When something travels at relativistic speed, time passes slow (clock runs slower) for that thing (proper time, time dilation).
When Flash travels at near light speed, the world is going fast and he is stationary from his point of view. So the world clock's 5 seconds is equal to his own clock's 50 seconds (say), so the earthlings (lol sorry) think that the Flash is doing 50 seconds worth of stuff in 5 seconds. That's great.
However, from the world's frame of reference, shouldn't the Flash be feeling only 5 seconds has passed in his clock (he was running so his clock got slow) when in fact 50 seconds has passed in the world's clock? Why is this argument wrong?
What am I missing, length contraction maybe? I get confused by relatively no matter how well I study and understand it each time.
Best Answer
I like this question and I think you're right on the time aspect, but not thinking about the question the right way.
Short answer. You're right on time dilation being reverse because it limits the speed at which things can move, it doesn't provide more time (relatively) to the very fast moving person, it does the opposite. If you need 7 days to complete a homework assignment that's due in 12 hours, accelerating close to the speed of light won't help you. What you need to do is accelerate your teacher to close to the speed of light so their 12 hours becomes 7 days. It can't be done the other way around.
Long answer:
Lets say an average Joe can perform 1 simple task in 1 second.
The Flash, if you've ever seen him work at his computer or put a puzzle together or decorate a Christmas tree, he can perform, lets be conservative, 10,000 simple tasks in 1 second.
Now lets say the Flash is running around at .9 the speed of light (which would cause a lot of problems, see here, his clock is slowed down about 2.3 times relative to the non-meta-humans. So now he can only perform 10,000 simple tasks in 2.3 seconds (to a person watching), still some 4,000 times faster than normal.
He'd have to move really fast, like .999999995c to slow down by a factor of 10,000 (relativity calculator), and at that speed, he'd be time-dilated to the point where he gets just 1 simple task per second, the same as a normal person, but if he wants to get stuff done faster, all he needs to do is slow down a little.
That, however, is an oversimplified scenario because presumably, the faster he can run, the faster he can do tasks, so . . . (and this is where it gets fun).
Lets say his speed depends on how many steps or strides he takes per second. And lets go crazy and say, he's got the speed force, he can take as many steps as he wants per second. No limits.
An average man has a stride of 7 feet 9 inches over an 800 meter run. Source. Lets say the Flash is above average, so, estimate 3 meters (to make the math easier). And light travels at 300,000,000 meters per second (299,8... but lets just use 300 million). So, as we step up the orders of magnitude:
10,000 strides per second: 30 km/s (about the speed the Earth orbits the sun). At this speed, if you ignore air resistance, he would run right off the Earth. Gravity doesn't hold onto objects moving that fast on the surface of Earth. Time dilation is effectively negligible.
100,000 strides per second: 300 km/s (time dilation still negligible)
1,000,000 strides per second: 3,000 km/s (relativistic effects still minimal, a 5 hour run would slow his watch down by almost 1 second to his friends back at star labs.)
10,000,000 strides per second: 30,000 km/s. (dilation is now 1/2 of 1%, his speed is actually 29,850 km/s to someone watching him. His watch will lose 1 second every 3 minutes and 20 seconds.)
100,000,000 strides per second: 300,000 km/s. Here's where relativity takes over. By his watch, he's taking 100,000,000 steps per second, but his stride shortens, he can't cover as much ground with each step. The world has shortened too, so maybe he writes this off as a trick of the light (we're assuming he doesn't know his relativity very well). But when he's finished 1 second of running by his watch and he's run around the world 7 1/2 times, the friends timing him say he was only going .707 the speed of light. Their watches were moving 1.414 times faster than his. (square root of 1/2)
So, Flash, determined to break the speed of light, this time runs a billion strides per second. By his watch, he circles the Earth 75 times by the time his watch strikes 1 second but his stride (and the Earth) were very much shortened and to the people watching him it took him 10.05 seconds to cover that distance (10 light seconds or 75 trips around the Earth). (Square root of 1/1.01). I see a pattern, but I'd rather not work out the formula.
There's no limit to the number of steps the flash can take per second, from his point of view, but the faster he takes his strides, each 3 meters, to someone watching him, he never gets to the 100,000,000 strides per second he needs to reach the speed of light and from his point of view, as his steps approach infinity per second, the distance of each step gets smaller and smaller, so neither party observes him going the speed of light, no matter how fast he moves his legs and feet.
I that's the answer to your question, no matter how fast he moves, relativity still maintains the limit of the speed of light, but the more steps he can take per second, the faster he goes from the point of view of every observer, even with time dilation "slowing him down". If you think about it, that's the only way it makes any sense. Nobody can ever move so fast that they move slowly. That's silly. As long as arm speed (ability to do tasks) and running speed (strides per second) are tied together. The faster he can move, then, the faster he gets things done and the faster he appears to move, but time dilation and relativity lead to diminishing returns. Past a certain point he's just adding numbers to the right of the decimal point and not adding speed very much.
I'm, of-course, ignoring acceleration and assuming all observations are instantaneous.