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Just to make sure: you typed the code in the box in the form ventuz said, clicked "Add code" button, selected it in the list to activate it then clicked OK and it didn't work?
Assume you could apply an unlimited amount of energy to a particle inside the event horizon of a black hole. Could it be used to slow down the time of the particle so much that it would delay its inevitable fall into the singularity by more than the lifetime of the black hole itself? Would the GR equations support/allow this to happen (at least in theory)?
No, for several reasons.
First: In pure GR, the lifetime of a black hole is infinite. The particle's time would have to stop for it to happen, which won't happen for any massive particle.
Second: inside the event horizon, all timelike and null geodesics end* at the singularity. For this reason, the singularity is called 'timelike': it is in the future of all geodesics, and is unavoidable.
Third: In GR, as well as in SR, the unaccelerated ("inertial") geodesics are those with maximal lapse of proper time. If you happen to fall into a black hole, you will live longer by not accelerating towards or away from it. With an unlimited amount of energy at your disposal, you will simply reach the singularity as fast as you want to. Of course, if anyone from outside the event horizon could see anything, the unaccelerated observer would reach the singularity faster.
The only "way" to escape from the singularity is to shift to a spacelike geodesic; this is as impossible in GR as it is in SR.
* Technically, there are several timelike and null geodesics that apparently "start" at the singularity and move away towards the event horizon, never reaching it but never falling back into the singularity. These geodesics can, however, be consistently interpreted as the analytic continuation of the geodesics of infalling particles from the outside: they "cross" the event horizon at "t = +infinite", "double-back in time" and fall into the singularity. Note the double-quotes, I am being deliberately imprecise here for the sake of clarity.
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nitsuja wrote:
There are lots of obvious differences too, more than some of the previous updates, but there are definitely parts that at least appear to have been left completely alone (e.g. almost all of AIZ and CNZ1).
And all of Sky Sanctuary; and a few bits and pieces here and there. I tried to improve on these, but kept coming short, so I decided to leave them alone; which is why I listed you both as coauthors. The GMV I sent, however, did have you and Upthorn listed as co-authors; it was a publisher oversight, as OmniPotentEntity stated.
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Warp wrote:
The General Theory of Relativity predicts that if a particle enters the event horizon of a black hole, it would require an infinite amount of energy to stop it from falling into the singularity at the center.
Not quite; ignoring for a moment the distinct types of black holes, and assuming a Schwarzschild black hole (non-rotating, uncharged), then no amount of energy will allow you to scape once you get past the event horizon. This is because any amount of energy pumped into acceleration (even "infinite" energy) will still leave you moving inside the local light cone. Infinite energy is also a terrible thing in GR -- energy curves space-time too, and having an infinite amount of it means you are a black hole...
Moreover, contrary to normal intuition, the harder you try to accelerate away from the singularity, the faster you are going to hit it according to your proper time.
Other than that, rhebus is on the money: the biggest problem in Physics today is that the best theories it has for explaining the Universe are mutually incompatible at a fundamental level.
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That one is a lot more complicated. I did the math once for the initial part of a similar system when I was teaching undergraduates in a lab -- and it is ugly. The system in question was regarding an experiment with an inclined plane of variable inclination and stacks of cylindrical weights (they had a hole in the middle and a stick to prevent the top ones from sliding from the other cylinders). The students had to analyze the sliding angle of the cylindrical weights when changing the number of stacked cylindrical weights, and the tilting effect caused the higher stacks to slide at lower angles than would have been expected otherwise, as the tilt reduced effective friction.
I don't know where my notes are (it was a few years back), but it was done with the method of Lagrange multipliers in classical variational calculus. But the gist is that the cube will tilt first due to the inclination and friction; this will reduce the normal force, which reduces the friction. This will cause the cube to slide. If the friction is still high enough (or the plane inclined enough), the cube will tilt faster than it slides. From then on, the cube will bounce on the inclined plane when it finishes the first 90-degree rotation, so it depends on the elastic properties of the cube and of the inclined plane. I didn't go that far in the math, though, as I had already seen what I wanted -- the reduction of the sliding angle for the cylinders.
