Contradiction in special relativity?

[curiousBos]

ok so according to special relativity, light moves at a fixed speed C in every frame of reference. Special relativity also makes clear that any two things moving at a constant velocity along side each other, will perceive each other to be motionless. In other words, in any specific frame of reference, there is absolutely no difference between constant velocity motion and no motion at all, according to that frame of reference. So my question is as follows:

If two photons are traveling along side each other, what do they perceive the other photon to be doing? One would intuitively think that the answer is that they would appear motionless, as special relativity states. Yet it also statest that light moves at C according to ANY reference frame. The reference frame of a photon should be as justified as every other reference frame. So what's going on here?

People have answered this question by stating that photons cannot "perceive" and the question is meaningless. But this misses the point. Surely there reference frame is as valid as any other, even tho they can't technically "perceive."

To take this one step further, let's assume, as I've been told before, that light cannot evade itself at speed C because it's already moving at speed C. So the answer is that the two photons would indeed "perceive" each other as motionless. Well...ok, but now that raises another question. As special relativity states, an object (lets call it Object A) is traveling at .9999...%C will still perceive light to evade it at speed C. Thats fine, but then let's say that Object A could reach speed C and become a photon. Now it perceives light to evade it at speed zero (motionless). So doesn't that mean that special relativity should state that light moves at a fixed speed C in every reference frame EXCEPT the small fraction between .9999...%C and C?

My apologies for the length of the question. Also, I've heard over and over to watch the terms i use, such as "perceive." But clearly you understand the overall point to my argument. I also understand that an everyday object, such as Object A, would never be able to reach speed C and become a photon because it would require an infinite amount of energy. But surely there must still exist a gradual succession of speeds between .999999999...%C and C. Please help me out! Thanks

 

[Doc Al]

Sorry, but photons do not have a reference frame in relativity. Treating them as if they do will lead to all sorts of nonsense.

 

[curiousBos]

hey doc thanks. this seems to me to be the only logical answer to my question. but it is really not a satisfying answer at all. I thought one of the fundamental postulates of special relativity was that ALL frames of reference are equally valid. The fact that we could say "yes all of them are valid EXCEPT the reference frame of a photon" seems to be an answer that is simply there to make the theory work. do u agree?

 

[Fredrik]

ok so according to special relativity, light moves at a fixed speed C in every frame of reference.

That should be "in every inertial frame".

People have answered this question by stating that photons cannot "perceive" and the question is meaningless. But this misses the point. Surely there reference frame is as valid as any other, even tho they can't technically "perceive."

There are coordinate systems in which the photon is stationary, but none of them is an inertial frame.

As special relativity states, an object (lets call it Object A) is traveling at .9999...%C will still perceive light to evade it at speed C. Thats fine,

Yes, it is.

...but then let's say that Object A could reach speed C and become a photon.

That's impossible. If it moved at a speed <c before, it had a positive mass. If it did, it can't ever move at the speed of light (which requires it to be massless).

I also understand that an everyday object, such as Object A, would never be able to reach speed C and become a photon because it would require an infinite amount of energy.

If you understand that, I don't see why you assumed it to be false in your argument above.

But surely there must still exist a gradual succession of speeds between .999999999...%C

The speed is ≥0 and <c, but there are no other restrictions on it.

You may find the velocity addition law useful, if you don't know it already. In 1+1 dimensions:

$$u\oplus v=\frac{u+v}{1+\frac{uv}{c^2}}$$

 

[Doc Al]

I thought one of the fundamental postulates of special relativity was that ALL frames of reference are equally valid. The fact that we could say "yes all of them are valid EXCEPT the reference frame of a photon" seems to be an answer that is simply there to make the theory work. do u agree?

No, I don't agree. One of the postulates of relativity is that light travels at speed c with respect to every inertial frame. That means there is no inertial frame in which a photon is at rest.

 

[Fredrik]

I thought one of the fundamental postulates of special relativity was that ALL frames of reference are equally valid.

They are. Even the non-inertial ones. (But the postulate I think you have in mind only mentions the inertial frames). However, the speed of light is only c in the inertial frames. We can make the speed of light whatever we want it to be by choosing the appropriate (non-inertial) coordinate system.

