Is Simultaneity Absolute in Einstein's Theory of Relativity?

[George Jones]

Please explain how you measure proper time between events with clocks alone.

I'm a little confused by this. For me the proper time between two events is, by definition, the time recorded on a clock that experiences both events, i.e., both events are on the clock's worldline.

See the second paragraph of https://www.physicsforums.com/showpost.php?p=1230123&postcount=19" by robphy. Even though it's meant for non-physicists, I think that you might enjoy reading General Relativity from A to B. I certainly did.

The interval along a given curve is a geometric object, one that is independent of any particular choice of coordinate or metric.

This also confuses me. For me, the metric and the interval are the same thing.

 

[bernhard.rothenstein]

proper time

As I understand it then, dt is the coordinate time and g is the reciprocal of the time component in the metric tensor. It might be better to write it dt(0)=gdt, in which case g is the metric component. In special relativity g=1.

Read please
dt=(dtau)/(1-VV/cc)^1/2

 

[country boy]

I'm a little confused by this. For me the proper time between two events is, by definition, the time recorded on a clock that experiences both events, i.e., both events are on the clock's worldline.

See the second paragraph of https://www.physicsforums.com/showpost.php?p=1230123&postcount=19" by robphy. Even though it's meant for non-physicists, I think that you might enjoy reading General Relativity from A to B. I certainly did.

Thank you for the reference.

I didn't mean that you couldn't measure the proper time between a particular two events with a single clock and no rods. What I meant was that an observer stuck an arbitrary frame measuring the proper time between an arbitrary pair of events would need to use both rods and clocks, or some equivalent. He would need to have both the positions and the times at the two events. For a particular pair of events, he would, of course, get the same answer as the single clock moving between the two events.

 

[country boy]

Read please
dt=(dtau)/(1-VV/cc)^1/2

Ah, I see. But the dt in this equation is not the coordinate time as usually defined. It is just the time between two events. And dtau is the time measured in the co-moving frame (the frame that puts the two events at the same place.)

 

[George Jones]

Thank you for the reference.

I didn't mean that you couldn't measure the proper time between a particular two events with a single clock and no rods. What I meant was that an observer stuck an arbitrary frame measuring the proper time between an arbitrary pair of events would need to use both rods and clocks, or some equivalent. He would need to have both the positions and the times at the two events. For a particular pair of events, he would, of course, get the same answer as the single clock moving between the two events.

Here is how an inertial observer establishes an inertial coordinate system using her wristwatch and light signals. Suppose P is any event in spacetime. The observer continually sends out light signals, and suppose the light signal that reaches event P left her worldline at time t1 according to her watch. Upon reception of this signal, P immediately sends (reflects) a light signal back to the observer, which she receives at time t2. If x is the spatial distance of P from the observer's worldline, then, since the light goes out and back, the light travel a distance 2x in a time (t2 - t1). Thus

2x = c (t2 - t1) or x = (t2 - t1)/2 with c = 1.

The light spends half the time going out, and half the time coming back. Therefore the time coordinate of event P is the same as the the event on the observer's worldline that is halfway (in time) between the observer's emission and reception events. Consequently,
the time coordinate of P is

t = (t2 + t1)/2.

It is easy to convince oneself that this operational definition establishes a standard inertial coordinate system.

 

[bernhard.rothenstein]

coodinate time interval

Ah, I see. But the dt in this equation is not the coordinate time as usually defined. It is just the time between two events. And dtau is the time measured in the co-moving frame (the frame that puts the two events at the same place.)

Thanks. Please let me know how do you define the coordinate time interval. Is there a special reason for using coordinate time and not coordinate time interval? As far as I know the coordinate time interval equates the difference between the readings of two synchronized, distant identical clocks located at the points where two events take place when they occur respectively.

