Proper time of a 'half-moving object'

[Dale]

In the first posts on this thread it was written that the worldtube of a partially moving object cannot have its proper time (neither can it have a rest frame), but now you're saying that we can define proper time for any worldtube, even if it's the one I've been mentioning throughout this thread.

He didn't say that you can define a unique proper time for a half moving object. He said that you could specify any world tube in terms of invariants like proper time.

Consider specifying the location of your house. You could say something like "my house is at latitude 35 N longitude 100 W" or you could say "my house is on Winding Road 2.37 km from where it intersects with Fast Freeway". The former specifies the location in terms of coordinates, the latter in terms of invariants.

Each event in a world tube can be located in terms of invariants. That is completely independent of the fact that not all worldlines have the same length.

 

[Nugatory]

In the first posts on this thread it was written that the worldtube of a partially moving object cannot have its proper time (neither can it have a rest frame), but now you're saying that we can define proper time for any worldtube, even if it's the one I've been mentioning throughout this thread.

Back to post #25 in this thread:
"Proper time" is a property of the path between two points in spacetime and the path between them. It's not something that you "define for a worldtube", and it's not something that a worldtube "has". So there's no contradiction when everyone is telling you that a partially moving object doesn't have "proper time".

I have a nagging suspicion that you are confusing proper time with the notion of simultaneity, thinking that proper time is somehow related to the way that we can speak of "the cat's nose and the cat's tail right at this moment".

 

[DrGreg]

An analogy: two people are walking along a road. Laura walks on the left side of the road. Rob walks on the right side. Whenever they come to a curve in the road, the one on the outside of the curve has to walk further than the one on the inside of the curve. So by the time they get to the end of the road, one is likely to have walked a little further than the other. You can't accurately specify a precise length of the road, because one side of the road is a little longer than the other side.

The effect could be even worse if you imagine a weird road where one side of the road is straight and the other side is wiggly.

In this analogy, "length" is analogous to proper time.

 

[pervect]

I'd like to point out, quickly, that relativity is just space-time geometry.

And, going back to Euclidean geometry as presented in high school, the axiomatic elements of geometry are points and lines.

It's pretty much the same for space-time geometry, aka relativity.

So the situation as I see it is that SR is space-time geometry, defined with points (which we call events) and lines (which we call worldlines). And the OP is obsessed with tubes for some reason, and not listening when we try to explain the axioms of geometry in terms of points (events) and lines (worldlines), which is how the textbooks actually present them, and keeps asking about "tubes".

It's not like tubes are totally outside the realm of geometry - it's just that they're not a good way to learn it, they aren't the axiomatic elements of geometry.

It also doesn't seem to be helping to point out that tubes can be defined in terms of lines.

Anyway, this may be my last post on this thread for a while, I'm feeling a bit burnt out.

 

[durant]

Back to post #25 in this thread:
"Proper time" is a property of the path between two points in spacetime and the path between them. It's not something that you "define for a worldtube", and it's not something that a worldtube "has". So there's no contradiction when everyone is telling you that a partially moving object doesn't have "proper time".

I have a nagging suspicion that you are confusing proper time with the notion of simultaneity, thinking that proper time is somehow related to the way that we can speak of "the cat's nose and the cat's tail right at this moment".

As I've red in articles about SR, time is local. So the basic impression is that all the person stages are connected, despite the motion of some parts and the rest of others. In this thread it has been stated that only parts with a specific state of motion can have proper time, so is it meaningless to talk about person stages (that all observers will agree upon) when only parts have well-defined stages, and their combination gives us the worldtube.

 

[ghwellsjr]

As I've red in articles about SR, time is local. So the basic impression is that all the person stages are connected, despite the motion of some parts and the rest of others. In this thread it has been stated that only parts with a specific state of motion can have proper time, so is it meaningless to talk about person stages (that all observers will agree upon) when only parts have well-defined stages, and their combination gives us the worldtube.

Time is what a clock measures. A clock, of necessity, must be made out of parts that take up some volume of space. And yet we never bother to show the spatial extent of a clock. Rather we conceptualize it as having no spatial extent whatsoever. That's because we are lazy and because we assume that everyone knows that we are taking all kinds of shortcuts to make things manageable when we are depicting a situation or a scenario in which we are trying to teach someone the basics of Special Relativity.