The fact that we could say "yes all of them are valid EXCEPT the reference frame of a photon" seems to be an answer that is simply there to make the theory work.

There is no such thing as a "reference frame of a photon". There's a natural way to associate an inertial frame with a massive object moving at constant velocity. You define the time axis to be the object's world line. You choose the scale on the time axis that agrees with the proper time along the world line. Then you choose the spatial axes so that the speed of light is 1 (or c). (This specifies the frame up to a rotation in space).

There's no way to do this to a photon. If you let the world line of a photon be the time axis of a coordinate system, you'd have to use something other than proper time to set the scale on that axis, since proper time is zero along a null line. There's also no way to choose the spatial axes so that the speed of light is 1.

 

[curiousBos]

thanks fredrik, i see your points.

I've asked the question many times and i understand why people respond the way they do. The answers you give are correct, according to the theory. But the reason i say things like "an object A becoming a photon" even though i know it can't, is because i am trying to think without the restrictions of the theory.

By no means am i putting down special relativity. i understand its precise predictions. But it always seems like the question can't be answered because of the restricitons of the theory. is it wrong to ask a question based on special relativity that doesn't require an answer that special relativity gives?

Why can't i say ..."Hypotheticaly, if object A could reach speed c, then doesn't there have to be a small fraction where light does not escape that object at speed c, during the period between 99.9999c and c?" ...maybe it is just meaningless because the only way to answer it is through relativity and that in itself makes the question lose validity.

I do strongly feel that there is a reason to speak of a photons reference frame, even though it is not an inertial one. This is why i am looking for an answer beyond SR, which is obviously one hard to come by.

 

[Dale]

this seems to me to be the only logical answer to my question. but it is really not a satisfying answer at all.

I am sorry that you consider a logical answer to be unsatisfying, but what can you possibly expect anyone else to be able to do about that?

the reason i say things like "an object A becoming a photon" even though i know it can't, is because i am trying to think without the restrictions of the theory.

When you deliberately go outside a theory then you cannot reasonably ask what the theory predicts. You would have to develop a whole new theory on your own.

It seems to me that you know the theory well enough to understand that it is logically consistent and experimentally valid, but you just don't "like" it. Somehow it feels aestetically unpleasing for no logical reason, and even though you know it intellectually it is strange enough to bother you anyway. If this is the case, I truly understand. I don't think there is anyone here who didn't grapple at some point with the wierdness of it all.

 

[Fredrik]

But the reason i say things like "an object A becoming a photon" even though i know it can't, is because i am trying to think without the restrictions of the theory.

If you make assumptions that contradict the theory, then your conclusions won't be about SR. They will be conclusions about a theory in which your assumptions are true. (Unless your argument is of the reductio ad absurdum type).

By no means am i putting down special relativity. i understand its precise predictions. But it always seems like the question can't be answered because of the restricitons of the theory.

In this case, it can't even be asked in the framework of SR.

is it wrong to ask a question based on special relativity that doesn't require an answer that special relativity gives?

If you ask a question based on SR and get an answer that contradicts SR, then there's a problem with the logic somewhere.

Why can't i say ..."Hypotheticaly, if object A could reach speed c, then doesn't there have to be a small fraction where light does not escape that object at speed c, during the period between 99.9999c and c?"

If it could reach c, then it's either massless or we're not using Minkowski space as a model of space and time. I don't understand the rest of your question. You'd probably have to specify what specific model of space and time I'm supposed to use before I can even attempt to answer it.

I do strongly feel that there is a reason to speak of a photons reference frame, even though it is not an inertial one.

You'd have to define what "a photon's reference frame" means before you can really speak of it. Finding a frame in which it's stationary is no problem. Pick any inertial frame and make the coordinate change t'=x-ct, x'=x+ct. (Note that this turns the world line of a photon moving to the right into the new time axis, and the world line of a photon moving to the left into the new spatial axis).

 

[curiousBos]

hey dalespam,

Well yes, this is pretty much the case for me. I'm not a physics student or a teacher. I've just read a whole bunch of books on it. I feel like i have a good grasp on it and understand it pretty well considering.

"When you deliberately go outside a theory then you cannot reasonably ask what the theory predicts. You would have to develop a whole new theory."