 

[country boy]

Here is how an inertial observer establishes an inertial coordinate system using her wristwatch and light signals. Suppose P is any event in spacetime. The observer continually sends out light signals, and suppose the light signal that reaches event P left her worldline at time t1 according to her watch. Upon reception of this signal, P immediately sends (reflects) a light signal back to the observer, which she receives at time t2. If x is the spatial distance of P from the observer's worldline, then, since the light goes out and back, the light travel a distance 2x in a time (t2 - t1). Thus

2x = c (t2 - t1) or x = (t2 - t1)/2 with c = 1.

The light spends half the time going out, and half the time coming back. Therefore the time coordinate of event P is the same as the the event on the observer's worldline that is halfway (in time) between the observer's emission and reception events. Consequently,
the time coordinate of P is

t = (t2 + t1)/2.

It is easy to convince oneself that this operational definition establishes a standard inertial coordinate system.

Thank you for the clear description. This is Einstein synchronization, right? The time is thus established at each point in the frame, independent of the spatial coordinates. And if the speed of light is used, the distance to each point is established. These two types of measurements generate the space-time coordinate system. Then, for two events at different places and different times, a determination of the proper time (or, more generally, the invariant interval) between the events requires the separations in both space and time. This will agree with a clock located at a moving point that happens to be at both events when they occur.

 

[country boy]

Thanks. Please let me know how do you define the coordinate time interval. Is there a special reason for using coordinate time and not coordinate time interval? As far as I know the coordinate time interval equates the difference between the readings of two synchronized, distant identical clocks located at the points where two events take place when they occur respectively.

I'm not familiar with that definition. To me, coordinate time is the time measured by a clock located at the event. The coordinate time interval is, then, the diffence in readings of that clock between to events at its location.

Please tell me if I misunderstand this.

 

[pervect]

I'm not familiar with that definition. To me, coordinate time is the time measured by a clock located at the event. The coordinate time interval is, then, the diffence in readings of that clock between to events at its location.

Please tell me if I misunderstand this.

Coordinate time is not necessarily the time measured by a clock located at the event.

Example: The schwarzschild 't' coordinate, which can be thought of as the time read by a "clock at infinity" rather than a clock at the particular location. Note that what is usually called "gravitational time dilation" makes the two clocks (the clock at infinity and the clock at a particular location) run at different rates when compared by usual methods (such as light signals). The coordinate time is different than the time given by a clock at a particular location - not even the rate is the same.

There is actually no requirement for coordinate time other than that every point have a time coordinate. (One may also desire that the time coordinate be unique. Requiring time to be unique in some cases such as the Rindler metric limits the size covered by the coordinate system, however. There may be some differences in conventions here, so be careful about assuming uniqueneness).

 

[bernhard.rothenstein]

radar detection

Here is how an inertial observer establishes an inertial coordinate system using her wristwatch and light signals. Suppose P is any event in spacetime. The observer continually sends out light signals, and suppose the light signal that reaches event P left her worldline at time t1 according to her watch. Upon reception of this signal, P immediately sends (reflects) a light signal back to the observer, which she receives at time t2. If x is the spatial distance of P from the observer's worldline, then, since the light goes out and back, the light travel a distance 2x in a time (t2 - t1). Thus

2x = c (t2 - t1) or x = (t2 - t1)/2 with c = 1.

The light spends half the time going out, and half the time coming back. Therefore the time coordinate of event P is the same as the the event on the observer's worldline that is halfway (in time) between the observer's emission and reception events. Consequently,
the time coordinate of P is

t = (t2 + t1)/2.

It is easy to convince oneself that this operational definition establishes a standard inertial coordinate system.

The interesting fact is that considering the same procedure in I' and taking into account that t and t' are related by the Doppler Effect, we can derive the Lorentz transformations.

 

[robphy]

The interesting fact is that considering the same procedure in I' and taking into account that t and t' are related by the Doppler Effect, we can derive the Lorentz transformations.

Yep.
That's the Bondi k-calculus, which, in my opinion, should be referenced in http://arxiv.org/abs/physics/0703002.

 

[bernhard.rothenstein]

clock reading, time interval

I'm not familiar with that definition. To me, coordinate time is the time measured by a clock located at the event. The coordinate time interval is, then, the diffence in readings of that clock between to events at its location.

