Proper time of a 'half-moving object'

[durant]

The part that I've bolded above makes no sense as written, so I'm reasonably sure that you've misunderstood something. Point to a specific example of such an article and we may be able to tell you how you've misunderstood it.

I was quick on my keyboard, so I wrote this nonsense. What I actually meant was the difference between proper and coordinate time. For instance, we all know that for moving observers with respect to Earth clocks on Earth tick slower, but inside their reference frame they measure the proper time. So only the coordinate time slows down/speeds up, right?

Here's a quote from an article:
"We sometimes speak of time dilation by saying time itself is “slower,” but time isn’t going slower in any absolute sense, only relative to some other frame of reference. Does time have a rate? Well, time in a reference frame has no rate in that frame, but time in a reference frame can have a rate as measured in a different frame, such as in a frame moving relative to the first frame."

 

[PeterDonis]

Shouldn't they accumulate an equal amount of proper time, but different values of coordinate time?

Proper time between which pair of events? That's the key element you appear to be leaving out. Proper time is not well-defined unless you specify which pair of events it's between. More precisely, if you hvae an extended object, with parts that may be in relative motion, each part has its own worldline (one of the family of curves making up the world tube of the object as a whole), and between any two events on one particular part's worldline, there is an elapsed proper time.

Now, consider two parts of your body (say your head and your left foot) which are in relative motion. The question "do they accumulate an equal amount of proper time?" is meaningless as it stands; in order to make it meaningful, you have to specify a pair of events on each part's worldline, and the two pairs of events have to "match up" in some way you're interested in. Otherwise there's no way to make a comparison.

For example, suppose that you start out with your entire body motionless, so your head and your left foot are at rest relative to each other. And suppose that you are six feet tall, so it takes light six nanoseconds to travel from your head to your foot. (To be precise, suppose you're six feet tall as measured in the rest frame of your head.) At some event A, your head receives a light signal indicating that your foot has started to move relative to your head. Then, at some later event B, your head receives another light signal indicating that your foot has stopped moving relative to your head. (Suppose the motion is such that the distance from your head to your foot doesn't change, as measured in your head's rest frame.)

Now, we have a way to pick out pairs of events on the worldlines of your head and your foot. For your foot, it's easy: we pick the two events, A' and B', at which the light signals were emitted that your head receives at events A and B. For your head, we pick the two events A'' and B'', which are each six nanoseconds earlier, by your head's clock, than events A and B. Because light takes six nanoseconds to travel from your foot to your head, in your head's rest frame, events A'' and B'' will take place at the same time (coordinate time) as events A' and B'. So if we compare your head's proper time between A'' and B'', and your foot's proper time between A' and B', we will be making a meaningful comparison. And we will find that your foot has less elapsed proper time between A' and B', than your head does between A'' and B''.

 

[PeterDonis]

we all know that for moving observers with respect to Earth clocks on Earth tick slower

No, that's not what we know. What we know is that the clocks *of moving observers* appear, to observers at rest, to tick slower than clocks of observers at rest. For example, in the scenario I just posted, your foot's clock appears, to your head, to tick slower than your head's clock.

 

[durant]

No, that's not what we know. What we know is that the clocks *of moving observers* appear, to observers at rest, to tick slower than clocks of observers at rest. For example, in the scenario I just posted, your foot's clock appears, to your head, to tick slower than your head's clock.

Sorry, but I simply don't understand this. What appearence? And what do you mean by less accumulation of proper time. If time 'flows' locally at the same rate, and simply varies in different inertial frames with respect to that object, then comparing the two worldines we may conclude that at some point they accumulated the same amount of proper time.
This is perhaps the most confusing thing that I've red on this thread. Isn't the proper time of both worldines invariant?

 

[Nugatory]

If time 'flows' locally at the same rate, and simply varies in different inertial frames with respect to that object, then comparing the two worldines we may conclude that at some point they accumulated the same amount of proper time.
This is perhaps the most confusing thing that I've red on this thread. Isn't the proper time of both worldines invariant?

