I think the characters would be slower, for one — as you approach the speed of light, going faster is more difficult. Your characters would be struggling as they walk.
I'm not sure what exactly would remain constant ; my guess is on the momentum. Assuming the character's mass is m0 and his non-relativistic speed is v0 = 1.5 m/s (30ft/round), his momentum is :
p0 = m0v0The relativistic momentum is given by :
p0 = m0v / sqrt(1 - v2/c2)Thus the relativistic speed is :
v = 1 / √(m02/p02 + 1/c2) = 1 / √(1/v02 + 1/c2)If c = 2 m/s (for instance), then :
v = 1 / √(1/1.52 + 1/22) = 1.2 m/s = 24 ft/round
As for the time dilation... I'm not sure at all, but I think a stationary observer will indeed see events unfolding more slowly. Relativity plays fast and loose with simultaneity.
For example, if two character are standing close to each other, then one ("A") walks away for 1 round while the other ("B") stays in place :
- As we've seen (assuming I'm correct), A will walk 7.2 m in 6 s.
- With my values for v and c, we've got t' = 9.487 s (rounded).
- Let's not forget that the light will take 3.6 s to come back to B.
In hindsight, that second part wasn't helpful at all.
edited 10th Jul '15 10:13:26 AM by Aetol
Worldbuilding is fun, writing is a chore
I am aware that setting this low a speed of light would, realistically, mean that the characters would have a very hard time just walking — I mostly intend to ignore this, along with the irradiation problem, for the sake of fun (and weirdness).
I am still mostly interested in what would be observed and the difference in observation. That second part was actually somewhat helpful, as I'd forgotten to consider how long it would take the light to reach B from A — even though I was having the characters moving close to the speed of light, in my head light was still travelling at the actual speed of light.
I think — I /think/ — I might have gotten some things wrong, though, in trying to understand this? If an object is moving a distance of 6 light seconds at .8c, then the time it should take to travel that distance would be 8 seconds, but the reduction in elapsed time experienced by the object would be the reciprocal of the lorentz factor — sqrt(1-v^/c^2) = 0.6 for v=.8c — so the object itself would experience 3.6 seconds, but an observer in a stationary frame of reference will still experience 8 seconds, as they should...? Which would mean that if A were moving at a speed such that they would move 30 feet in 6 seconds, B would observe them having moved for 6 seconds, but they would have experienced less time. (Right? But that still assumes essentially instant measurements, and it doesn't take into account how long it takes light to travel... Presumably, if A were moving away from B, the journey would seem to take more than 6 seconds from B's perspective and A would be redshifted, but if A were moving towards B the journey would seem to take less than 6 seconds from B's perspective, and A would be blueshifted, so in the first case A would (seem to) move slower and in the second case would (seem to) move faster?)
(6 ls at 0.8c takes 7.5 s...)
I think you may be right. It would even account for the speed dilation, in fact :
-
A sees himself moving at his normal speed v, for a dilated duration t' = t/gamma (gamma is the Lorentz factor). -
B sees A moving at a dilated speed v' = v/gamma, for a duration t.
Both agree that A moved a distance d = v*t/gamma.
This is wrong. See my post below for the corrected version.
As for the light lag problem... I think now that the time dilation only depends on the frame of reference, not the point of observation. So the time the light takes to reach B should be added to (or removed from) the dilated time.
EDIT : I'm also noticing right now that the Lorentz factor gamma, above, should be calculated from the stationary point of view, by putting the speed v', not v, in the formula. To find v' from v, see my above equation.
Conclusion : good luck defining a round.
edited 10th Jul '15 1:55:19 PM by Aetol
Worldbuilding is fun, writing is a chore
Hmm, yes, thank you. This has been helpful. As for defining a round — as long as everyone has fun, I would count this a success! Unless I'm feeling really ambitious, this is mostly going to end up as flavor for a short section of a dungeon, though — I don't think I want to have a combat where relativistic effects need to be taken into account.
edited 10th Jul '15 12:39:00 PM by Sekantti
Wait, I think I made a mistake. Here is what I meant :
- Alice sees herself moving
at his normal speed v, for a dilated duration t' = t*gamma (gamma is the Lorentz factor). - Bob sees Alice moving at a dilated speed v' = v*gamma, for a duration t.
Both agree that Alice moved a distance d = v*t*gamma.
Example :
- c = 2 m/s
- v = 1.5 m/s (Alice's normal speed)
- t' = 6 s (one round from Alice's POV)
This gives us :
- v' = 1.2 m/s (as detailed above)
- gamma = 0.8
- t = 7.5 s
And I forgot something else : space dilation. Alice does not, in fact, think she walked the same distance as Bob thinks she did. Why ? Because from Alice's POV, the world around her is also moving at 1.2 m/s : there is no special frame of reference, the "stationary" one is entirely arbitrary. The v = 1.5 m/s figure represents in fact the effort Alice puts into her movement (so if you don't want to deal with that, you can ignore it).
So Alice and Bob actually agree on her speed. But Alice thinks her trip was quicker than Bob does, so it must have been shorter. From Alice's POV, the world is compressed along the direction of her movement (which must look very weird when she starts turning). Conversely, from Bob's point-of-view, Alice is compressed along the direction of her movement.
Let's say Alice starts next to Bob and walks away from him.
- In Alice's frame of reference : she walks at 1.2 m/s for 6 s and ends up 7.2 m away from Bob. When she stops, this distance suddenly dilates (or rather, decompresses) to 9 m.
