this is to work with constants of motion directly. There are also six, and they are closely related to orbital elements above. Energy, angular momentum, z-component of angular momentum, time of periapsis passage, argument of periapsis, and longitude of the ascending node. The best part is that they are constants of unperturbed Hamiltonian, they have rather simple laws of motion. In fact, they are just Hamilton's Equations on perturbation potential. For example, the equation for the rate of change of energy is dE/dt = -∂U/∂T. Here, U is effective potential of the applied force. In other words, what you really know is that ∂U/∂x = -Fx, and so on. If you apply the chain rule a whole bunch of time, you'll find that dE/dt is equal to dot product of velocity and force, which is exactly what you expect. Some of the other results are a bit less trivial, and they are almost nightmarish to derive. On the other hand, once you have these equations derived, you can apply forces directly to orbital elements, and then you don't have to worry about your energy running away, like it does with Euler method (or even higher order methods, to lesser extent) or oscillating, like it does in KSP.

AngelLestat, that's a bunch of speculation on your part with not a shred of evidence. Again. Please, learn to distinguish between scientific method and making stuff up. Nitpick accepted, conditionally. The limit is clearly zero volume at zero time. But whether exact zero time actually exists is a separate question. Singularities are, well, singular. Infinitesimal volume at infinitesimal time from Big Bang is less flimsy.

How do you plan to detect the O17? At uni, I would have done mass-spec or NMR, but these aren't exactly at-home options. I suppose, you can just straight up detect oxygen, but it'd be hard to guarantee that it's not just from leaks.

That is a fallacy, as I've pointed out. Universe could have had zero or finite size at Big Bang and then inflate to infinite size during Big Bang. In fact, all of our observations seem to confirm that universe started out as a single point. Another thing, we have evidence that suggests boundless universe. That doesn't mean infinite. Universe can be closed on itself and have a finite volume. That is entirely consistent with every observation that we've made.

If you spend a little bit of time actually studying measure theory, and then take a good look at modern cosmology, you'll understand that infinite universe is entirely within realm of possible.

Bad wording on my part. "At the exact instance off..." would be more correct. Though, with a finite universe, there are some interesting interpretations of "before". In a finite universe, Big Bang can work like a center of a sphere, with each instance of time being a new layer. It automatically has expansion built in, and has some other nice features. Most importantly, the question of what's "before" or "outside" disappears naturally. These bits are just speculation, though. Both finite and infinite universes are consistent with observations and theory.

On macro-scale, you get self-interference, which in practice, reduces probability amplitude. Arrival of the time traveler becomes a random event conditional on that random event, and time traveler's own time-line obviously has an opposite outcome. How you interpret it is up to you. From perspective of MWI, time traveler arrived from alternate reality. Other interpretations will look at it differently, but the effect is the same.

Big Bang and infinite universe do not contradict each other. It all goes to question of how an infinite universe can expand. Expansion is not just boundaries, if they exist, moving out. Every point of space experiences expansion. This is a feature of differential geometry our space has. And it goes back to analysis. Can infinitely many regions of space add up to zero volume before the Big Bang? They can, if initial expansion rate was a "greater" infinity. It is all about measures and limits.

By not having events. You have fields that satisfy certain conditions locally, which gives you local causality. But global structure is determined by the Lagrangian and boundary conditions. In the big picture, there isn't a cause-and-effect, because the entire structure is already predetermined. If you have a flat space-time, you can use the local causality to build up a causal timeline, which is basically what we're doing. But having exceptions to that simply isn't a contradiction. Timelines become path-dependent. Your timeline might not agree with everyone else's, but that isn't a contradiction either.

It's dangerous to deal with infinities alone. Take this.

You honestly don't need much, so yeah, I think that could be arranged.

Not by mass it isn't.

It's icky for the same reason pounds of mass are icky. And pounds of mass are the reason we have a factor of g in rocket equation. Do you want a factor of 1/g in your rocket equation? Because that's what you'll get with kilograms of force.

It's fairly trivial to prove that no uniform material is strong enough to build a vacuum balloon. There are some designs that involve composite honeycomb and Mylar that are just barely strong enough to allow for positive net buoyancy. That makes for an extremely expensive and extremely inefficient construction. You are better off building a hot air balloon than a vacuum one.

Your equation isn't correct either. Momentum is given by p = γmv, where γ = 1/sqrt(1-v²/c²) is the Lorentz Factor.

Not how it works. Any FTL guarantees that you can travel into the past and change it. Can be impractical, but no FTL scheme makes it impossible. Time traveling paradoxes must resolve themselves. And they do in field theory. It is a non-issue. Grandfather paradox is no more paradox than EPR is.

Euler does both. Energy will increase, and argument of periapsis will precess. One can be more noticeable than the other.

FTL flight in one coordinate system is time-travel in another. It's really as simple as that. Now, constructing a loop, which you need for practical time travel, might be a bit more complicated, but these are purely practical limitations. If FTL is possible, reality you live in must at least allow for time travel. And physics as we know it does allow for time travel. Global causality is not a requirement at all.

Abacus is right. You have to do a definite integral from t0 to t1, which gives you the ln(m0/m1) in the answer.