Proper time is the time elapsed on a particular path through spacetime between two events.

Proper time is invariant, meaning that all observers, regardless of coordinate system and state of motion, will agree about the proper time elapsed on any given path between two events whether they are traveling that path or not.

However, proper time may be different on different paths even if the paths connect the same events. This is the essence of the famous "twin paradox" in which the two events are "Twin A says 'goodbye' to twin B, gets into a spaceship and flies off" and "The spaceship returns to earth, Twin A steps out and says 'hello again' to twin B". More proper time will have passed on B's path through spacetime than A's, so A will be aged less than B at their reunion. None of this has anything to do with reference frames or coordinate time.

 

[PeterDonis]

Sorry, but I simply don't understand this. What appearence?

I agree that "appears" is not a very good word to describe time dilation, but unfortunately we don't have a better one. The best way to describe it is with math and/or spacetime diagrams, but you have said you're not very familiar with them.

Let me try rephrasing what I said. Consider the scenario I described, with your foot moving relative to your head. Your head receives light signals from your foot; since your foot is at a constant distance from your head (as measured in your head's rest frame), the arrival time of those light signals at your head can be adjusted for the light-travel time (six nanoseconds) to obtain the times at which the signals were emitted from your foot. Suppose the signals are emitted, according to a clock moving with your foot, once per nanosecond. Then the time between the signals, according to a clock moving with your head, will be *greater* than one nanosecond. This is what is normally referred to as "time dilation".

And what do you mean by less accumulation of proper time.

The elapsed proper time for your foot, between events A' and B', is less than the elapsed proper time for your head, between events A'' and B''.

If time 'flows' locally at the same rate, and simply varies in different inertial frames with respect to that object, then comparing the two worldines we may conclude that at some point they accumulated the same amount of proper time.

Between which pairs of events? And why would you choose those particular pairs of events? Sure, I can find some pair of events on any worldline I like that have a particular amount of proper time elapsed between them, but what does that prove?

For example, in the scenario I described, as I just noted, the proper time for your foot, between events A' and B', is less than the proper time for your head, between events A'' and B''. But I can find *some* event, C', on your foot's worldline, which will be to the future of B', such that the proper time for your foot between events A' and C' is the same as the proper time for your head between events A" and B''. But what does that prove? Why should anyone care? (Or, I could find some event C'' on your head's worldline, which will be to the past of B'', such that the proper time for your head between events A'' and C'' is the same as the proper time for your foot between events A' and B'. Again, what does that prove?)

This is perhaps the most confusing thing that I've red on this thread. Isn't the proper time of both worldines invariant?

Once again, you are missing the key point, which I'm now going to emphasize: proper time is only well-defined between a specific pair of events on a specific worldline. In so far as proper time is invariant, it is only invariant once it's been defined that way. In other words, if you specify a worldline, and a pair of events on that worldline, then the proper time along that worldline between those two events is invariant: all observers will agree on it, regardless of their state of motion. But that does not mean that the proper time will be the same along a different worldline, or between a different pair of events.

I suggest that, rather than thinking about proper time in general terms, you force yourself to specify, every time you use the term "proper time", which worldline, and which pair of events on that worldline, you are using to define it. For example, in the scenario I specified, as I said above, the proper time along your foot's worldline, between events A' and B', is less than the proper time along your head's worldline, between events A" and B''. Both of these proper times are invariants--all observers agree on them. But they are not the same, because the worldlines and the pairs of events are different.

 

[Dale]

Durant, if you want to really understand relativity you should start thinking geometrically. Relativity is nothing more than geometry with a different formula for "distance". Geometrically, point particles are lines in spacetime, and points in spacetime are events.

The Euclidean distance between two points is $ds^2=dx^2+dy^2+dz^2$ and in relativity the interval between two events is $ds^2=-c^2 dt^2+dx^2+dy^2+dz^2$. Everything else stems from that.

The slope of a worldline is its speed and the length of a worldline is its proper time. When you rotate (boost) a line (worldline) you change its slope (speed) but not its length (proper time). Every effect in relativity has a geometric analogy.

