Contradiction in special relativity?

[curiousBos]

Hey JesseM,

I am not a mathematician and i trust that you are correct. That is interesting and i didn't know that was possible. I don't really see how though. All the times are the same as the distances (1/2 a second for 1/2 a meter, 1/3 of a second for 1/3 of a meter...etc.) If you say the distances are infinite, why wouldn't the time be as well. How can you add up all the times, if they are infinite as well?

 

[JesseM]

Hey JesseM,

I am not a mathematician and i trust that you are correct. That is interesting and i didn't know that was possible. I don't really see how though. All the times are the same as the distances (1/2 a second for 1/2 a meter, 1/3 of a second for 1/3 of a meter...etc.) If you say the distances are infinite, why wouldn't the time be as well. How can you add up all the times, if they are infinite as well?

The sum of the distances is not infinite, it's just that you've broken up a finite distance (2 meters) into an infinite number of ever-decreasing chunks. The infinite series 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... always has a sum of 2*, regardless of whether it's a sum of ever-decreasing distances or a sum of ever-decreasing times.

*The sum of an infinite series is defined as the limit as the number of terms approaches infinity, meaning that if you add any finite number of terms in the series--a trillion terms, say--you'll always get an answer less than 2, although you can get arbitrarily close to 2 by adding more terms. See infinite series for more info.

 

[curiousBos]

Well yes but just becaues you can put 1 + 1/2 + 1/4 ... = 2, doesn't mean that you can actually get to 2. You said yourself "you'll always get an answer less than 2, although you can get arbitrarily close to 2 by adding more terms." But we don't want to get close, we want to reach 2. So according to calculus, you can only get close because 2 is defined as the limit, right?

 

[JesseM]

Well yes but just becaues you can put 1 + 1/2 + 1/4 ... = 2, doesn't mean that you can actually get to 2. You said yourself "you'll always get an answer less than 2, although you can get arbitrarily close to 2 by adding more terms." But we don't want to get close, we want to reach 2. So according to calculus, you can only get close because 2 is defined as the limit, right?

If we're actually calculating the ongoing sum on a calculator, we can't ever get to 2 by adding successive terms in sequence, but that's because it doesn't take us an ever-decreasing time to add each term. If you could add with ever-decreasing time--taking 1 second to add the first term, 1/2 seconds to add the second, 1/4 seconds to add the third, etc., then you would be able to add an infinite number of terms in 2 seconds (for any finite number N, no matter how incredibly large, it's easy to prove that in 2 seconds you could add more than that number of terms, so obviously if the number of terms you add in 2 seconds is larger than any finite number it must be infinite).

 

[Dale]

Hi curiousBos,

From the recent conversation it is pretty clear that your problem is not a problem specific to relativity, but a general misunderstanding of infinite, infinitesimal, and limits. These are difficult concepts to understand, but the difficulty does not imply that they are in any way illogical or contradictory.

 

[curiousBos]

But even if hypothetically you COULD add with ever decreasing time, you CAN'T ever get to 2 seconds. So isn't this back to where we started?

 

[curiousBos]

Hey DaleSpam,

Yes i think you're right. The very reason the question about SR bothered me was because of the fact that it places an unreachable boundary on somehting real in the universe, which is essentially similar to infinity. It's a limit, as we've stated. But i think the answer is that our fundamental description of time is flawed in that, we think it can always be cut down further and further.

 

[JesseM]

But even if hypothetically you COULD add with ever decreasing time, you CAN'T ever get to 2 seconds. So isn't this back to where we started?

I don't understand what you mean by "can't ever get to 2 seconds". Normally the words "can't ever get to" would mean you never reach something in time, but that doesn't seem to make any sense here, since naturally the time for you to reach 2 seconds will be...2 seconds. So if you're not talking about time, what are you talking about?