Please tell me if I misunderstand this.

I have learned from Einstein that time t is what a clock reads. That time is used in order to define the time coordinate of an event that takes place in front of the clock when it reads t. In order to become opperational the clocks of a given inertial reference frame should be synchronized in order to display the same running time. If we can find out the space time coordinates of an event using the readings of a single clock I think that clock synchronization could be avoided.
The time interval dt is associated with the time separation between two events.
Consider two events E(1)[x(1),y(1),z(1),t(1)] and E(2)[x(2),y(2),z(2),t(2)]
where t(1) and t(2) represent the readings of clocks C(1)[x(1),y(1),z(1)]and C(2)[x(2),y(2),z(2)] when the mentioned events take place in front of them. If x(1)=x(2); y(1)=y(2),z(1)=z(2), then t(2)-t(1) represents a proper time interval. If the two events take place at different points in space then how could we call (t(2)-t(1) in this case?
I find some definitions of time intervals in Thomas A. Moore A travelers guide in space-time (Mc.Graw-Hill Inc. 1995) who distinguishes three time intervals. He defines the coordinate time (I would say coordinate time interval) as:
Coordinate time: "The time measured between the events either by a pair of synchronized clocks at rest in a given inertial reference frame (one clock present at each event or by a single clock at rest in that inertial reference frame (one clock present at each event) or by a single clock at rest in that inertial frame ( (if both events happen to occur at that clock in that frame) is called the coordinate time between the events in that frame The symbol dt is used to represent the coordinate time between the events. It is surprising for me that the Author uses time and not time interval.
Proper time: The time between two events measured by any clock present at both events is called a poper time between those events. We will use the symbol d(tau) to represent a proper time between two events. A proper time measured by a given clock is an absolute quantity independent of reference frame.
The Author also defines the concept of space-time interval which is not in an easy reach for me. I think that he has in mind the possibility to express cctt-xx=cc(tau)2 as a function of the proper time interval.
My problem is if the definition of proper time is not included in the definition of the coordinate time?
In conclusion I think that besides the fact that we should or we should not synchronize clocks, it is important to have an unique conception about what they measure.
Please let me know your oppinion.

 

[bernhard.rothenstein]

Yep.
That's the Bondi k-calculus, which, in my opinion, should be referenced in http://arxiv.org/abs/physics/0703002.

Thanks. I will in its final version!

 

[country boy]

Coordinate time is not necessarily the time measured by a clock located at the event.

Example: The schwarzschild 't' coordinate, which can be thought of as the time read by a "clock at infinity" rather than a clock at the particular location. Note that what is usually called "gravitational time dilation" makes the two clocks (the clock at infinity and the clock at a particular location) run at different rates when compared by usual methods (such as light signals). The coordinate time is different than the time given by a clock at a particular location - not even the rate is the same.

There is actually no requirement for coordinate time other than that every point have a time coordinate. (One may also desire that the time coordinate be unique. Requiring time to be unique in some cases such as the Rindler metric limits the size covered by the coordinate system, however. There may be some differences in conventions here, so be careful about assuming uniqueneness).

I agree. "Coordinate time" is, strictly speaking, the time measured at a spatial coordinate, and there is one of these at every spatial coordinate (even at infinity). My use of "event” was just to signify that something is being measured. Even the reading of a clock is an event.

 

[bernhard.rothenstein]

time

I agree. "Coordinate time" is, strictly speaking, the time measured at a spatial coordinate, and there is one of these at every spatial coordinate (even at infinity). My use of "event” was just to signify that something is being measured. Even the reading of a clock is an event.

As I see you consider that "coordinate time" is "what a clock reads". Then "coordinate time interval" is the difference between the readings of two such clocks?

 

[MeJennifer]

As I see you consider that "coordinate time" is "what a clock reads". Then "coordinate time interval" is the difference between the readings of two such clocks?

Clocks read proper time not coordinate time.