 

[ghwellsjr]

Please answer my questions. Then I'll answer yours.

I've done it...

Good. Can you please post your answers?

 

[durant]

Good. Can you please post your answers?

I wrote it on the word document which I didn't save, but I got the 'almost 1' result, or to say 0.99 and some more numbers.

 

[ghwellsjr]

I wrote it on the word document which I didn't save, but I got the 'almost 1' result, or to say 0.99 and some more numbers.

Please do it again and post the exact number that you get out of your calculator along with the answers you selected for the two multiple-choice questions.

 

[durant]

Please do it again and post the exact number that you get out of your calculator along with the answers you selected for the two multiple-choice questions.

Is it necessary? I still don't understand it. You're behaving like there are no objects in the outside world just numbers and abstract coordinate systems :/

 

[Dale]

You're behaving like there are no objects in the outside world just numbers and abstract coordinate systems :/

Nonsense. If he used only English to describe objects in the outside world would you say "You're behaving like there are no objects in the outside world just words and pages in a book"? Your complaint is even sillier than that.

Math is the best means that we have of describing the behavior of objects in the outside world; we are interested in math precisely because it is far better at describing the outside world than English. There is a good reason why physics teachers assign homework that involves doing math.

 

[Fredrik]

You're behaving like there are no objects in the outside world just numbers and abstract coordinate systems :/

This is a fairly common complaint, but it's based on a common misconception. The only good answers to questions about the real world are those given by theories, and theories are defined using mathematics, so we don't have a choice. We have to use mathematics.

 

[Nugatory]

Is it necessary? I still don't understand it. You're behaving like there are no objects in the outside world just numbers and abstract coordinate systems :/

It's probably NOT necessary or useful until you understand why performing these calculations will lead to the answers that you're looking for.

A key point here is that the behavior of an object is correctly described by the behavior at each individual point. Your foot is traveling a different path through spacetime than your head, and therefore a bacterium on your toes and another one on your scalp will observe very small levels of time dilation and length contraction between them. It really doesn't matter whether your head and your feet are connected by the rest of your body or not.

Two notes:
1) Although each point on an object follows its own path through spacetime, these paths cannot diverge too much or the object will be torn apart. When you are walking and your foot pushes away from the surface of the earth, your muscles and skeleton transmit the forces to the rest of your body so the worldlines of each piece of your body stay reasonably close to each other. But if the forces involved were much greater, the bones and muscles would fail and something unpleasantly medieval would happen to you.

2) You will see textbook after textbook and thought experiment after thought experiment citing examples of objects (Einstein's trains, spaceships, cars on a road, your cat, elevators, ...) as if it is a single indivisible whole, all subject to the exact same clock speeds and simultaneity. Always, without exception, these examples are making a simplifying assumption that the all parts of the body are in uniform motion, not the "partial motion" that kicked off this thread. Loosely speaking, this comes down to stipulating that the world lines of each point are parallel (that's "loosely speaking"!). In any partial motion situation, you have to analyze the movement of each part separately... Same math, but more of it. (And that's roughly what ghwellsjr was telling you you have to do).

 

[ghwellsjr]

Is it necessary? I still don't understand it. You're behaving like there are no objects in the outside world just numbers and abstract coordinate systems :/

It isn't necessary if you don't want to learn anything about Special Relativity but if that's the case then you shouldn't be posting on this forum because that's its stated purpose.

You asked questions about a problem in your OP but you didn't provide any specifics:

As I've red, we can measure the proper time of an object with a clock that is at rest with respect to the object. So, how would we measure the proper time of an object that is partially moving and partiall at rest. For instance if I'm moving my head and the rest of my body is at rest, how would this situation be measured? Or how would the rest frame of me in that 'situation' be defined?