...this is sort of what I'm getting at. I'm not trying to develop a new theory, or ask anyone else to in order to answer my question. I understand the question becomes meaningless (according to relativity) for reasons we've mentioned. You say i "cannot reasonably ask what the theory predicts"...you're right, i can't for this question, but that doesn't mean that the question can't be pondered and thought about.

I just feel like even though (accroding to SR) nothing with any rest mass can ever reach light speed, this doesn't mean that we can't talk of something that does. It's more of a thought provoking question than a practical one, and i just find it unsettling that the theory can't answer it because it doesn't allow it.

 

[curiousBos]

"If you ask a question based on SR and get an answer that contradicts SR, then there's a problem with the logic somewhere."

...maybe there is a problem with the logic of SR. But like i said before, I'm not trying to hate on the theory, which is why I've read so much about it. but it seems to me like there IS something wrong there. but what do i know I am just an animator lol.

 

[Dale]

I'm not trying to develop a new theory, or ask anyone else to in order to answer my question.

Unfortunately, yes you are.

Anyway, I understand your discomfort. It is not rational, but you already knew that. You really should be no more upset by this than by the fact that division by zero is undefined, and once you see the connection it may be less uncomfortable.

 

[neopolitan]

One of the postulates of relativity is that light travels at speed c with respect to every inertial frame. That means there is no inertial frame in which a photon is at rest.

Does this not also imply that photons are not inertial?

I guess that opens up questions about what is meant by inertial, but let me try to cut off one discussion ... while inertia usually implies mass, I don't believe that "inertial frame" as used in SR implies mass.

According to http://en.wikipedia.org/wiki/Inertial_frame" :

(A)n inertial frame of reference is a frame of reference which belongs to a set of frames in which physical laws hold in the same and simplest form.

Inertia itself, again according to http://en.wikipedia.org/wiki/Inertia" :

... is the tendency of all objects to resist a change in motion

Photons certainly seem have that (or the splitting of white light into a spectrum would not happen). The inertia of photons is just related to the frequency (a measure of energy) rather than mass (another measure of energy). So, at least in one sense, photons are inertial.

Is there a sense in which they are not?

cheers,

neopolitan

 

[curiousBos]

thanks for the responses

 

[atyy]

Also, I've heard over and over to watch the terms i use, such as "perceive." But clearly you understand the overall point to my argument. I also understand that an everyday object, such as Object A, would never be able to reach speed C and become a photon because it would require an infinite amount of energy. But surely there must still exist a gradual succession of speeds between .999999999...%C and C. Please help me out! Thanks

There is a gradual succession of speeds between 0.99999999999c and c. A massive object can reach any speed between 0.99999999999c and c, but it cannot reach c itself. If a massive object is traveling at constant velocity 0.999999999999999999999c, it is just as good as stationary. So the speed of light is very special sort of boundary. It reminds me of Escher's "Angels and Devils" http://www.artprints.com/-ap/Angels-and-Devils-Posters_p78503_.htm . Between some point in the disc and the edge, there are an infinite number of angels and devils. According to Thurston, someone on the disc trying to get to the edge would never get there because the edge is "at infinity" http://books.google.com/books?id=9kkuP3lsEFQC&printsec=frontcover#PPA56,M1.

As for what light "perceives", my favourite crackpot says "a photon would not 'experience' any passage of time at all!" http://books.google.com/books?id=oI0grArWHUMC&printsec=frontcover#PPA255,M1.

 

[Fredrik]

"If you ask a question based on SR and get an answer that contradicts SR, then there's a problem with the logic somewhere."

...maybe there is a problem with the logic of SR.

That's not what I meant. I meant that a person who finds an internal contradiction in SR has made a mistake. It's very obvious that there are no internal inconsistencies in SR. The mathematical model it uses consists of the set $\mathbb R^4$ and some functions. There's no way that it could contain a contradiction that isn't already present in the axioms of set theory. If SR contains a contradiction, then all of mathematics would fall with it.

 

[curiousBos]

That's not what I meant. I meant that a person who finds an internal contradiction in SR has made a mistake. It's very obvious that there are no internal inconsistencies in SR. The mathematical model it uses consists of the set $\mathbb R^4$ and some functions. There's no way that it could contain a contradiction that isn't already present in the axioms of set theory. If SR contains a contradiction, then all of mathematics would fall with it.