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There are way to many possible results depending on the friction between the ball and the surface.
If there is no friction, the ball will not roll and will travel at the same speed as the cube did.
If there is friction, the ball will roll, and part of the kinetic energy will be rotational -- thus slowing the ball. It then depends on whether or not the friction is high enough to prevent sliding or not.
If the ball does not slide as it rolls down, the ball will be at its slowest. And it is the easiest to compute. Lets call the side of the cube a = (m/d)^(1/2) and the radius of the ball r = (3*m/4*pi*d)^(1/3).
Now, since the ball did not slide, then the speed of the point of contact of the ball and ramp must be purely radial (a tangential component would be sliding).
With all that, we can set up our coordinate system. The best choice is one in which the ball moves along one axis (say, towards the positive the x direction) parallel to the ramp. The slanted "vertical" (y) direction will have no motion, and we can ignore it. The other important variable is the radial angle q that describes the rolling motion of the ball.
The no-sliding condition can be written as
r*dq/dt + dx/dt = 0
The motion on the x direction comes from Newton's equation:
d^2x/dt^2 = g*sin(alpha) - Fn/m
And so does the rolling:
I*d^2q/dt^2 = r*Fn
Here, I = (2/5)*m*r^3 is the moment of inertial of the ball around its center and Fn is the frictional force, which acts as a torque. Putting it all together to eliminate Fn, we have
(I/m*r)*d^2q/dt^2 = -d^2x/dt^2 + g*sin(alpha)
d/dt {(I/m*r)*dq/dt + dx/dt)} = g*sin(alpha)
Using the no-sliding condition,
((I-m*r)/m*r)*d^2q/dt^2 = g*sin(alpha)
Integrating once in time, we have
dq/dt = (m*r/(I-m*r))*g*sin(alpha)*t + c1
where c1 is a constants of integration; if the ball starts with zero angular speed, c1 = 0. Returning to the no-sliding condition, we have
dx/dt = m*r^2*g*sin(alpha)*t/(I-m*r)
x = (1/2)*m*r^2*g*sin(alpha)*t^2/(I-m*r) + c2
where c2 is a constants of integration; if the ball starts at x = 0, c2 = 0.
We know that the ball has traveled a distance L along the x axis; hence, we can solve for t:
t = sqrt{ 2*L*(I-m*r)/m*r^2*g*sin(alpha) }
I will leave substituting I and r back, as well as comparing with the cube, for you.
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moozooh wrote:
Also, a semi-unrelated question: what would be Tails's role in a glitchless/low-glitch run aside from killing bosses faster?
Take a look at the end of Angel Island 1 and all of Death Egg 1 for examples: boosting Sonic into higher places so as he gets there faster.
Re: 1: Hydrocity 1, Marble Garden 2 to get to Knuckles' boss.
Re: 2: Indeed this requires a definition; in particular, the top of the screen will change in boss fights. In most cases, it would be best to limit going over the top of the level, and disallowing going over the top of the screen in vertically wrapping levels (or sections of levels).
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Zeupar wrote:
I completely agree. I would leave out all glitches as well as any trick that can't be done in real time.
So what you are saying is that we can't take out most of the glitches then :-p
Because so far, the only glitches that non-TAS speed-runners haven't done in real-time are:
Hydrocity 1 zip (it is possible -- I myself have done it many times, and I am no speed-runner -- but hard to do reliably) and transition glitch (easily possible if you pull of the zip, you just need to know when to brake);
Marble Garden 1 zip (although only technically; they use another, present in earlier S3&K runs, which involve vertical screen wrapping abuse and a monitor; ends up in the same place);
IceCap (far to easy to get killed by crushing sprites along the way, far too hard to pull off the zip at the end);
Launch Base 2 zip into Knuckles' boss area (the zip into regular boss has been done);
Mushroom Hill 1 zip and glitched start on 2 (no camera hack makes the latter next to impossible);
Flying Battery 2 (although only technically; they use another zip with the same result);
Lava Reef 1 zips (the first one requires ridiculous precision; the other requires Tails to lift you up so as to not to be slower);
Lava Reef 2 ground entrance (although the boss skip has been done);
Hidden Palace: expedited Knuckles' fight;
Death Egg 1 zip (they usually don't need to -- the slightly slower start makes the light bridge vanish before they get there).