In subsequent answers, it was pointed out to you that each part that is moving (the head) with respect to other parts (the body) has its own Proper Time and that there is not one Proper Time that applies to all parts. In your example, you want to consider the body to be at rest. That means that we can use the rest frame of the body as our coordinate system. Since the Proper Time of any object/clock is determined by its speed in our chosen coordinate system and since the body's speed is zero, we can treat the Proper Time of the body to be the same as the Coordinate Time of the system and then all we have to do is decide on the speed of the head with respect to the body to determine how much slower its Proper Time is compared to the body's Proper Time. Finally, we have to decide how long we want this to go on for so that we can calculate a total aging difference between the head and the body.

I'm doing all this because these are the questions you asked about. I'm taking your request seriously and I want to help you learn how to arrive at the answers to your questions. I don't think it is fair for you to ask these questions and then after I (and others) invest so much time in providing answers for you to give up. I realize that it still may be confusing to you and that is why I tried to make it as easy as possible for you when I provided specifics in post #46:

In the case of a human being moving his head but not his body, there is no way to actually measure the difference in Proper Time between them no matter how much he wags his head around. But you could calculate the difference if you define the exact motion you want to consider and you have a calculator with enough precision. Perhaps it would be useful if you would provide these details for a scenario you find interesting. Let's assume the human's body is at rest in the negative portion of a coordinate system along the z-axis (all the parts of his body have negative coordinates in the z-axis). Then let's say that his head is one foot high and he nods his head back and forth along the x-axis a total of one foot (plus and minus six inches). And let's say that he stretches his neck as he does this so that the top of his head only has motion along the x-axis (the y- and z-axis parameters are constant). And let's say that the speed of the top of his head is constant with instant reversal of the motion. Now describe how many times per second you want him to complete each cycle of this motion and for how long you want this to go on for and see if you can calculate the difference in the aging of his head relative to his body, in the rest frame of his body. It's really a very simple problem. Can you do it?

Then, to make it even easier, I provided step by step directions for you to follow in post #51:

... You know now to operate a computer. I'm sure on your computer is a calculator that includes a square root function. For a simple problem, you don't have to do an integral. But first you have to define your problem. I defined most of it for you. I just left it up to you to provide two numbers. I'll make it real easy, multiple choice:

1) How many times per second do you want him to complete each cycle of moving his head back and forth?

a) One cycle per second
b) Two cycles per second
c) Five cylces per second
d) Ten cycles per second

2) How long do you want this to go on for?

A) One minute
B) One hour
C) One day
D) One month
E) One year
F) One decade
G) One century
H) One millennium

Now here's what you need to do:

First you need to calculate the speed of the tip of his head. You know that it moves a total of two feet per cycle. Based on your answer to the first question, you need to divide two feet by the number of seconds per cycle but since the answer is given in cycles per second, you need to multiply two feet per cycle by the number cycles per seconds to get the speed in feet per second. But since we are using units of speed in terms of feet per nanoseconds, you need to divide that answer by 1 billion (1000000000). This will be the speed of the tip of the head in terms of beta, β, the speed as a fraction of the speed of light.

Now you have to calculate the reciprocal of gamma, 1/γ, according to the formula:

1/γ = √(1-β²)

If you have Windows on your computer and you are using the provided calculator, make sure it is in the Scientific mode by selecting it under the View menu.

So take whatever answer you got for beta and square it by hitting the [x^2] button. Subtract 1 from it [-],[1],[=]and change the sign of the answer by hitting the [+/-] button. Now take the square root of the answer by checking the [√] Inv box and hitting the [x^2] button. You should have a number that is slightly less than 1 (a decimal point with a bunch of nines after it and then maybe some more numbers).

Now multiply this result by what ever answer you provided for question 2. Since they are all 1, you won't have to do anything except understand that Proper Time of the man's head will be slightly less than that of the rest of his body by that factor.

Tell me your answers to the two questions and the result of the calculation. I know you can do it.

 

[durant]

It isn't necessary if you don't want to learn anything about Special Relativity but if that's the case then you shouldn't be posting on this forum because that's its stated purpose.