Of course coordinate time and proper time could overlap for a particular group of observers, but they cannot be the same for two or more observers who are moving with respect to each other.

 

[bernhard.rothenstein]

clock and time

Clocks read proper time not coordinate time.

Of course coordinate time and proper time could overlap for a particular group of observers, but they cannot be the same for two or more observers who are moving with respect to each other.

Thanks. With all respect, I do not understand your point of view. IMHO when I speak about the time displayed by a clock I am not able to distinguish if it is proper or coordinate time. Please have a look at my post 47 and tell me where my point of view is wrong. A correct understanding of the problem is of big importance for me, not being very familiar with the Anglo-American nomenclature and probably for others as well. Also please comment the definitions given by Thomas Moore A Traveler's Guide to Spacetime according to whom:
"Coordinate time :The time measured between two events either by a pair of synchronized clocks at rest in a given inertial reference frame (one clock present at each event) or by a single clock at rest in that inertial reference frame (if both events happen to occur at that clock in that frame) is called the coordinate time between the events in that frame.

 

[MeJennifer]

Thanks. With all respect, I do not understand your point of view. IMHO when I speak about the time displayed by a clock I am not able to distinguish if it is proper or coordinate time. Please have a look at my post 47 and tell me where my point of view is wrong. A correct understanding of the problem is of big importance for me, not being very familiar with the Anglo-American nomenclature and probably for others as well. Also please comment the definitions given by Thomas Moore A Traveler's Guide to Spacetime according to whom:
"Coordinate time :The time measured between two events either by a pair of synchronized clocks at rest in a given inertial reference frame (one clock present at each event) or by a single clock at rest in that inertial reference frame (if both events happen to occur at that clock in that frame) is called the coordinate time between the events in that frame.

Some comments that hopefully clear things up.

I have learned from Einstein that time t is what a clock reads. That time is used in order to define the time coordinate of an event that takes place in front of the clock when it reads t. In order to become opperational the clocks of a given inertial reference frame should be synchronized in order to display the same running time.

This is certainly one way of setting up a coordinate system, but not the only way.
In the setup you mention it is true that for the clock, which we assume is inertial, proper time overlaps coordinate time.

Now consider another clock in relative motion with this clock, could we say that this clock measures coordinate time? I think the answer is no, the coordinate time and the proper time no longer overlap and we need a Lorentz transformation to calculate the difference.

The reason that the proper time direction no longer overlaps the coordinate time is that the second clock is semi-rotated in the first clock's space-time coordinate system and, as a consequence, the direction of the proper time line is now rotated away from the coordinate time direction. And due to the metric of space-time, such a rotation will shorten any line segment and thus the proper time interval will be smaller compared to the coordinate time interval.

It seems that Moore defines as coordinate time, the condition in which proper time and coordinate time overlap.

Note that the physical meaning of coordinate time becomes more problematic when we consider curved space-time as well. In curved space-time coordinate time is no longer guaranteed to be ortho-normal to the spatial coordinates.

 

[bernhard.rothenstein]

Some comments that hopefully clear things up.


This is certainly one way of setting up a coordinate system, but not the only way.
In the setup you mention it is true that for the clock, which we assume is inertial, proper time overlaps coordinate time.

Now consider another clock in relative motion with this clock, could we say that this clock measures coordinate time? I think the answer is no, the coordinate time and the proper time no longer overlap and we need a Lorentz transformation to calculate the difference.

The reason that the proper time direction no longer overlaps the coordinate time is that the second clock is semi-rotated in the first clock's space-time coordinate system and, as a consequence, the direction of the proper time line is now rotated away from the coordinate time direction. And due to the metric of space-time, such a rotation will shorten any line segment and thus the proper time interval will be smaller compared to the coordinate time interval.

It seems that Moore defines as coordinate time, the condition in which proper time and coordinate time overlap.

The physical meaning of coordinate time becoms more problematic when we consider curved space-time as well. In curved space-time coordinate time is no longer guaranteed to be ortho-normal to the spatial coordinates.

