A Question Concerning Time Dilation

[J.C.MacSwell]

Relativity tells us that different inertial frames measure time and distance differently. So it doesn't make sense (to me, anyway) to speak of two observers being in the same frame and yet disagreeing on x and t.

 

I understand what you mean, but to me (present interpretation) they are in each others frames, but do not share the same rest frame.

 

The two observers still exist in each others time and space. I think sometimes people learning SR get some mysterious sense that they are separate, and it is reinforced when they are told they are not in the same frame as each other and misinterpret the context.

 

 

 

An accelerating frame is by definition not inertial.

 

The frame is not accelerating. The object at rest (zero velocity) with respect to it's coordinates is.

 

An inertial frame is said to be one in which physics takes it's simplest form. Obviously this does not just refer to the physics of fixed objects so context is everything.

 

 

IOW, if you are stationary with respect to the coordinates, you are in that frame.

 

I think I misinterpreted this (bolded). So you are saying an accelerating object is not in an inertial frame, not even one that it has zero velocity in.

 

I'm OK with your interpretation, though I don't think it is universal in physics and do not see where it is taught or defined (prior to here) It seems unnecessarily non intuitive to say my arm is not in the frame of my body, or something inside a rocket is not in the frame of the rocket. At other times it makes more sense, though using phrases like "at rest in or "with respect to" makes the context more clear.

Edited September 4, 2011 by J.C.MacSwell

[DrRocket]

Hmm. I just came up with another scenario.

 

Let's say Sam and Mary are both in space stations, orbiting the earth at the same altitude; Sam is several degrees ahead of Mary.

In this scenario, Sam and Mary are not travelling in the same direction linearly, in the galilean/newtonian sense; however, they are travelling in the same direction geodesicly, in the einsteinian sense (at least, based on my understanding of the geodesy of spacetime according to Einstein, which I'm willing to bet is probably even worse than my understanding of time dilation).

 

It seems to me that, in the curvature of spacetime by the earth's mass, Mary is following directly in Sam's path, and therefore her experience of time relative to him ought to be the same as if they were both travelling in a straight line, one after the other. Are they in the same reference frame in this scenario? Or am I totally misinterpreting the concept of spacetime geodesy?

 

Both are in free fall. Both have world lines that are spacetime geodesics. Both will experience the maximum propertime between any two points on their world lines.

 

The idea of a "reference frame" in general relativity is purely a local concept. As an approximation, and only as an approximation, if Stan and Mary are close one could consider them as being in the same reference frame -- essentially floating near one another like two astronauts in the space station.

 

In a pure sense one can only compare the readings of two clocks at points of intersection of their spacetime world lines and cannot compare two clocks at different locations. Time is also a local concept.

[Elen Sila]

In a pure sense one can only compare the readings of two clocks at points of intersection of their spacetime world lines and cannot compare two clocks at different locations. Time is also a local concept.

 

For reference, when I say "in the same frame", I mean "perceiving time at the same rate", as measured using the signal-timing method I described before in this thread.

[IM Egdall]

I think I misinterpreted this (bolded). So you are saying an accelerating object is not in an inertial frame, not even one that it has zero velocity in.

 

 

An object in uniform motion will stay in uniform motion, unless it is acted upon by an outside force. That is the Law of Inertia. The key here is uniform motion, where the speed of the object and the direction of the object do not change. So an inertial frame of reference is one that is in uniform motion.

 

As soon as some outside force makes the object change speed and/or direction, it is no longer in uniform motion -- it is accelerating. (In physics, any change in speed or direction is defined as acceleration.) So once an object changes speed and/or direction, it is no longer in an inertial reference frame.

 

Hope this helps.

Edited September 5, 2011 by IM Egdall

[J.C.MacSwell]

An object in uniform motion will stay in uniform motion, unless it is acted upon by an outside force. That is the Law of Inertia. The key here is uniform motion, where the speed of the object and the direction of the object do not change. So an inertial frame of reference is one that is in uniform motion.