Everything else has been done in unassisted runs.
Edit: @Noob Irdoh: the purpose for the list of glitches is to try to reach an agreement as to what constitutes a "small" glitch. For "no" glitches, most of the ultra high jumps, as well as slope jumping, would have to be removed as they are technically a form of abusing the game physics.
Oh, I also disagree that the switch confusion can't be done without TAS techniques. It can't be done reliably without TAS techniques, but if you know when to brake, you can pull it off on occasion. And anticipating your objection: most of the ultra high slope jumps also cannot be done reliably unassisted; so if you want to take out one, you have less reason not to take out the other.
Also
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Hydrocity 1: Wall zip.
Carnival Night 2: technically, 2 small zips, one of which has long-lasting consequences (ultra high speed).
Launch Base 1: Moves objects off-screen to go through them. Glitches into Knuckles' boss.
Sandopolis 1: Goes over the top of the screen to reach wall zip.
Sandopolis 2: Uses the fact that Tails does not interact with some off-screen objects. Also, abused incorrect assumption in boss programming to get the hitbox to return faster than it should. Moves objects off-screen to go through them.
Lava Reef 1: Uses the fact that Tails does not interact with some off-screen objects to open up a faster route. Moves objects off-screen to go through them.
Lava Reef 2: Moves object off-screen to go through it.
Hidden Palace: Moves object off-screen to go through it.
Sky Sanctuary: Moves object off-screen to go through it.
Death Egg 2: Moves objects off-screen to go through them.
Moreover, several levels feature jumps into locations that should be impossible for Sonic to do, but are managed anyway with high speed and slope jumping (Hydrocity 1, Marble Garden 1 & 2, Mushroom Hill 2, Sandopolis 1, Death Egg 1).
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Oh, and here I thought that this movie and this movie collected all Chaos Emeralds... guess I was wrong. [/sarcasm]
I won't speculate on the legality of that, though.
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boct1584: your list leaves off: scrolling of objects offscreen to go through them, vertical level wrapping abuse (see Sky Sanctuary), going over the top of the screen (e.g., in Marble Garden 2 one can reach the Knuckles boss without having to zip by walking over the top), anti-gravity glitch (might be useful even if no zips), manipulation of object placement. Are you sure some of these are "low" glitch?
@True: At least in Ubuntu, with the past several versions from the WineHQ ppa, I can get sound with no issues. I can't dump neither video nor audio, but everything else works perfectly (though the Wine binaries in main Ubuntu repos aren't as outdated as they used to be...). I have been using the Alsa setting in Wine, which goes through PulseAudio in Ubuntu.
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moozooh: I have also been wondering about a glitchless run for much of the same reasons -- i.e., showing some actual gameplay. It would require defining "glitchless" more precisely, though: technically, even slope jumping can be considered a glitch if taken to abusive levels as is done in TASes.
Pointless Boy: I half feel that there should be additional encodes with the camera hack centered on Sonic and Tails, as well as the one with the HUD, just so people can see a little bit better what is going on.
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Full game WIP. Death Egg 2 is indeed 20 frames faster than the published run, and finished at 2:18::06. Full movie time is 29:51.20, compared to 32:05.52 from the published run -- a difference of 02:14.32. In in-game time, the difference is smaller: 01:28.48. This is due to the Knuckles cutscene skips in Angel Island 2 and Launch Base 2, but mostly due to the latter -- the latter comes at an expense of 11 seconds in in-game time, but gains more than twice as much in real time.
Unless someone has any ideas, I think this is the version I will submit -- I don't think I can get anything else from Death Egg.
