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For reference, the setting is urban fantasy, with a bit of hard science thrown in. One character can manipulate time; in this setting that means he can speed up his own movement through time.

One of the things he's capable of is punching. Now, let's suppose he throws a punch which should take 1 second, but he speeds up the punch to a quarter of a second. The question is, is the force magnified fourfold, or (and this is how I think it works) since he's affecting time directly (and not just moving faster), the force is magnified by 16, because F = M*A, and A is really just m/(s^2)?

(Side note: A human punches with 1000 Newtons, so the difference here is between 4,000 (boxer's punch) or 16,000 (shatter your own fist).)

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    $\begingroup$ See this question (which might be a duplicate) and this answer and this answer. $\endgroup$ – JBH May 14 at 17:41
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    $\begingroup$ I feel like this question might be too broad as written. Time Acceleration can work a lot of different ways at the Worldbuilder's whim, so it could be EITHER 1000 Newtons OR 4000 Newtons, or lots of other values depending on how your time manipulation works. E.g. does it let you violate the laws of thermodynamics? If not, even if time is slowed down, it still takes the same amount of force to move the fist, so if you want to move it four times as fast, your protagonist has to put four times as much power in to do it. $\endgroup$ – Morris The Cat May 14 at 17:49
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Given that you're stating that this is a hard-science setting (not a hard-science question) then let's break this down a little to explain the interaction between force and time.

Your Premise
F=ma, therefore time is a consideration in the amount of force. If a person can throw a punch at normal speed BUT their localised frame of temporal reference is lower than their environment, to an outside observer the punch is faster, therefore delivers more force.

My Premise
This is just an exotic way of punching faster, delivering more force.

One interesting question I get when explaining relativity to others is why the astronaut who's travelling at close to the speed of light ages so much more slowly than his twin back on earth - if relativity is based on the observer, then the speed of both men is close to the speed of light relative to each other, right?

Right?

Well the problem with this thinking is that relativity is not about relative speed, it's about relative energy. The astronaut has a much higher kinetic energy than the earth, meaning his ageing is slower.

Thing is, it's the same with your person's fist. Your person's ability to slow down time is really just a fancy way of saying that he or she can speed up his or her own matabolism to a point where he or she is moving faster. Whether you slow down time or speed your movements up, you're still introducing the same amount of extra energy. That is to say, you're hero is going to have to increase their metabolism to expend the same amount of extra energy either way.

If we take relativity into account, speeding one's movements and slowing outside time are basically the same thing. In point of fact, moving faster introduces more energy, which by the astronaut example literally does slow down time. This is because space and time are basically the same thing expressed different ways, hence our 4 dimensional concept of spacetime.

In short, if you want it to be hard science, then being able to accelerate time is a confusing distraction for most people - your hero can simply accelerate his or her own movements, which amounts to the same thing. Either way, your hero has to be able to expend massive amounts of energy in a short time which also means he or she is going to get very hungry after doing this for a bit.

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  • $\begingroup$ So, to clarify your position, you are saying there is functionally no difference between moving faster and using magic to adjust the user's time state so he moves at the exact same speed for him, and moves 4 times as fast from his opponent's perspective? Because, fundamentally, force in Newtons is measured in kilograms*meters/seconds^2, so it seems like an adjustment to seconds should be exponential, as opposed to a moving faster, which (as meter/second) is only linear. $\endgroup$ – Halfthawed May 15 at 3:08
  • $\begingroup$ @Halfthawed, You're right that of the two, time is the bigger lever. But, it would also therefore take the larger amount of energy to pull. Adjusting time, if it gives an increase of the square of the adjustment, will also require the square of the increase in energy that adjusting the speed would. Which ever way you do it, the amount of energy you're adding should (in theory) result in the same net increase in force. The choice is really between adding a lot of speed, or making a small adjustment to time. $\endgroup$ – Tim B II May 15 at 3:29
  • $\begingroup$ "Well the problem with this thinking is that relativity is not about relative speed, it's about relative energy." This isn't true, at least within the confines of special relativity. If both the person on Earth and the person in the spaceship continue moving away from each other quickly, each will see the other as moving slower than themselves. The reason the twin on the spaceship ages less when you bring him back to Earth is because the symmetry of the problem is broken because the spaceship accelerates and isn't in an inertial frame. $\endgroup$ – el duderino May 15 at 5:48
  • $\begingroup$ ...If you could somehow strap enough rockets on the earth that the Earth accelerated away from the twin on the rocket and then back to it while the rocket remained in an inertial frame, you would find that instead the twin on Earth would have aged less. Presence/flux of energy and mass does affect the passage of time in general relativity, but that's a whole other, much more complicated story that isn't related to the twin paradox. $\endgroup$ – el duderino May 15 at 5:54
  • $\begingroup$ @elduderino but if you're accelerating, aren't you just adding more energy? Strapping rockets to the earth to make it accelerate gives is more energy, not the rocket which is now standing still, or at least not having energy added to it through acceleration. I understand what you're saying about an inertial frame but I thought it's the change of inertia which fundamentally adds or reduces energy that affects the temporal frame of reference. Is this wrong? $\endgroup$ – Tim B II May 15 at 6:06
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Without violating the conservation of energy, 1,000 Newtons has to be the output. Since distance and force are unchanged... this means that time dilation will need to affect the speedster's mass. So, a if your time moves at a higher speed then your relative mass becomes smaller and acceleration increases maintaining the same force.