Okay ghwellsjr, I appreciate your help, but it's extremely hard for me to understand how come, for instance, the human body doesn't function as an unity. Or to say, that it has stages. In my mind, it seems like you and the other members 'decomposed' everything that I counted as entities with proper times into smaller bits that count as holders of timelike related events. I appreciate your help and I hope you will continue to show me patience as I'm very interested in relativity and I'm hopeful I'll have the same level of knowledge as you guys in some time.

Maybe the concept of the space-time interval would be of some help here. Can you please describe how we describe the interval and measure it (if you could please do it with a clock example, like the description how to we measure proper time)? How could we measure the space time interval between objects which have different state of motion, for instance one is at rest with respect to the Earth and another is moving with respect to it?

 

[PeterDonis]

it's extremely hard for me to understand how come, for instance, the human body doesn't function as an unity.

Because it takes a finite time for the different parts of any extended object, like your body, to interact. No extended object can possibly be a unity, because its parts cannot instantaneously respond to each other. For many practical purposes, the time it takes for the parts of an object like your body to interact can be ignored, but that doesn't mean the interactions don't take time; it just means the time is short enough to be ignored for that particular purpose.

For example, it takes light a few nanoseconds to travel the length of your body. Interactions between the individual atoms of your body are not much slower than that, since they are basically electromagnetic interactions. But nerve impulses from your foot to your head take tens of milliseconds to travel (because your nerves are very slow transmitters of electrical impulses), and it takes hundreds of milliseconds for your brain to consciously evaluate the signals it receives. So as far as your brain is concerned, your body is a unity, because the interactions between its parts are so fast compared to your brain's processing time. (And any relativistic effects of those interactions, like time dilation, are much smaller still.)

This is why you commonly see people talk as if macroscopic objects, like human bodies, or rocks, etc., are unified single objects--the error involved in doing so is small enough to be ignored for everyday purposes. But you are asking here about fundamental theory, and as far as fundamental theory is concerned, all interactions take a finite time, and you can't ignore that, and that means you can't treat an extended object as a unity, as far as fundamental theory is concerned.

 

[durant]

Because it takes a finite time for the different parts of any extended object, like your body, to interact. No extended object can possibly be a unity, because its parts cannot instantaneously respond to each other. For many practical purposes, the time it takes for the parts of an object like your body to interact can be ignored, but that doesn't mean the interactions don't take time; it just means the time is short enough to be ignored for that particular purpose.

For example, it takes light a few nanoseconds to travel the length of your body. Interactions between the individual atoms of your body are not much slower than that, since they are basically electromagnetic interactions. But nerve impulses from your foot to your head take tens of milliseconds to travel (because your nerves are very slow transmitters of electrical impulses), and it takes hundreds of milliseconds for your brain to consciously evaluate the signals it receives. So as far as your brain is concerned, your body is a unity, because the interactions between its parts are so fast compared to your brain's processing time. (And any relativistic effects of those interactions, like time dilation, are much smaller still.)

This is why you commonly see people talk as if macroscopic objects, like human bodies, or rocks, etc., are unified single objects--the error involved in doing so is small enough to be ignored for everyday purposes. But you are asking here about fundamental theory, and as far as fundamental theory is concerned, all interactions take a finite time, and you can't ignore that, and that means you can't treat an extended object as a unity, as far as fundamental theory is concerned.

You're again confusing me. I know that the parts need time to respond to each other (in a causal manner), but you're debunking the whole concept of an object here. After all, events that don't ineract causally may be regardered as simultaneous from some frames. And simultaneous implies some kind of unity. How else would we speak of proper time if not as an local aspect of the object?
This is blasphemic towards almost all of metaphysics

 

[Nugatory]

How could we measure the space time interval between objects which have different state of motion, for instance one is at rest with respect to the Earth and another is moving with respect to it?

The space-time interval is measured between points in spacetime, not objects.

An object whether moving or not, is continually changing which point in spacetime it's at, but that doesn't interfere with our ability to talk about the interval between points in spacetime.

 

[Dale]

Okay ghwellsjr, I appreciate your help, but it's extremely hard for me to understand how come, for instance, the human body doesn't function as an unity.