 

[curiousBos]

Yes but this is my whole argument in the first place. The universe we live in let's time flow... 2 seconds will take 2 seconds. But if you use math, you could argue that in order to get to 2 seconds you have to first get to 1 second and in order to get to one second you have to get to 1/2 a second and so on and so on. It's the same paradox as stated earlier. Thats why i think that the fundamental unit of time must be discreet, which would allow you to stop cutting it in half, and eventually, reach 2 seconds. The infinite cutting of time is what prevents you mathematically from ever reaching 2. It sets a limit. If time is discreet (if there's a fundamental smallest unit of time), that issue doesn't exist.

So when i say you can never reach 2 seconds, I mean mathematically.

 

[JesseM]

Yes but this is my whole argument in the first place. The universe we live in let's time flow... 2 seconds will take 2 seconds. But if you use math, you could argue that in order to get to 2 seconds you have to first get to 1 second and in order to get to one second you have to get to 1/2 a second and so on and so on. It's the same paradox as stated earlier.

Yes, you could argue that, but there'd be no problem as long as you accept that 1 + 1/2 + 1/4 + 1/8 + ... has a sum of 2, as is assumed in calculus. I still don't see what your actual argument against this would be.

 

[curiousBos]

Well i think you answered your own question. It's "assumed" in calculus, yet we can't actually ever do it, even hypothetically, because of what i just argued.

I don't want to go in circles here, but i do enjoy the friendly debate.

 

[MeJennifer]

The sum of the distances is not infinite, it's just that you've broken up a finite distance (2 meters) into an infinite number of ever-decreasing chunks. The infinite series 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... always has a sum of 2*, regardless of whether it's a sum of ever-decreasing distances or a sum of ever-decreasing times.

*The sum of an infinite series is defined as the limit as the number of terms approaches infinity, meaning that if you add any finite number of terms in the series--a trillion terms, say--you'll always get an answer less than 2, although you can get arbitrarily close to 2 by adding more terms. See infinite series for more info.

Note that this is not an issue with relativity. If the distance between A and B is 2 as measured by a light signal it is guaranteed that the actual distance traveled by a traveler from A to B is always less than 2, in fact the faster he goes from A to B the less distance he travels from A to B.

 

[JesseM]

Well i think you answered your own question. It's "assumed" in calculus, yet we can't actually ever do it, even hypothetically, because of what i just argued.

But you haven't argued anything! You've just said it's impossible without presenting an argument. I see no logical reasoning why the idea that you can divide a 2-second interval into an infinite sum of smaller intervals should mean you can't ever "reach" 2 seconds, or what that even means if you aren't talking about the time needed to reach it. You seem to be just asserting that somehow this would be impossible because you find it counterintuitive, but there's no actual argument as to why.

 

[JesseM]

Note that this is not an issue with relativity. If the distance between A and B is 2 as measured by a light signal

What does "2 as measured by a light signal" mean? Are you referring to the same weird idea about there being a "true" (frame-independent) distance between objects which is "defined by light" that you were talking about on this thread? If so, I note that as usual when I pressed you for the specifics of how you were defining things, you responded with vague generalities and then ducked out of the thread.

 

[curiousBos]

Here's my argument:

If me and you are in a race. I run 10 m/s and you run 1 m/s. But you get a 10 meter head start. So after 1 second, I'm at the 10 meter mark (where you started), but in that 1 second, you ran 1 meter (now your at the 11 meter mark). So after 1/10 of a second, I run 1 meter (now I'm also at the 11 meter mark). But in that 1/10 of a second, you ran 1/10 of a meter (so you're at 11 and 1/10 meter mark). I assume you see where this is going, as it is zeno's paradox.

There is nothing wrong with this logic mathematically, yet for some reason I, who'se running 10 times as fast, can't surpass you. The problem is with how we are giving ourselves a limit in time because we keep cutting it down. If you take a calculus perspective, you can easily say well after 10 seconds I ran 100 meters and you ran 10, so clearly I've passed you. But this would be missing the point, as I think you are.

If we now look at time as having a smallest unit, we can see how the paradox is resolved. Let's hypothetically say that the smallest unit of time is 1/10 of a second. Since we can't ever reach anything smaller than that, clearly you can see how I will now be able to pass you, as I would in the real world. The point is that if we assume that time can be continuously cut down to an infinitely small level, I truly wouldn't be able to surpass you. Yet when we use discreet units, fundamentally small units of time, the paradox goes away. This is my point.