Logically it would seem that mass would also have to stay the same since you're not gaining or losing any matter, but mass is a quantity that is solely dependent upon the inertia of an object; so, a better way of putting is that as you speed up time, your inertia decreases.

There is one advantage here though for performing a speedster punch. While you still only get 1000 Newtons, they will be applied over a shorter period of time. So instead of experiencing a force of 1000 Newtons spread over the course of 1 second you might take it all in .25 seconds. So, your opponent will not go flying back any more than from a normal hit, but you are more likely to create a force over time that is high enough to overcome material strengths and cause injuries.

Now, if you assume that your mass stays constant and you can in fact violate the conservation of mass and energy by speeding or slowing time, then force will be a 1-to-1 ratio not exponential gain. In (F = M*A) speeding up time x4 speeds up acceleration by x4 so your outcome would look like (4F = M*4A), but you will also divide your force over 1/4 of the time so you will knock your opponent back with 4x times the force, but cause the structural damage of something with 16x the force.

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  • $\begingroup$ I'm assuming that the law of conservation of energy is being broken. The way you set up the situation would keep everything working within the normal laws of physics in a way that's pretty clever. Unfortunately, I have no desire to do that, as it simply creates a character who uses magic to achieve what you can just do at the edge of human ability. $\endgroup$ – Halfthawed May 14 at 21:15
  • $\begingroup$ Not necessarily, with enough time dilation he could still dodge bullets and shatter steel with his bare hands, he just would not send a tank flying in the process. $\endgroup$ – Nosajimiki May 14 at 21:29
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The time it takes to launch the punch has no effect on the result.

If we model the punch with an object of mass M being launched at velocity V against a target, the impact force of that object will be given by the relationship $F \cdot \delta t = M \cdot \delta V$.

Note that $\delta t$ is the time it takes for the object to stop interacting with the target. There is nowhere to be seen the time it took to accelerate the object to V.

If you character can reduce by a factor C the above time, then the impact force will be multiplied by the same factor: that is, reducing by 2 the impact time doubles the impact force.

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    $\begingroup$ If he's reducing the time, he has to increase the power he's putting into moving his arm too though, or he's violating Newton's Third Law and creating energy out of nowhere. $\endgroup$ – Morris The Cat May 14 at 17:50
  • $\begingroup$ @MorrisTheCat in the punch-thrower's frame of reference, he's putting as much into the punch as he would without the time shift. You're talking about the receiver's frame of reference. Now that I think about it, there may be no difference in the force of the punch at all. 1,000 joules in, 1,000 joules out. Conservation of momentum. There should only be a difference if the speed-up is actually in the receiver's reference frame. $\endgroup$ – JBH May 14 at 17:53
  • $\begingroup$ @JBH Exactly, it depends on how the time dilation works. Yes, relativity is an issue, but accelerating time doesn't let you just create energy for free out of nowhere. The punch-thrower's arm still has the same amount of inertia whether time is accelerated or not. Think of using this on a rocket. You still have X mass vs Y thrust so the rocket is moving the same distance on the same amount of fuel either way. You can get to orbit in less relative time (from an outside observer), but you still use the same amount of fuel (e.g. energy.). $\endgroup$ – Morris The Cat May 14 at 18:06
  • $\begingroup$ @MorrisTheCat, that's my point. $\endgroup$ – JBH May 14 at 18:07
  • $\begingroup$ @L.Dutch Correct me if I'm wrong, but in the equation you gave, the velocity would be quadrupled as well. If we assume the punch to travel 1 meter, it's now traveling at a speed of 1m/.25s instead of 1m/1s. $\endgroup$ – Halfthawed May 14 at 19:20
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Inspired by @Morris_The_Cat comment to L.Dutch, I feel like you're the only one that can answer to the following question :

If your hero took a car and used his power on it while traveling, what'd happen :

  • Would the travel go perfectly normal, the hero experimenting nothing unusual, the car consuming the normal amount of carburant for the trip but, in the end, just get earlier to his destination that would have been possible ?

  • Or would the car burn his carburant faster that should be, the car shaking from the air friction ?

If I were you I'd go with the first solution, it's the more magical one but allows you to ignore a lot of side effects and inconsistencies. So your hero is going to give normal punches, people just won't be able to avoid them.

Else, at every turn you'd need to think about air friction, mass acceleration... lot of hassle. Just imagine your hero completely freezing time, his hearbeat only would make him vibrate faster than the speed of light (relatively) and he would explode on the spot through friction with the air. It wouldn't make a very good superpower...