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New WIP, up to just before the first mini-boss of Death Egg 2. I am so far 20 frames faster than the published run, mostly from minor optimizations here and there. I didn't feel like doing the boss yet, as I still have to double check if it is possible to improve a bit further. Sadly, the possible shortcuts I tried end up leaving me stuck where I don't want to be, and like the predecessor, this level will be mostly a frame war.
I also edited all of Death Egg's input to maintain consistency, as I will other levels later on.
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New WIP, fully resynched up to Death Egg 1.
After the previous improvement, I also optimized the capsule hit on Hydrocity 2. This caused desynchs -- guess which levels?
Yeah. marble Garden 2 desynched; resynching led to no loss or gain. Then Carnival Night 2 desynched (I hate those balloons). On the flip side, I managed to get Tails slightly earlier, leading to an improvement of 3 frames (final time: 0:41::41).
Launch Base 1 desynched because the level-swap lag frame at the score tally happened 1 frame later than the previous WIP. That was easily fixed, but still it cost me a frame in real time. There was an extra lag frame at the start of a level, but I haven't paid attention which one.
Mushroom Hill 1 desynched at the mini-boss -- Tails got hit 1 frame later, the reverse of what happened earlier. I replayed the fight so as to prevent this from happening again (Tails goes head on into the flame now), and this led to an improvement of 11 frames, down to 0:32::50. With some subpixel manipulation, I managed to resynch Mushroom Hill 2 without needing to replay it.
Then the Lava Reef 1 mini-boss desynched because it was further right (by 10 pixels). After resynching it, Sky Sanctuary desynched at Mecha Sonic. I added a small (1 frame) pause to manipulate luck, as the pattern was the slowest.
Then the Death Egg 1 mini-boss desynched. I replayed it, and ended up gaining 1 frame in the fight -- so the level is down to 1:04::03.
All told, this is 132 frames of improvement from the previous WIP up to this point. I am, of course, comfortably sub-30 min now.
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Argh. Thanks to blooper #12 in this post (need to be signed in to see it), I now have an improvement of 49 frames in Hydrocity 2 (and it could even be a few more frames, I am still trying). This will probably require resynching a lot of things again. Damn you, mike89! :-p
Edit: New WIP, back to Hydrocity 2; the level is now finished in 0:23::37, an improvement of 107 frames over the published run. Now back to resynching the rest of the run... again...
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New WIP,New WIP featuring Death Egg 1 done in 1:04::04 -- an improvement of 43 frames over the published run.
Edit: replayed boss battle to eliminate 24 unnecessary frames of lag.
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The problem with Sky Sanctuary is that I can find nothing to improve it. I suppose the initial waiting could be changed to something else, but all else is nigh unimprovable. Unlike Death Egg 1 -- it has turned out to be a tough nut to crack, but I managed to find some 30-something frames of improvement.
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A new WIP which is partway through Hidden Palace. This is up to the end of the Knuckles fight, which I just had to share -- better manipulation of Tails lead to a 59 frame improvement over the newgame+ run or 103 frames faster than the published any% run, for a final time of 0:17::27. Knuckles is down before his music begins playing :-)
Edit: New WIP, finishing Hidden Palace and Sky Sanctuary. Try as I might, I found nothing to improve in Sky Sanctuary. I even tried glitching Mecha Sonic with Tails to kill him in one pass, but the setup for it was slower than I gained from doing it. So I just kept the old input.
Death Egg 1 and 2 are not looking promising in terms of improvements either, though I just begun. I tried reproducing the newgame+ shortcuts in Death Egg 1 using a bubble shield, but the bubble bounce causes me to fall too far down and get stuck in the wall in every single one. In Death Egg 2 there are a couple zips if you start with a bubble shield -- they are slower if you have to get it -- but I don't think it will be possible to finish act 1 with one without losing far more time than could be gained in act 2.
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New version is up, featuring:
Added instashield timer.
Improved Tails' flight icon.
Fixed Marble Garden 1 mini-boss HUD.
Compatibility update for latest version of Amy Rose in Sonic 1.
When disabling or enabling the original HUD in Sonic 1 and Sonic 1-based hacks through the configuration menu, you no longer have to wait for the next act for it to be restored.