 

As soon as some outside force makes the object change speed and/or direction, it is no longer in uniform motion -- it is accelerating. (In physics, any change in speed or direction is defined as acceleration.) So once an object changes speed and/or direction, it is no longer in an inertial reference frame.

 

Hope this helps.

 

Physics is not the problem. It is the definition of what can be considered to be in a frame.

[DrRocket]

Or am I totally misinterpreting the concept of spacetime geodesy?

 

 

 

You are misinterpreting several things, particularly how time is handled in general relativity and the relationship to special relativity.

 

Maybe this will help.

 

Time, Time Dilation, etc.

Throughout this discussion we will make the assumption that time is accurately described by general relativity. This is purely an assumption, but since general relativity is the best currently available thery of space, time, and gravity it is a reasonable assumption. We also choose units in which the speed of light, c, is 1. This is convenient and common in texts on relativity.

 

By “clock” we mean any idealized device that accurately records time.

 

 

Special Relativity

 

Special relativity is general relativity on flat Minkowski space, i.e. spacetime in the absence of gravity. Alternately it is a local approximation to general relativity; i.e. general relativity on the tangent space to the spacetime manifold.

 

Minkowski space is \mathbb R^4 endowed with a non-degenerate metric of signature (+,-,-,-) [or alternatively (-,+,+,+) but we choose the former.] A vector x in Minkowski space is called timelike if <x,x> >0, spacelike if <x,x> < 0 and null if <x,x>=0. Note that the line joining two spacetime points having identical spatial coordinates (in some fixed but arbitrary coordinate system) is timelike, and the difference in time coordinates is what would be recorded by a stationary clock at that location.

 

A curve in Minkowski space is timelike if the tangent vector to the curve is everywhere timelike. The arc length of a timelike curve in Minkowski space is the proper time associated to that curve. It can be shown that the proper time of a world line (timelike curve) is the time that would be recorded by an accurate clock having that world line. Clocks record the proper time of their world line and that is all that clocks record.

 

Because the spacetime of special relativity is flat Minkowski space there exist global coordinates, “inertial reference frames” that provide global notions of time and of space. What constitutes “time” and what constitutes “space” are dependent on the reference frame (aka observer). Lorentz transformations preserve the Minkowski inner product, which is invariant (indepenfdent of the observer) and provide a correspondence between the “time” and “space” of one observer to the “time “ and “space” of another observer. Thus in special relativity the global time coordinate of one observer is relatable to the global time coordinate of a second observer, the time at one point in a single reference frame is relatable to the time at a second point in the same reference frame (which is equivalent to an ability to synchronize clocks in a given reference frame) and the time of a “stationary” clock in a given reference frame is the same as the proper time associated with the world line of the clock, which is an invariant quantity. All of this is baked in to the Minkowski metric and the Lorentz transformations that preserve it. (For the sophisticated, one can also consider translations, which also preserve the metric and deal with the Poincare group or the inhomogeneous Lorentz group, further specializing to the orthochronous elements).

 

“Time dilation” results directly from an application of the Lorentz transformation group to the time coordinate of any given reference frame. But it is critical to note that one must start with a reference frane.

 

 

General Relativity

 

General relativity adds gravity and acceleration to the mix in the form of curvature of the spacetime manifold. In doing this the existence of global time and spatial coordinates are given up. There is no such thing as a reference frame in general relativity, only local approximations – this is the critical difference between a manifold and an affine space or vector space. In general relativity “time” and “space” are only local/approximate concepts. But the Minkowski metric generalizes to the Lorentzian metric tensor, and one can still talk sensibly about timelike tangent vectors and arc length of timelike curves. The reasoning from special relativity aqnd Minkowski space applies directly in general relativity to timelike curves (world lines) on the spacetime manifold, and arc length translates directly to proper time. Clocks still measure the proper time of their world lines.

 

Lacking global time or space coordinates there is no obvious way to compare time at one spacetimepoint with time at another point, and no meaning to “time dilation”. Either the familiar time dilation due to relative motion or the “gravitational time dilation” sometimes encountered. So what do people mean when they talk about time dilation in general relativity ?