- In Bob's frame of reference : Alice walks at 1.2 m/s for 7.5 s and ends up 9 m away from him. Furthermore, Bob sees Alice arriving 4.5 s late, so from his POV the trip lasted 12 s and she walked at 0.667 m/s.
Now, Alice walks back toward Bob :
- In Alice's frame of reference : she walks 7.2 m at 1.2 m/s for 6 s. Again, the world around her compresses when she starts and dilates when she stops.
- In Bob's frame of reference : Alice walks 9 m at 1.2 m/s for 7.5 s. Furthermore, Bob sees Alice departing 4.5 s late, so from his POV the trip lasted 3 s and she walked at 3 m/s.
edited 10th Jul '15 4:43:31 PM by Aetol
Worldbuilding is fun, writing is a chore
Some other thoughts.
Space Dilation 2 : Electric Boogaloo
Now, let's consider that Alice is not a spaceship drifting through space at constant speed, as is the case with most thoughts experiments on relativity. She is walking ; in other words, taking steps. Why is that a problem ? Let's compare the viewpoints :- In Alice's frame of reference : let's say Alice takes 60 cm steps. She will reach her destination (7.2 m away, for her) in 12 steps.
- In Bob's frame of reference : Alice is compressed along the direction of her movement. Her steps should be 48 cm long. Yet she does reach her destination (9 m away, for him) in 12 steps, and her feet touch the ground every 75 cm.
Mass Increase
As we know, the mass of a body accelerating at relativistic speed increases. Let's say Alice has a rest mass m = 60 kg ; while she is moving, her mass will appear to be m' = m/gamma = 75 kg. And a quick googling confirms that yes, it does affect her weight on top of her inertia. Trying to run too fast will probably make her tumble.Important disclaimer : my knowledge of special relativity is cursory and self-taught. I do not claim to be an expert on the subject. Everything I've written in this post and above may or may not be bullshit ; at the very least I've probably mangled terminology. If someone who have actually studied the subject could provide feedback, it would be much appreciated.
edited 10th Jul '15 4:45:41 PM by Aetol
Worldbuilding is fun, writing is a chore
I'm pretty sure you need to be moving at relativistic speeds to actually gain a noticeable amount of mass. Even in a reality with a moderately smaller speed of light, it probably won't affect your characters that much in that specific way.
I postulated c = 2 m/s. Even walking would be relativistic speed.
Worldbuilding is fun, writing is a chore
A whole lot of other things would break down at a light speed that low, including most of the things that allow us to live :P.
Reality is for those who lack imagination.
While it is certainly true that the speed of light being that low would cause all sorts of other things as well, I'm mostly going to ignore that — I'm the DM, I can make the laws of physics if I so desire.
Actually, with some honest-to-god physicists in the group, the best thing you could probably do is not get it right. The thing about the Far Realm is that it has no internal consistency, there is no logic behind it. Just changing a constant around might have dramatic results, but everything would still follow the same basic laws. The point of the Far Realm is to take the players way out of their comfort zone. Putting in something that is utterly impossible even if you change all the constants around could be more effective than just saying "oh, this operates on a somewhat modified set of rules".
Reality is for those who lack imagination.
The one-off isn't actually set in the far realm, rather, the "plot" (such as it is) involves a warlock opening a portal to the far realm. So having the entire dungeon be weird and confusing might not necessary, but having weird stuff in the dungeon would be fun. I'm planning on trying to incorporate escheresque geometries in the layout, and I just felt that having the characters experiencing relativistic effects might be interesting.
I agree with Matt. If you have actual, real-life physicists in your group, and you're not one of them, don't bother. Just do whatever seems cool and don't bother explaining it. If the players ask, just say it's magic, or reality breaking down or something similarly vague. Anything else will just raise more questions than it answers.
edited 13th Jul '15 1:42:09 PM by Tungsten74
I am writing a one-off for D&D. It involves stuff to do with the far realm, so I thought it would be fun to introduce changes to fundamental laws of physics and/or alien geometries. (I ended up somewhat scrapping the latter idea after deciding that trying to describe the 4D appearance of a klein bottle and then setting the boss fight on it would be too much effort — for both myself and the players.)
However, I think it would be fun to have them stumble across an area where the speed of light is significantly lowered — so that the characters' running speed approached the speed of light, and the characters could experience relativistic effects.
Now, I have a pretty good idea of what a character moving at relativistic velocities would see in terms relativistic doppler effects and other relevant phenomena (having played A Slower Speed of Light).
My question, then, is what would a stationary character observe? I know the formula for relativistic time dilation — t'=t/√(1-v^2/c^2). At 95% of the speed of light with t=6 (the length of a round in D&D — let's assume the moving character did, in fact, move 30 feet), the result you get out for t' is about 19 seconds. My understanding is that 19 seconds is how long the stationary character would say it took the moving character to move — is this correct? I know that Lorentz contraction is a thing, and I have a rough idea about what its effects are, but I have only a very fuzzy idea about what a stationary character would observe of the moving character (regarding time dilation, lorentz contraction and relativistic doppler effects), let alone how the experiences of the stationary observer and the moving character would compare. (Is this even the correct forum to be posting this in?)
Additionally, one of the players has a bachelor's degree in physics, and another a master's, so I really want to get this right.
Let's pretend the moving character wouldn't be irradiated by the light that has been blue-shifted into the gamma ray range, as that would be no fun at all.