I'm not saying that there's any mathematical inconsistencies in SR, but that doesn't make it perfect. Just because a theory is mathematically consistent and proven to high degrees of accuracy, doesn't mean it describes the full picture. Quite like Newtons laws weren't wrong at all, they just required a deeper understanding such as that of SR.

I think that to simply say something can reach every speed before C, but never in fact reach C itself, is sort of like sweeping the problem under the table. For me, its the same issue i have when physicists or mathematicians consider a problem meaningless or undefined when the answer turns out to be infinity. Maybe there's nothing wrong with the question, but there's something wrong with the answer.

By giving such answers to my question like "it's a strange feature of the universe" or "anything with a rest mass can NEVER reach C" is, in my opinion, missing the point. It's the same as if one were to ask Newton HOW gravity transmits its effects and him replying "it just does." I feel like those are the type of answers I am getting on this question and that's what is a little frustrating. If SR wants to claim that we can reach every single speed except C itself, that's fine for theory. But for reality, I think when we come across things like infinity or "a boundary that can never be reached (such as C)," I think it's much more likely that it is the restrictions of the math that lead to such vague answers, rather than the true nature of how the universe works.

I don't want to come across as attacking SR by any means and I'm certainly in no position to do so. As i said before, I'm no physicist. But just as the concept of infinity is hard to swallow because we don't have a complete understanding of it, I feel like the unreachable boundary of C is hard to swallow for the same reason. Just because the math in the theory says it can't be reached, doesn't mean that it truly can't. It means that up until this point, we have no other way of describing why something can't ever reach C other than saying that it's because that "something" would require an infinite amount of energy. But since the concept of infinity is so hard for mathematicians to cope with, we answer the question by saying that it simply CAN'T. This is the essence of the question I was asking.

 

[DaveC426913]

curiosBos. I understand the desire to set up thought experiments, such as 'what would a photon experience'?

But if you do so, you cannot pick and choose the implications of that what-if. Buy into it all the way.

Consider: whether it started off life moving at c because its a photon, or whether it started off at less than c and was (somehow, say, by becoming massless) accelerated to c, there is a consequence upon the object that you haven't factored in:

It ceases to experience time.

And that's why its frame of reference makes no sense.

Two photons traveling next to each other, are, in fact, not traveling at all. They are simply two static lines.

Reformulate your initial questions removing all references to time (and this must therefore include all references to movement and speed - which are merely distances per unit time) and you will see that the paradox falls away.

 

[Mentz114]

Just because the math in the theory says it can't be reached, doesn't mean that it truly can't.

Maybe SR tells us that we cannot observe any mass traveling at c or greater. This makes sense because the object in question would go behind a horizon, wherefrom no signal could reach us. For example a test particle traveling from rest at infinity(!) towards a Schwarzschild BH would not reach c ( wrt a distant observer) until the very moment it reaches the event horizon.

Also in the FLRW expanding universe model, there is a horizon at a certain distance in every direction, caused by bodies exceeding c wrt to the observer.

So nature appears to be censoring this from us. But what do we know ? Experiment and/or observation is the only actual truth we can rely on.

 

[curiousBos]

curiosBos.
It ceases to experience time.

And that's why its frame of reference makes no sense.

I am well aware that a photon does not experience the flow of time. But that is not the issue at hand. The only point I'm trying to make is that there must exist some speed at which light does not evade something at light speed. And that speed must be some extremely small fraction between 99.999...c and C itself. The only reason there is no answer is because currently we know of nothing that can surpass this boundary. It's either a photon or it's not. But like i have said multiple times, I'm asking a thought provoking question about reality, not about practicality. Just because our math gives us a boundary to get extremely close to C but never quite reach it, does not mean its the true nature of how the universe works.

 

[atyy]

The only point I'm trying to make is that there must exist some speed at which light does not evade something at light speed. And that speed must be some extremely small fraction between 99.999...c and C itself.

it has nothing to do velocity. constant velocity 0.99c is as good as zero. It has to do with change in velocity or acceleration. look up the horizon of the relativistic rocket or the Rindler horizon.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

 

[Fredrik]

I feel like those are the type of answers I am getting on this question and that's what is a little frustrating.