 

Note that in general relativity we still have the fundamental notion of proper time. Clocks measure the proper time of their world line, and the only viable operational definition of time is “time is what clocks measure”. So to talk about any sort of time dilation we need to relate it to proper time.

 

Special relativity, as the local approximation to general relativity provides the answer.

 

Conceptually a manifold is very different from a Euclidean or Minkowskian space. But locally a manifold is describable as just such a space – just as flat maps describe the surface of the earth over a small area. QA manifold is pieced together with a set of local “charts” that in the case of general relativity are Minkowskian, together with more complex relationships that describe how the charts are “sewn together”. But locally one can approximate the manifold with a single chart and ignore the additional terms, or approximate them simply.

 

In general relativity speed related and gravitational time dilation are simply approximations, in a local coordinate system, of effects that are only 100% describable in more abstract terms. So, in short, in curved spacetime “time dilation” due to either speed or curvature and indeed comparison of “time” at spatially separated points, is dependent on a somewhat arbitrary choice of a local coordinate system, particularly for precise quantification.

 

So why do we standardize time and coordinate standards at different locations on earth ? First, note that the earth’s gravity is fairly weak, say compared to that of a black hole near or inside the event horizon. Spacetime in the vicinity of the earth is nearly flat. A local chart is very accurate over earthly distance scales, as are minor corrections made in terms of the local coordinates for actual curvature (gravity). The local charts determine “coordinate time” and it is coordinate time that is the subject when discussion turns to time dilation. Coordinate time is only a local concept, somewhat artificial, but an effective surrogate for proper time over small distances in weak gravitational fields. Using these local charts and approximations one can make approximate sense of comparing clocks at separate locations, and hence one can talk sensibly about “time dilation” always remembering that we are only talking about approximations to proper time – and that clocks measure proper time and nothing but proper time.

 

For people who are engaged in the development of very sophisticated clocks for the standardization of time these issues are relatively minor. Remember that the discussion has thus far involved only ideal clocks, which measure proper time with arbitrary precision. The scientists who develop atomic clocks do not have the luxury of ideal clocks. They are forced to use real clocks, made of real materials and subject to real quantum effects, which we have thus far ignored. Those clocks must be compared to one another (for a clock in isolation is useless) and hence the real distinction between coordinate time and proper time is blurred. The focus is on development of exquisitely accurate clocks that accurately measure their own proper time and verifying that accuracy by calculation and comparison with other clocks of comparable accuracy, which requires the use of coordinate time, which itself is based on somewhat arbitrary choices and agreements as to what coordinates are to be used.

 

It is important to recognize that the “time” of everyday experience, in extreme conditions or when measured to extreme levels of precision is not as clearly and unambiguously defined as one might naively suppose. The time of general relativity is a local concept. Comparison of time at spatially separated locations requires approximation, and very sophisticated approximations when extreme precision is required. Naïve questions often run aground when the nature of the approximations is not recognized and people start talking about different notions of “time” without recognizing that they are doing so.

 

 

[csmyth3025]

Thanks for the article DrR.

 

Although much of General Relativity still eludes me, at least now I have a better understanding of why you say that time is a local concept.

 

Chris

[IM Egdall]

Physics is not the problem. It is the definition of what can be considered to be in a frame.

 

Please give us more details on your question. In the meantime, I'll try to help.

 

As I understand it, a reference frame refers to a co-ordinate system. Think of three long wooden pointers assembled so they are all perpendicular. They are all attached to a central point. One pointer points up, one points to the right, and one points towards you. This represents an x,y, z co-ordinate system for the three dimensions of space. Now let's put a clock at the center of this contraption. Now we have a space and time co-ordinate system. We can measure a location in space of an event relative to the center of this "co-ordinate system" in the three axes, x, y, and z. We can also measure the time duration from event to event using our clock (assuming this contraption is in uniform motion - no change in speed or direction - and there is no gravity). We can think of this as an "inertial" reference frame.

 

Say you are in a car moving along a highway at 60 miles an hour going north. Your car is in uniform motion. You could imagine the above contraption inside your car. It represents an inertial reference frame because of its uniform motion. (And for all practical purposes, the gravitational field is uniform).