You are already well aware that it is not functioning as "an unity", it was part of your initial specification of your problem not only that it was spatially extended, but also that different parts were moving differently.

You seem to be making 0 effort to understanding the excellent information you have received here, instead preferring to waste time arguing every time that the correct analysis doesn't immediately fit right into your preconceptions. You haven't even responded to any of the geometry.

 

[durant]

The space-time interval is measured between points in spacetime, not objects.

An object whether moving or not, is continually changing which point in spacetime it's at, but that doesn't interfere with our ability to talk about the interval between points in spacetime.

So can we measure, for instance, the space time interval between two thunders?

 

[durant]

You are already well aware that it is not functioning as "an unity", it was part of your initial specification of your problem not only that it was spatially extended, but also that different parts were moving differently.

You seem to be making 0 effort to understanding the excellent information you have received here, instead preferring to waste time arguing every time that the correct analysis doesn't immediately fit right into your preconceptions. You haven't even responded to any of the geometry.

Okay, I was aware of that. You seem to be making 0 effort in understanding how hard the transition between common-sense and relativity is. Your behaviour is like everybody's born as an Einstein. If I wasn't making efforts I would stop discussing, but I'm taking my time to learn the concepts.

 

[Nugatory]

entities with proper times

Entities do not have proper time, and speaking about "an entity with proper time" makes about as much sense as speaking about "the square root of my dog".

I'm not going to repeat the definition of proper time that I've posted (in #25 of this thread and at least once since then), but I'm begging you - go back and read it again, and keep working at it until you understand what proper time IS so that you can use it in a sentence that makes sense.

It occurs to me... There is some possibility that you are not clear on what a "point in space-time" is; that would explain much of the confusion here, especially because "proper time" is defined in terms of these points. If so, we can work on clarifying that definition.

 

[durant]

Entities do not have proper time, and speaking about "an entity with proper time" makes about as much sense as speaking about "the square root of my dog".

I'm not going to repeat the definition of proper time that I've posted (in #25 of this thread and at least once since then), but I'm begging you - go back and read it again, and understand what proper time IS so that you can use it in a sentence that makes sense.

It occurs to me... There is some possibility that you are not clear on what a "point in space-time" is; that would explain much of the confusion here, especially because "proper time" is defined in terms of these points. If so, we can work on clarifying that definition.

I apologize for the misconception. Please, can you define the point in space-time to me. And proper time relates events on the worldtube of the object, that's what you meant I guess.

 

[WannabeNewton]

It can't be made any clearer than this. Seriously.

You could define a world tube as a collection (a timelike congruence, to be technical) of worldlines. The nature of a timelike congruence is that one and only one worldline passes through any point of the congruence.

With such a congruence, you can meaningful talk about the proper time of any point in the congruence as being the proper time along the unique worldline passing through that point - at least as long as you define some initial set of points in the congruence that have a proper time of zero.

However, given a worldtube, timelike congruences are not unique - you could specify several different congruences that "cover" some particular worldtube. So there isn't any unique or meaningful way of talking about the proper time in a worldtube without specifying a particular congruence.

Also, as an aside, it is quite comical to say "This is blasphemic towards almost all of metaphysics" considering the existence of metaphysics is in and of itself a blasphemy ;)

 

[durant]

It can't be made any clearer than this. Seriously.



Also, as an aside, it is quite comical to say "This is blasphemic towards almost all of metaphysics" considering the existence of metaphysics is by itself a blasphemy ;)

I won't mention metaphysics again, that is blasphemic on this forum I guess.

 

[Dale]

This is blasphemic towards almost all of metaphysics

And another thread locked.

This forum is for science, not philosophy, you are aware of this. "Blasphemy" against any philosophy or religion is not a valid counterargument in science.

You are clearly wasting everyone's time trying to re-shape relativity so that it fits into your metaphysical agenda. You would be far better served to FIRST learn how the universe actually works and THEN try to build a metaphysical view which is compatible with nature.