Thank you for your help and for the kind way to answer.

"Now consider another clock in relative motion with this clock, could we say that this clock measures coordinate time? I think the answer is no, the coordinate time and the proper time no longer overlap and we need a Lorentz transformation to calculate the difference."

My problem is related to the clocks of the same reference frame in a state of rest relative to each other and synchronized. I am confused by the fact that Authors make not a net distinction between time and time interval So far SR is not involved. It becomes when I consider a clock C' which moves with constant V relative to the clocks mentioned above, reading zero when it is located in front of a stationary clock which reads zero as well. After a given time of motion it reads t' being located in front of a stationary clock. Then by definition
t-0 represents a coordinate time interval
t'-0 represents a proper time interval
related by (t-0)=gamma(t'-0)
The Lorentz transformation becomes involved in the case when in both inertial reference frames the observers measure coordinate time intervals.
Do you consider that such a way of teaching a beginner is correct?

 

[MeJennifer]

My problem is related to the clocks of the same reference frame in a state of rest relative to each other and synchronized.

Then in this coordinate time overlaps proper time.

But note that, in general, this is not the case and certainly will give problems when you consider cases with curved space-time.

I am confused by the fact that Authors make not a net distinction between time and time interval

I agree with you that, when appropriate, adding the term interval is better.

So far SR is not involved. It becomes when I consider a clock C' which moves with constant V relative to the clocks mentioned above, reading zero when it is located in front of a stationary clock which reads zero as well. After a given time of motion it reads t' being located in front of a stationary clock. Then by definition
t-0 represents a coordinate time interval
t'-0 represents a proper time interval
related by (t-0)=gamma(t'-0)
The Lorentz transformation becomes involved in the case when in both inertial reference frames the observers measure coordinate time intervals.

Do you consider that such a way of teaching a beginner is correct?

It is certainly not incorrect.

If it is the best way of teaching, well, I certainly have an opinion on it, but I feel that it is not proper to "vent" my opinon here in this topic.

Note that in this case the coordinate system is only valid for a particular group of observers, namely those who are at relative rest to it.

 

[country boy]

Reply to post 47:

I have learned from Einstein that time t is what a clock reads. ...

In conclusion I think that besides the fact that we should or we should not synchronize clocks, it is important to have an unique conception about what they measure.
Please let me know your oppinion.

My idea of coordinate time is, I believe, more specific than what you describe. If two events happen at different locations and the coordinate times are recorded at both points, the difference of the two coordinate times is delta t. If the two events happen at the same location then only one clock is needed and delta t = delta tau.

The "time" of an event is equivalent to the "time interval" between two events, one of which is at the clock's zero reading.

"Proper time" is a special case of the invariant "space-time interval." Proper time refers to invariant intervals that are on or inside the light cone.

A lot of this discussion seems to be about definitions, but that is okay because it leads to clearer understanding (speaking for myself, anyway).

 

[bernhard.rothenstein]

clock and time

Reply to post 47:

My idea of coordinate time is, I believe, more specific than what you describe. If two events happen at different locations and the coordinate times are recorded at both points, the difference of the two coordinate times is delta t. If the two events happen at the same location then only one clock is needed and delta t = delta tau.
That is what I mentioned in one of my intervention. I consider that we should make a net distinction between coordinate time which IMHO represents the reading of a clock when an event takes place at its location abd I use it to define the time coordinate of the event. I would aggree with you if instead of using coordinate time would use coordinate time interval.

The "time" of an event is equivalent to the "time interval" between two events, one of which is at the clock's zero reading.
It is a little confusing for me. If the clock are synchronized that condition is automatically fulfilled.

"Proper time" is a special case of the invariant "space-time interval." Proper time refers to invariant intervals that are on or inside the light cone.
I would add: If in one of the reference frames dx=0

A lot of this discussion seems to be about definitions, but that is okay because it leads to clearer understanding (speaking for myself, anyway).
for me too

With thanks for the participation