So when we refer to the Proper Time of a clock at a particular location or following a particular trajectory, we are taking shortcuts because it really has multiple Proper Times at different locations on the clock itself. In its own rest frame, assuming that it is inertial and never accelerates, all the Proper Times can be considered synchronized, although technically speaking, that is impossible to do but we can still conceptualize it. Then when we transform this clock into another inertial frame moving with respect to the first one, the different parts of the clock are no longer synchronized. But nobody cares because we pretend that the clock occupies zero space and we don't want to spend the rest of our lives embroiled in the nitty gritty details that would more closely correspond to reality but would not add anything to our understanding.

So it would make a whole lot more sense and be a whole lot easier for you to focus on the parts of a simple light clock which consists of just two reflectors and an impulse of light bouncing between them along with a magic counter on one of the reflectors that marks off the Proper Time of the light clock while ignoring the Proper Time of the other reflector and the Proper Times of the infinite number of locations between the two reflectors and to see how this simple scenario plays out in different reference frames instead of jumping first into the Proper Times of a person moving his head around with respect to the rest of his body.

Understand this: if you want to deal with all the details of a human body articulating in a random manner, it can be done. You can define each point on the body in terms of its coordinate location in some inertial frame at a starting coordinate time of zero and assign a Proper Time of zero to each of these points and then define any kind of coordinate motion to the different parts as a function of Coordinate Time and this will enable you to determine the Proper Times of all these parts as a function of the Coordinate Time of your inertial frame. Then if you want, you can transform all the coordinates of all these events into the coordinates of any other inertial frame moving with respect to your first one and get a whole new set of coordinate values for each event. And then you can do it again for a thousand other inertial reference frames, each one moving in at a different speed and/or direction with respect to the first one.

Why do you want to do this? Why isn't it enough to treat a simple problem so that you can grasp the concepts of Special Relativity and be done with it? All you are doing with these complex scenarios is compounding the process without increasing any understanding.

 

[durant]

Why do you want to do this? Why isn't it enough to treat a simple problem so that you can grasp the concepts of Special Relativity and be done with it? All you are doing with these complex scenarios is compounding the process without increasing any understanding.

Because it isn't that simple for me... I'm a beginner in this and these complex scenarios aren't really that complex, they are like everyday examples. And, sincerely, it's killing me knowing that when I write something nobody will approve it, in fact I cannot get a straight-forward answer to the stuff that's been bothering me.

I know that we can divide a human body into parts with their own coordinate systems, but why can't we then speak of the human body in terms of a space-time worm, as it is called in metaphysics of time. A 4 dimensional object (worldtube) which stages are fixed, and all observers will agree on the sequence of events or person stages on that object's worldtube? Seems to me that we can only speak of the stages of the head, stages of the hands and so on, but we can't 'find' a unique state of body which all observers will agree upon.

 

[durant]

According to the posts here, when an object has a state of motion as a unity, it has a local time, but if it changes to the state where one part is moving and another is at rest, then the local time of the object no longer exists, but there still exist local times of the parts of that worldtube.

 

[ghwellsjr]

Because it isn't that simple for me... I'm a beginner in this and these complex scenarios aren't really that complex, they are like everyday examples. And, sincerely, it's killing me knowing that when I write something nobody will approve it, in fact I cannot get a straight-forward answer to the stuff that's been bothering me.

I'll tell you what's bothering me: ten minutes after I make a rather lengthy post, you have posted a reply. That tells me you haven't read and studied my post. You haven't had time. And the evidence follows:

I know that we can divide a human body into parts with their own coordinate systems,

I never said we can divide a human body into parts with their own coordinate systems. I said we "can define each point on the body in terms of its coordinate location in some inertial frame". One frame for all, not one frame for each part. In other words: ONE FRAME. Got it? ONE FRAME. ONE FRAME. ONE FRAME.

This is evidence that even though you are given straight-forward answers to your questions, you are not taking the time to read and study them.

but why can't we then speak of the human body in terms of a space-time worm, as it is called in metaphysics of time. A 4 dimensional object (worldtube) which stages are fixed, and all observers will agree on the sequence of events or person stages on that object's worldtube? Seems to me that we can only speak of the stages of the head, stages of the hands and so on, but we can't 'find' a unique state of body which all observers will agree upon.

All observers will never agree on anything unless it is precisely spelled out to begin with. I have given you a straight-forward answer to this question but you fly right past it and claim that I haven't answered your question.