 

[Fredrik]

Well i think you answered your own question. It's "assumed" in calculus, yet we can't actually ever do it, even hypothetically, because of what i just argued.

I think you have misunderstood both the argument and the counterargument.

The argument is that going from 0 to 1 in steps of 1/2, 1/4, 1/8,...(meters) will take an infinite time because of the infinite number of steps.

The counterargument is to point out that this conclusion is false by showing that the sum of the times isn't infinite. If we can show that 1/2+1/4+1/8+... (seconds) is less than or equal to 1, we have successfully proved the argument wrong. We can do that e.g. by proving that the partial sum with n terms is =(1-(1/2)ⁿ) and that's ≤1 for all n.

So the fact that all of the partial sums are less than the sum of the infinite series is the reason why the argument fails.

 

[Fredrik]

I see that you want to do this without involving the time of each step. It's possible that this is actually closer to the original version of these paradoxes than the version that I just disproved. But the claim that the faster runner won't catch up because of the infinite number of steps is simply wrong. It's trivial to prove that. Just write down the equations of the two runners' paths through spacetime, and you can easily verify that they are straight non-parallel lines in the x-t plane. That implies that they must meet somewhere. That version of the paradox is so trivial that we usually don't even mention it.

 

[curiousBos]

I don't know if you've read this whole thread, but I've mentioned that I'm an animator, not a physicist or a mathematician. I just have a passion for it and read a lot about it. Excuse me if I sound trivial, but i don't know of all the mathematical proofs and different variations on each paradox. I don't quite see why my logic is "simply wrong" because i don't know the equations to use for the runners' paths through spacetime.

"That version of the paradox is so trivial that we usually don't even mention it."

Thats interesting because I've had multiple physicists tell me that that specific paradox is indeed still a paradox and is not fully understood with our current mathematics and concepts of fundamental time.

 

[JesseM]

Here's my argument:

If me and you are in a race. I run 10 m/s and you run 1 m/s. But you get a 10 meter head start. So after 1 second, I'm at the 10 meter mark (where you started), but in that 1 second, you ran 1 meter (now your at the 11 meter mark). So after 1/10 of a second, I run 1 meter (now I'm also at the 11 meter mark). But in that 1/10 of a second, you ran 1/10 of a meter (so you're at 11 and 1/10 meter mark). I assume you see where this is going, as it is zeno's paradox.

Sure, and after 10/9 seconds you'll catch up with me (because after 10/9 seconds you've gone 100/9 meters, and I've gone 10/9 meters beyond the original 90/9 meters I started at, so we're both at 100/9 meters). The fact that you can subdivide this 10/9 second period into the infinite sum 1 + 1/10 + 1/100 + 1/1000 + ... still isn't an argument as to why there's any problem with you catching up with me, once again you're just restating the fact that you find it counterintuitive that we can pass through an infinite series of ever-decreasing time intervals in a finite time, as you did in the earlier post.

There is nothing wrong with this logic mathematically, yet for some reason I, who'se running 10 times as fast, can't surpass you.

Why can't you? Once again you haven't presented any argument as to why you can't pass through an infinite series of ever-decreasing time intervals in finite time. If you add any finite number of terms from the series 1 + 1/10 + 1/100 + 1/1000, you'll always get a time less than 10/9, so how could it possibly take you any longer than 10/9 seconds to pass through every finite number of intervals in this series?

The problem is with how we are giving ourselves a limit in time because we keep cutting it down.

I don't understand what you mean by this sentence. What is the "limit in time" in this example, is it the 10/9 seconds that is defined as the sum of the series in calculus? If so, what is the "it" that we keep cutting down, is it also the 10/9 second interval that we can see it should take you to catch up with me?

If we now look at time as having a smallest unit, we can see how the paradox is resolved.

You haven't given us any paradox. All you've done is assert that there is some problem with examples where we pass through an infinite series of shorter and shorter time intervals in a finite time, but you haven't given any rational explanation as to what that problem might be, it seems that it's just a matter of you finding it counterintuitive.