Then please ask your question in a way that makes sense.

The only point I'm trying to make is that there must exist some speed at which light does not evade something at light speed.

This is a good example of something that doesn't make sense. It actually fails to make sense in at least two different ways. It refers to a speed that's clearly <c and at the same time =c. It also suggests that there's a logical way to arrive at this conclusion even though there isn't.

You should probably talk about curves and their tangents instead of particles moving at different speeds, and if you're using something other than Minkowski space as a model of space and time, then you should explain what and why.

And that speed must be some extremely small fraction between 99.999...c and C itself.

It sounds like you just can't accept mathematical limits. Do you also feel that Zeno proved that motion is impossible?

 

[DaveC426913]

...there must exist some speed at which light does not evade something at light speed.

Why?

If I were in a rocketship moving at .9999999999999999999999999999c, I would still see light leaving my headlights at c (as would external observers) because my FoR time would be passing at a glacial 1 / .9999999999999999999999999999 compared to external observers.


Externally, they'd see light take 1 second to travel 186,000 miles My ship following closely behind it has traveled 185,999 miles, so they see my ship only one mile behind the beam of light.


But internally, a mere 1/186,000th of a second has passed on my ship, so it would make sense to me to see the light having traveled only 1 mile.

 

[curiousBos]

curiosBos.

It sounds like you just can't accept mathematical limits. Do you also feel that Zeno proved that motion is impossible?

He didn't PROVE that motion is impossible and he never will (because it IS possible). He simply proved that the language in which we describe motion is flawed.

 

[curiousBos]

Why?

If I were in a rocketship moving at .9999999999999999999999999999c, I would still see light leaving my headlights at c (as would external observers) because my FoR time would be passing at a glacial 1 / .9999999999999999999999999999 compared to external observers.


Externally, they'd see light take 1 second to travel 186,000 miles My ship following closely behind it has traveled 185,999 miles, so they see my ship only one mile behind the beam of light.


But internally, a mere 1/186,000th of a second has passed on my ship, so it would make sense to me to see the light having traveled only 1 mile.

I agree with your logic, but that's not what I'm questioning. Let's hypothetically say that you were able to reach C (even though I'm aware that in SR you cannot). Since now you are moving at C, you (now being a photon), would see another photon as motionless because your are now moving at the same speed. So now that light is not evading you at all (since you are actually light), at what point does it go from evading you at speed C, to evading you at speed zero? You can't just snap from light evading you at C to suddenly not evading you at all (since your now moving at C). There must be some small gradual change where light evades you at less than C, and then eventually zero when you become light.

Again this is a hypothetical thought provoking question with the assumption that there isn't an unreachable boundary of C.

 

[DrGreg]

But since the concept of infinity is so hard for mathematicians to cope with, ...

As a former mathematician with a PhD (the "Dr" in "DrGreg" isn't an affectation) I strongly reject this.

In my own (biased) opinion, mathematicians are better equipped than anyone else to cope with infinity. They have quite rigorous definitions of how to interpret any sentence that has the word "infinite" or "infinity" in it.

When we say "it takes an infinite amount of energy to accelerate a massive particle to the speed of light" that means "no matter how much energy you supply to a massive particle, it isn't enough to reach the speed of light". No more and no less. There is nothing vague about this.

I think that to simply say something can reach every speed before C, but never in fact reach C itself, is sort of like sweeping the problem under the table.

Consider an analogy. Suppose I am manufacturing a collection of paperweights. The first is 1 kg, the second 1/2 kg, the third 1/4 kg, the fourth 1/8 kg, etc, the nth $1/2^{n-1}$ kg. How many of these paperweights do I need to make to reach a total mass at least 2 kg? If you choose any target mass slightly less than 2 kg (e.g. 1.9999999999999999 kg), I will eventually be able to make enough paperweights to exceed that target. But no matter how hard I try, I'll never actually reach or exceed 2 kg. By stating that, am I "sweeping a problem under the table"?