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  • $\begingroup$ The car would get there faster, but experience more air friction. Without air friction the car would expend the exact same amount of energy. This is because to move the car needs a certain amount of energy, provided by the engines. From the car's perspective nothing is changed aside from the world moving seemingly slower. The engines suck in fuel and burn in at twice the speed, meaning twice the newtons per second is provided for thrust. This will accelerate the car to what the car experiences as 100km/h but with twice the wind against it. $\endgroup$ – Demigan May 15 at 14:13
  • $\begingroup$ essentially, if the car was driving with zero wind in the area against another car, then the sped up car would drive as if he had wind against him as high as his velocity. So at 100km/h it would seem like the car was driving with 100km/h winds compared to the unaccelerated car. $\endgroup$ – Demigan May 15 at 14:14
  • $\begingroup$ @Demigan "you're the only one that can answer this question" meant "what version do you prefer" and is directed at the OP. By changing the frame of his question it can help him understand/chose what the effects of the punch will be. $\endgroup$ – Echox May 15 at 14:31
  • $\begingroup$ My answer addresses this issue perfectly. Since the OP wants time-dilation to violate newtonian physics, the car moves faster with no difference in inertia. So the resistance of each m^3 would be the same as no dilation. Since the total resistance between point A and point B is the same, then you can conclude the car would feel the same resistance as though he was just driving faster. If time-dilation does not violate newtonian physics, then the decrease in inneria would make the air resistance more difficult to overcome, as you speed up the air would begin to feel like moving through water. $\endgroup$ – Nosajimiki May 15 at 16:24
  • $\begingroup$ Since everything inside the car is all in the same time-bubble, only outside forces will cause issues. So, the motor won't have to worry about functioning differently, but if you don't reduce inertia, then the vibrations of the roads will feel the same as just driving faster, you experience 25kph but your car interacts with the road as though it is moving 100kph. If you reduce inertia, then your car is not pushing back as hard meaning your car will only take the wair and tair of the speed you are experiencing, you just need to exert enough energy to overcome the extra air resistance. $\endgroup$ – Nosajimiki May 15 at 16:33
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The currently accepted answer is wrong, and what both @nosajimiki and @L.Dutch said are the correct answer.

The person punches with 1000 newtons. It doesn't matter if we slow him down or speed him up, the person will exert the same amount of force and feel the same amount of fatigue afterwards. There is no magical extra energy simply because you punch faster because there is no relativistic mass connected to the object.

Imagine this: You are in a 1 ton car going 100km/h. You have a set amount of force based on these parameters. If you hit the brakes and stop in 10 seconds you feel a certain amount of decelleration force. If you hit the brakes and stop in 5 seconds you feel more decelleration force. If you brake by slamming into a solid wall you experience the most decelleration force and both the car and you will suffer damage from it... despite that in each situation you had the same starting energy! The key here is not the Newtons, but Newtons per second experienced.

The person who experiences the punch will therefore not experience a punch with F=m*(time accelerationV^2), but he'll just experience F=mV^2 amount of force, but have half as much time to stop the blow.

In effect, any punch will be experienced with F(normal punch)*time acceleration. You move twice as fast, the opponent experiences twice the force. You move 4x as fast he experiences 4x the force etc.

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  • $\begingroup$ Based on @L.Dutch 's response, you're missing out a key factor. The punch itself will only have 4x the force, true, but the impact will occur in 4x less the amount of time, which also acts as a force multiplier. Does that make sense? $\endgroup$ – Halfthawed May 15 at 17:57
  • $\begingroup$ The punch will not have 4X the force. The person might be accelerated, but in his reference frame he only puts 1000N into the punch. His muscles generate and provide the same energy regardless of him being accelerated or not. If the victim experienced 4000N force there would be 3000N of force that the muscles of the person did not create and are magically added. So the only difference can be that the time in which the punch is slowed is different. You double your time, you halve the time your victim has to stop the punch and double the impact force he experiences. $\endgroup$ – Demigan May 15 at 18:17
  • $\begingroup$ For the sake of this discussion, please assume that the law of conservation of energy are being ignored by the magic system. The magic system doesn't work like Pym particles (comic variant), where the user is swapped in a different dimension which is smaller/larger to our own. The user is legitimately speeding himself us, just with using magic to affect time, rather than boosting his own metabolism / physical strength to do so. $\endgroup$ – Halfthawed May 15 at 18:21
  • $\begingroup$ That is exactly why the force remains at 1000N. If the metabolism or other energy was used it might be possible, but the person is simply using 1000N of force. Lets put it from the other perspective: the victim moves twice as slow (or whatever you use as acceleration) compared to the person, including inertia. So while you have twice the speed to affect the world, the world also needs more of that speed to be affected. This means there is only less time to absorb the blow, rather than more energy. $\endgroup$ – Demigan May 15 at 18:43

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