Once again: Start with a single inertial reference frame. Start at coordinate time zero. I would suggest that you use units of feet and nanoseconds and define the speed of light to be one foot per nanosecond. Provide the coordinate locations of every part of the human body you want to consider. Make a big long list including each joint of each bone and each part of each internal organ and each part of the surface of the body and each hair follicle along with the extended shape of each hair to its end point(s). Then make another list for each item on the first list describing how you want it to move in three dimensions as a function of the coordinate time (which includes those at rest). You can make these lists as long or as short as you want. Now you can place observers anywhere you want moving any way you want in this same inertial reference frame and, if you are making a spacetime diagram, you can draw in light rays going from each event (which includes each moment in time) going to each observer and you'll have the answer to your question of what any observer will see of this human. If you don't make a drawing, you will have to mathematically determine how long it takes for the image of each event to reach each remote observer.

Now different observers will see different things at different times according to their own particular location and motion and therefore as a function of their own Proper Time. But they will all agree on what happens to the different parts of the human they are observing as function of the different Proper Times of all the different parts. There is not one Proper Time that applies to the human as a whole. In fact, there's not even a meaningful "average" Proper Time that applies to the human.

Now you can transform this entire scenario, including the complicated human and all the observers into any other Inertial Reference Frame moving with respect to the first one and you will get a new set of Coordinate Times and Coordinate Locations complete with the light rays going from each event of the complicated human to each observer and you will show that they still see exactly the same things according to their own Proper Times in terms of the Proper Times of the parts of the human that was depicted in the first Inertial Reference Frame.

Now I have repeated this same answer again for you. Are you going to say that I have not given you a straight-forward answer to your question? If you don't understand my answer, please explain where the problem is.

 

[durant]

I'll tell you what's bothering me: ten minutes after I make a rather lengthy post, you have posted a reply. That tells me you haven't read and studied my post. You haven't had time. And the evidence follows:

I never said we can divide a human body into parts with their own coordinate systems. I said we "can define each point on the body in terms of its coordinate location in some inertial frame". One frame for all, not one frame for each part. In other words: ONE FRAME. Got it? ONE FRAME. ONE FRAME. ONE FRAME.

This is evidence that even though you are given straight-forward answers to your questions, you are not taking the time to read and study them.

All observers will never agree on anything unless it is precisely spelled out to begin with. I have given you a straight-forward answer to this question but you fly right past it and claim that I haven't answered your question.

Once again: Start with a single inertial reference frame. Start at coordinate time zero. I would suggest that you use units of feet and nanoseconds and define the speed of light to be one foot per nanosecond. Provide the coordinate locations of every part of the human body you want to consider. Make a big long list including each joint of each bone and each part of each internal organ and each part of the surface of the body and each hair follicle along with the extended shape of each hair to its end point(s). Then make another list for each item on the first list describing how you want it to move in three dimensions as a function of the coordinate time (which includes those at rest). You can make these lists as long or as short as you want. Now you can place observers anywhere you want moving any way you want in this same inertial reference frame and, if you are making a spacetime diagram, you can draw in light rays going from each event (which includes each moment in time) going to each observer and you'll have the answer to your question of what any observer will see of this human. If you don't make a drawing, you will have to mathematically determine how long it takes for the image of each event to reach each remote observer.

Now different observers will see different things at different times according to their own particular location and motion and therefore as a function of their own Proper Time. But they will all agree on what happens to the different parts of the human they are observing as function of the different Proper Times of all the different parts. There is not one Proper Time that applies to the human as a whole. In fact, there's not even a meaningful "average" Proper Time that applies to the human.

Now you can transform this entire scenario, including the complicated human and all the observers into any other Inertial Reference Frame moving with respect to the first one and you will get a new set of Coordinate Times and Coordinate Locations complete with the light rays going from each event of the complicated human to each observer and you will show that they still see exactly the same things according to their own Proper Times in terms of the Proper Times of the parts of the human that was depicted in the first Inertial Reference Frame.

Now I have repeated this same answer again for you. Are you going to say that I have not given you a straight-forward answer to your question? If you don't understand my answer, please explain where the problem is.

The problem is that you're describing something that has nothing to do with what I asked. You are describing a relationship between a coordinate system and a specific object, in this case some part of the human body. I was referring to the fact that timelike events in SR are invariant, that is, all observers will agree of them. So in the case of a human, I don't understand how we can't define a unique state of its body 'experienced' by its body alone, as a whole. It's more of a thought experiment. Please explain the previous comment that I wrote, this one:

"According to the posts here, when an object has a state of motion as a unity, it has a local time, but if it changes to the state where one part is moving and another is at rest, then the local time of the object no longer exists, but there still exist local times of the parts of that worldtube."