 

[curiousBos]

Consider an analogy. Suppose I am manufacturing a collection of paperweights. The first is 1 kg, the second 1/2 kg, the third 1/4 kg, the fourth 1/8 kg, etc, the nth $1/2^{n-1}$ kg. How many of these paperweights do I need to make to reach a total mass at least 2 kg? If you choose any target mass slightly less than 2 kg (e.g. 1.9999999999999999 kg), I will eventually be able to make enough paperweights to exceed that target. But no matter how hard I try, I'll never actually reach or exceed 2 kg. By stating that, am I "sweeping a problem under the table"?

i apologize if you took offense to my comment about coping with infinity. As for your analogy, I am aware of Zeno's paradox and this is just a different take on one of them. To answer your question, no, you're not "sweeping them under the table." But you are not answering the question either. The problem with infinity is that it is the result of math, the very language we use to interpret the universe. Yet, the universe doesn't act like this.

Take for instance the famous paradox where you can never reach a wall if you keep going half the distance to it. If you take this further, you'd never actually be able to move, because in order to move any distance, you'd have to first take half of that distance and half of that and so on... so you could never even take the first step. Yet, our universe does not work like this. We can indeed move. The problem arises from the math and the fact that we keep cutting it in half. But maybe the universe doesn't keep cutting time and distance in half. If time comes in discreet units (a chunky universe), then the paradoxes are solved because we can eventually stop the cutting in half once we reach the smallest increment of time. Pete Lynds wrote an interesting paper on this.

Having a paradox of such that you describe doesn't tell me there's a limit in the real universe, it tells me there's a limit in the mathematics. Wouldn't you, being a mathematician agree?

 

[JesseM]

Take for instance the famous paradox where you can never reach a wall if you keep going half the distance to it. If you take this further, you'd never actually be able to move, because in order to move any distance, you'd have to first take half of that distance and half of that and so on... so you could never even take the first step. Yet, our universe does not work like this. We can indeed move. The problem arises from the math and the fact that we keep cutting it in half. But maybe the universe doesn't keep cutting time and distance in half. If time comes in discreet units (a chunky universe), then the paradoxes are solved because we can eventually stop the cutting in half once we reach the smallest increment of time. Pete Lynds wrote an interesting paper on this.

But calculus already solves Zeno's paradox without the need for discrete time (although it's still possible time is discrete, nobody knows). If the wall is 2 meters away, and it takes 1 second to travel the first meter, then traveling at the same speed it'll take an additional 1/2 a second to travel the next 1/2 a meter, and an additional 1/4 of a second to travel the next 1/4 of a meter, etc. So, even though we can break up the trip into an infinite number of steps this way, the time for each successive step is decreasing, and we know from calculus that an infinite series can have a finite sum, as in this case where if you add all the times 1 + 1/2 + 1/4 + ... you get the finite sum of 2 seconds, showing that Zeno was wrong that just because there are an infinite number of steps that automatically means it'll take an infinite time to make the journey.

 

[atyy]

Yes, if we observed something actually accelerating and reaching the speed of light, then special relativity would be wrong. In fact, we already know special relativity is wrong (but not about the speed of light being special), that's why we have general relativity, which we also know breaks down at some point. The question of what really is is settled by observation, not by what must be. Mathematicians invent many coherent structures, physicists pick those that are useful for succinctly describing what we observe. The rest that are not useful, we just think about for fun.

 

[DaveC426913]

I agree with your logic, but that's not what I'm questioning. Let's hypothetically say that you were able to reach C (even though I'm aware that in SR you cannot). Since now you are moving at C, you (now being a photon), would see another photon as motionless because your are now moving at the same speed. So now that light is not evading you at all (since you are actually light), at what point does it go from evading you at speed C, to evading you at speed zero?

You've used several words/concepts that make no sense. 'Seeing' requires passage of time. 'Motionlessness' requires passage of time.

You are trying to describe phenomena that have no time component to them as if they happen over time.

Again, a photon does not move from here to there, it only seems to because we experience one of the spacetime dimensions in a linear fashion. A photon is simply a straight and static line between two points in spacetime.



But that's not what you're looking for, I know. OK.

Let's look at this object that is moving at .99999999999999999999999999999999999999999999999999999999999999999999999999c.

As it accelerates, its relativistic mass is approaching the mass of the universe. It is using energies that are on the order of all the energy in the universe to cause it to accelerate further, and it is experiencing the history of the universe aging and dying in a fraction of a second.

Does it still sound like if you pushed it a little harder it would turn into a photon?

 

[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.