 

[Fredrik]

I don't quite see why my logic is "simply wrong" because i don't know the equations to use for the runners' paths through spacetime.

I assumed that you knew that their movements can be represented by straight non-parallel lines in the x-t plane, and that non-parallel lines in the same plane will always intersect somewhere. The details (what I'm about to explain) aren't really important.

The equations can be expressed as x=vt+a, where a is the position at t=0. We have v=10, a=0 for the fast runner and v=1,a=10 for the slow runner. So their equations are

x=10*t
x=t+10

That means that they will be at the same position at the time time t that satisfies 10*t=t+10. Solve for t. The result is t=10/9=1.1111... The position can then be calculated as 10*10/9 or 10/9+10. Either way the result is 11.1111...

Thats interesting because I've had multiple physicists tell me that that specific paradox is indeed still a paradox and is not fully understood with our current mathematics and concepts of fundamental time.

Maybe they just meant that time isn't fully understood (i.e. that even general relativity can't tell us all there is to know about time), and that we don't know what the resolution of Zeno's paradox will look like in "the next big theory". Its resolution in the framework of GR (or any other of the current theories) is very well understood though. I don't think those physicists will argue against that.

 

[curiousBos]

Hey thanks for all the info. and i hope I am not coming across as rude, but i honestly have been told what I'm telling you by teachers, and find it troubling that you can't even at least see how one could think this is a paradox.

This is in fact zeno's paradox. It's called a paradox for a reason (or at least it was, since you claim we've solved it.) So clearly I'm not the only one who has found this situation to be paradoxical.

You seem to have solved it the exact way i predicted you would. I had said one can easily say that after 10 seconds I've ran 100 meters and you've ran 10 meters, so clearly I have surpassed you. But that is because this logic, along with yours, starts the time that has passed AFTER 1 second. That is the key to the paradox. The example i just gave a sentence ago works because it starts at 10 seconds later. The example you gave works because it starts at 1.1 seconds.

The example i gave starts at 1 second and then goes through an infinite series of ever decreasing time intervals (1 second later, then 1/10 second after that, then 1/100 second after that, etc.). This is the where the "limit" I am talking about comes in. Mathematically, you can never go past 1.11111111111... seconds if you choose to keep moving in an ever-decreasing fraction of time.

I don't understand how you don't understand this point. You agreed with me before! :

Me: "Yes but this is my whole argument in the first place. The universe we live in let's time flow... 2 seconds will take 2 seconds. But if you use math, you could argue that in order to get to 2 seconds you have to first get to 1 second and in order to get to one second you have to get to 1/2 a second and so on and so on. It's the same paradox as stated earlier. Thats why i think that the fundamental unit of time must be discreet, which would allow you to stop cutting it in half, and eventually, reach 2 seconds. The infinite cutting of time is what prevents you mathematically from ever reaching 2. It sets a limit. If time is discreet (if there's a fundamental smallest unit of time), that issue doesn't exist.
So when i say you can never reach 2 seconds, I mean mathematically".

You: Yes, you could argue that, but there'd be no problem as long as you accept that 1 + 1/2 + 1/4 + 1/8 + ... has a sum of 2, as is assumed in calculus. I still don't see what your actual argument against this would be.

**The reason we don't agree is because you are ASSUMING that you can reach 2 seconds. I'm saying, in the real world, as we move through the flow of time, how do you reach the 2nd second if you can keep moving a smaller and smaller fraction of time forward (such as 1/10 of a second later, 1/100 of a second later, 1/1000 of a second). You are assuming you can reach the 2 seconds, but I'm asking HOW we actually do it in the real world. That is why I brought up the notion of discreet time and fundamental units.

 

[Dale]

Thats interesting because I've had multiple physicists tell me that that specific paradox is indeed still a paradox and is not fully understood with our current mathematics and concepts of fundamental time.

Either you misunderstood them or they are wrong. This is fully understood and completely resolved with freshman-level math.

It is certainly possible that time is discrete at some level, but it is also certainly not logically necessary that it be discrete. I.e. no paradoxes result if time is continuous.

 

[Doc Al]

I think this thread has gone on long enough.