 

[ghwellsjr]

According to the posts here, when an object has a state of motion as a unity, it has a local time, but if it changes to the state where one part is moving and another is at rest, then the local time of the object no longer exists, but there still exist local times of the parts of that worldtube.

If by "local time" you mean "Proper Time", then you should say "Proper Time". If you don't mean "Proper Time" then what do you mean by "local time"?

As I said in post #41, even the "Proper Time" of a clock is a shortcut and we use it because we are lazy. In SR scenarios, we talk about people traveling at high speed in rocket ships but we never worry about the details of their motion within the rocket ship because it detracts from the salient point that we want to make.

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?

 

[durant]

If by "local time" you mean "Proper Time", then you should say "Proper Time". If you don't mean "Proper Time" then what do you mean by "local time"?

As I said in post #41, even the "Proper Time" of a clock is a shortcut and we use it because we are lazy. In SR scenarios, we talk about people traveling at high speed in rocket ships but we never worry about the details of their motion within the rocket ship because it detracts from the salient point that we want to make.

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?

Unfortunately, I can't. I'm weak in this kind of mathemathics...
I don't understand why do you state it's so simple and then throw out all the possible coordinates and random lenghts. You clearly don't understand that I don't posses the same level of knowledge as you and things that look simple to you are extremely complicated to me.
I asked you a question and again you're not giving me an answer which eliminates all the calculating. If the things exist in a way you believe they do, explain it to me in a concrete way. You're behaving like I would if I was explaining integrals and derivations to a 5-year old.

 

[ghwellsjr]

The problem is that you're describing something that has nothing to do with what I asked. You are describing a relationship between a coordinate system and a specific object, in this case some part of the human body.

How did you get that from what I wrote? I said all the parts of the human body and all the remote observers are described according to a single Inertial Reference Frame and its coordinate system.

I was referring to the fact that timelike events in SR are invariant, that is, all observers will agree of them.

Events are not categorized as being timelike. (That term applies to certain pairs of events.) Where did you get that idea from? You should think of an event as having coordinates and these coordinates can be different in different coordinate systems, that is, in different Inertial Reference Frames. It is the Proper Time of an object/clock that can have different coordinate times (and locations) in different IRF's but they all will agree on the Proper Time of that particular object/clock. You can have two objects/clocks with different Proper Times colocated at the same location and at the same time so that they are associated with the same event.

So I'm having real trouble understanding your statement that timelike events in SR are invariant. The best I can surmise is that you mean all observers will agree on the Proper Time of a clock at all events of that clock. But didn't I make that very clear in my previous long posts? Yes, I did. I described how not only will all remote observers agree on the Proper Times of all the parts of the human body, even though they see them at different later times, they still will agree when we transform the entire scenario into a different IRF. Didn't I say that? Isn't that exactly what you are concerned about? So I did in fact address the issue you have in mind, you just are reading too fast and not studying what I write to take it in.

So in the case of a human, I don't understand how we can't define a unique state of its body 'experienced' by its body alone, as a whole. It's more of a thought experiment.

As I have repeatedly said, you can define any state of a human's body to any degree of complexity that you want but if you find it significant to take into account the nanoseconds of difference between different parts of the body then go ahead. But if by "experienced", you mean something that a human can actually detect within his body, then forget it. If we could detect relativistic effects within our own bodies, it wouldn't have been many thousands of years of human history before relativistic issues became known and it was only because of advances in technological instruments, not human bodies, that permitted precise enough measurements that lead to the recognition relativity and the development of the theory of Special Relativity.

Please explain the previous comment that I wrote, this one:

"According to the posts here, when an object has a state of motion as a unity, it has a local time, but if it changes to the state where one part is moving and another is at rest, then the local time of the object no longer exists, but there still exist local times of the parts of that worldtube."

I already did in the previous post. I'll see if it was adequate as I'm guessing you have probably already responded.

 

[durant]

Events are not categorized as being timelike. (That term applies to certain pairs of events.) Where did you get that idea from? You should think of an event as having coordinates and these coordinates can be different in different coordinate systems, that is, in different Inertial Reference Frames. It is the Proper Time of an object/clock that can have different coordinate times (and locations) in different IRF's but they all will agree on the Proper Time of that particular object/clock. You can have two objects/clocks with different Proper Times colocated at the same location and at the same time so that they are associated with the same event.

Then why do you say that a partially moving object doesn't have a proper time?

 

[ghwellsjr]

Then why do you say that a partially moving object doesn't have a proper time?

