Intriguing Questions Regarding Nature of Time in Special Relativity

[Curious6]

Hi,

I have been thinking about the philosophical implications of time and have in the process been analyzing the consequences of time as conceived of by special relativity. I have a couple of questions though, and I would really appreciate it if someone could offer their knowledge in an attempt to answer them:

1. How does time emerge in special relativity? Could we conceive as time as simply measuring the duration of a process?
i2. In an inertial reference frame, if two particles are at rest with respect to each other, is time passing for them?

Any insights on these questions would be very welcome indeed.

Thanks!

 

[Fredrik]

1. Two concepts of time are defined by the theory, proper time and coordinate time. It's postulated that a clock measures the proper time of the curve in spacetime that represents the clock's motion. (I'm not sure if that answers your question).

2. Yes. Their motions are represented by two straight and parallel lines in spacetime, but the time coordinate is not the same at any two points on the same curve, and the proper time integral is also non-zero along these curves. It's value increases as you consider longer segments of the curve.

 

[Curious6]

Thank you for your answers. Following on from them, I have another two questions:

1. Could you define a little more closely (perhaps with an example) the difference between proper time and coordinate time.
2. You say that the value of the time coordinate increases along the curves (which may be interpreted as space) so could we therefore say that time can be conceived of as being the extent of space? In other words, could we imagine time as being inherent or a property of space ?
3. Is time in a sense therefore just a measure of oscillations of space?

 

[Fredrik]

1. See e.g. this post. In 1+1 dimensions, units such that c=1, and in the coordinates of an inertial frame, the square root simplifies to $\sqrt{dt^2-dx^2}$. If you and I are initially at the same location, and I go for a run while you stay put, our paths through spacetime from when we got separated to when we meet again have different proper times. Your path has a greater proper time since there's no contribution from dx when we integrate along your path. This means that you have aged more than I when we meet again.

2. It's a property of a curve in spacetime. To be precise, it's the closest thing to a "length" of the curve that we can define. Proper time is often described as the "length" of the curve, but it's a bit counterintuitive. You might think that if you stay put, and I go for a run, my path is going to be longer, but my movement through space makes my path shorter, not longer, because of the minus sign in front of dx².

3. It's the "length" of a curve. The clock that measures this "length" might however do it by counting the number of some sort of oscillations.

 

[bernhard.rothenstein]

Thank you for your answers. Following on from them, I have another two questions:

1. Could you define a little more closely (perhaps with an example) the difference between proper time and coordinate time.
2. You say that the value of the time coordinate increases along the curves (which may be interpreted as space) so could we therefore say that time can be conceived of as being the extent of space? In other words, could we imagine time as being inherent or a property of space ?
3. Is time in a sense therefore just a measure of oscillations of space?

I enjoy the following definitions:
Proper time (interval) the time interval between two events as measured by a single clock that is present at both events. Its value depends on the world line that the clocki follows in getting from one event to the other. We can say that it is the time interval measured by the wrist watch of an observer.
Coordinate time (interval) the time interval between two events in an inertial reference frame by a pair of synchronized clocks, one present at one event the other present at the other event.
(Thomas A. Moore, A Traveler's Guide to Space Time)
Two clocks of the same inertial reference frame and standard synchronized display the same running time.

 

[Curious6]

Thank you both very much for your help and insights.

 

[neopolitan]

I enjoy the following definitions:
Proper time (interval) the time interval between two events as measured by a single clock that is present at both events. Its value depends on the world line that the clocki follows in getting from one event to the other. We can say that it is the time interval measured by the wrist watch of an observer.
Coordinate time (interval) the time interval between two events in an inertial reference frame by a pair of synchronized clocks, one present at one event the other present at the other event.
(Thomas A. Moore, A Traveler's Guide to Space Time)

So is this true:

(coordinate time*c)²=(proper time*c)²+(spatial separation between events)²=(proper time*c)²+(coordinate time*v)²

or this:

(coordinate time*c)²=(proper time*c)²+(proper time*v)²

or is there yet another equation?

The reason I ask is that it seems to me that there is one, and only one path in spacetime between two events. It can defined by proper time (in the rest frame in which the events are collocated) or by a combination of coordinate time and spatial separation.

It's just a little unclear as to how that separation should be defined. I think it is the latter equation above, but it is counter intuitive because it uses the time of a clock which can consider itself to be at rest and the speed at which another observer considers the clock to be moving.

cheers,

neopolitan

 

[Dale]

The reason I ask is that it seems to me that there is one, and only one path in spacetime between two events.

Careful here! Think "twins paradox". The separation and the reunion are two events and each twin takes a different path through spacetime between those events. Geometrically your statement is akin to saying that there is one and only one curve which joins two points in a plane.

 

[neopolitan]

Careful here! Think "twins paradox". The separation and the reunion are two events and each twin takes a different path through spacetime between those events. Geometrically your statement is akin to saying that there is one and only one curve which joins two points in a plane.

You are right. I didn't express myself well enough.

The two events I am thinking of are the two events from the situation which Bernard quoted. In other words you have one observer in one inertial frame for whom the events are collocated and you have another observer in one inertial frame (bye bye twin's paradox) for whom the events are not collocated. This would be akin to only one leg of the twin's paradox.

Geometrically, I would be thinking of a line drawn on a sheet of paper. It could be pointed "up" - corresponding to stationary. Or it could be pointed off to one side - corresponding to "as viewed by someone in motion". Would the magnitude of the line change depending on your perspective, or would it be the same, just with vector addition of time and spatial components?

cheers,

neopolitan

 

[Naty1]

Curious: you can see how Einstein explains time at
http://www.bartleby.com/173/

perhaps beginning at #8 ON THE IDEA OF TIME IN PHYSICS

where it seems to me he assumes in special relativity time in one location passes at the same rate as in another location. It's absolute in the Newtonian sense. In the next section he argues that time on a train passes differently viewed from an embankment and in the next that distance, too varies, with motion...he already had Lorentz- Fitgerald transformations to guide him so likely they were a key input..those are discussed in the next section, #11.

 

[Curious6]

Naty, thanks for that link.

I have come up with another question: If time is indeed a curve in spacetime, then is the reason we see time as passing due to the Earth moving along the curve?

 

[Dale]

If time is indeed a curve in spacetime, then is the reason we see time as passing due to the Earth moving along the curve?

Even a "stationary" object still "moves" in the time direction through spactime.