For the same reason that I just said that "You can have two objects/clocks with different Proper Times colocated at the same location and at the same time so that they are associated with the same event. Each object/clock that has a different history of motion will have a different accumulation of Proper Time. If you have what you are calling "a partially moving object" you really mean two or more independent objects with different histories of motion and therefore can have different accumulations of Proper Time.

So you could have a person slap himself on the head and while his hand is in motion, it accumulates less Proper Time than his head does during each slap. So now if you ask the question, what is the Proper Time of this person, it will have at least two answers, one for his hand and one for his body and head. (The different parts of his arm will have multiple other Proper Times because they are moving at different speeds.) Lot's of different answers for the different parts of his body because of their different histories of motion.

Do you think there can possibly be one answer to the question of the Proper Time of a partially moving object?

 

[ghwellsjr]

Unfortunately, I can't. I'm weak in this kind of mathemathics...
I don't understand why do you state it's so simple and then throw out all the possible coordinates and random lenghts. You clearly don't understand that I don't posses the same level of knowledge as you and things that look simple to you are extremely complicated to me.
I asked you a question and again you're not giving me an answer which eliminates all the calculating. If the things exist in a way you believe they do, explain it to me in a concrete way. You're behaving like I would if I was explaining integrals and derivations to a 5-year old.

But you're not 5-years old. If I recall correctly, you said you were 21. 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.

 

[Fredrik]

Then why do you say that a partially moving object doesn't have a proper time?

The motion of an extended object (i.e. not a point particle) is described by a set of curves in spacetime. Each of those curves has a proper time, just like every curve in space has a length. If you draw a bunch of curves on a piece of paper, you wouldn't be able to assign a single length to the set of curves, would you? Each curve would have a length, but the set of curves wouldn't.

 

[Dale]

Seems to me that we can only speak of the stages of the head, stages of the hands and so on, but we can't 'find' a unique state of body which all observers will agree upon.

Yes.

 

[Dale]

I was referring to the fact that timelike events in SR are invariant, that is, all observers will agree of them.

OK, I think you have a misunderstanding because you are using technical terms wrong. Timelike separation is not a property of an event, it is a relationship between two events. Take one event, and from that event draw all of the points in the future that could be reached by a flash of light emitted from that event, and all of the points in the past where a flash of light would reach that event. In 4D, this shape is a cone, called the light cone.

Any event which lies inside the light cone is timelike separated from the apex event. The spacetime interval between the two is measured by a clock, the two events can be connected by a worldline representing the motion of a massive particle. All reference frames agree on which event was first and which was second.

Any event which lies on the light cone is lightlike or null separated from the apex event. The spacetime interval between the two is 0 and cannot be measured by either a clock or a rod, the two events can be connected by a worldline representing the motion of a massless particle. All reference frames agree on which event was first and which was second.

Any event which lies outside the light cone is spacelike separated from the apex event. The spacetime interval between the two is measured by a ruler, and the worldline connecting the two cannot represent the motion of any particle. Reference frames disagree on the order.

An extended object will have events which are timelike, lightlike, and spacelike separated from each other.

 

[durant]

An extended object will have events which are timelike, lightlike, and spacelike separated from each other.

DaleSpam, can you please explain lightlike separated events, with a concrete example (if there is one)?

And also, from the point of view of the object (its rest frame), does there exist gravitational time dilation for its smaller parts, or the object from its rest frame subsumes the proper times of all of its smaller parts?

 

[Dale]

DaleSpam, can you please explain lightlike separated events, with a concrete example (if there is one)?

A camera's flash bulb, located 10 ft from my eye, emits a flash of light. 10 ns later, the flash arrives at my eye. The event of the flash is lightlike separated from the event of the light's arrival to my eye.

And also, from the point of view of the object (its rest frame), does there exist gravitational time dilation for its smaller parts, or the object from its rest frame subsumes the proper times of all of its smaller parts?

Let's not introduce gravitation. You are not ready yet.

 

[durant]

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.

Isn't it the case that the proper time is invariant, and the coordinate time for moving observers is variant? Please explain how can the man's head accumulate less proper time? Shouldn't they accumulate an equal amount of proper time, but different values of coordinate time?

 

[ghwellsjr]

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

 

[durant]

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

I've done it but I still don't understand why the proper time will be affected despite the fact that every article about time dilation on the internet clearly states that proper time goes normal in every rest frame, and only coordinate time gets slown down for a moving observer.

 

[Nugatory]

I've done it but I still don't understand why the proper time will be affected despite the fact that every article about time dilation on the internet clearly states that proper time goes normal in every rest frame, and only coordinate time gets slown down for a moving observer.

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.

 

[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.