 

Any change in speed and/or change in direction, is what physicists call acceleration. If your car accelerates, it no longer represents an inertial reference frame. It represents an accelerating reference frame.

 

Imagine two cars moving at the same constant speed and in the same constant direction. They are in the same inertial reference frame. All objects moving at the same speed and in the same direction are in the same inertial frame

 

But if we include gravity, the issue is more complicated. Say that you have two cars - one in Miami and one say in LA. Together they do not represent an inertial reference frame, even if they are going at the same speed and direction. Why? Because their respective gravitational fields are pointing in different directions. The gravity in both locations is approximately pointing towards the center of the Earth. If you visualize a globe, you'll see that the two gravity direction arrows are not parallel. So only locally, each car represents an inertial frame, but together globally the two cars do not.

 

This is a bit of a simplification, but I hope it helps visualize "frames". Please let me know.

Edited September 11, 2011 by IM Egdall

[J.C.MacSwell]

Please give us more details on your question. In the meantime, I'll try to help.

 

As I understand it, a reference frame refers to a co-ordinate system. Think of three long wooden pointers assembled so they are all perpendicular. They are all attached to a central point. One pointer points up, one points to the right, and one points towards you. This represents an x,y, z co-ordinate system for the three dimensions of space. Now let's put a clock at the center of this contraption. Now we have a space and time co-ordinate system. We can measure a location in space of an event relative to the center of this "co-ordinate system" in the three axes, x, y, and z. We can also measure the time duration from event to event using our clock (assuming this contraption is in uniform motion - no change in speed or direction - and there is no gravity). We can think of this as an "inertial" reference frame.

 

Say you are in a car moving along a highway at 60 miles an hour going north. Your car is in uniform motion. You could imagine the above contraption inside your car. It represents an inertial reference frame because of its uniform motion. (And for all practical purposes, the gravitational field is uniform).

 

Any change in speed and/or change in direction, is what physicists call acceleration. If your car accelerates, it no longer represents an inertial reference frame. It represents an accelerating reference frame.

 

Imagine two cars moving at the same constant speed and in the same constant direction. They are in the same inertial reference frame. All objects moving at the same speed and in the same direction are in the same inertial frame

 

But if we include gravity, the issue is more complicated. Say that you have two cars - one in Miami and one say in LA. Together they do not represent an inertial reference frame, even if they are going at the same speed and direction. Why? Because their respective gravitational fields are pointing in different directions. The gravity in both locations is approximately pointing towards the center of the Earth. If you visualize a globe, you'll see that the two gravity direction arrows are not parallel. So only locally, each car represents an inertial frame, but together globally the two cars do not.

 

This is a bit of a simplification, but I hope it helps visualize "frames". Please let me know.

 

It got a little OT, but it was to do with what can be considered "in a frame". The accelerations discussed were more with respect to the frame, not due to the frame itself accelerating.

 

If you have a link to a definition of being "in a frame" that would be helpful. I do not believe a definitive one exists, leaving it somewhat ambiguous and depending on context.

[IM Egdall]

It got a little OT, but it was to do with what can be considered "in a frame". The accelerations discussed were more with respect to the frame, not due to the frame itself accelerating.

 

If you have a link to a definition of being "in a frame" that would be helpful. I do not believe a definitive one exists, leaving it somewhat ambiguous and depending on context.

 

It helps me to substitute "in a frame" to "inside a vehicle". So in a frame in uniform motion is inside a vehicle in uniform motion. In an accelerating frame is inside an accelerating vehicle.

 

Inertial frames are frames in uniform motion. The tricky part is that, for uniform motion, any other "vehicle" moving at the same speed and in the same direction as the original vehicle is in the same inertial frame. All objects moving at that speed and direction are in that inertial frame.

 

See if the following link helps:

 

www.youtube.com/watch?v=ZSmz2XAjl1E

Edited September 14, 2011 by IM Egdall

[J.C.MacSwell]

It helps me to substitute "in a frame" to "inside a vehicle". So in a frame in uniform motion is inside a vehicle in uniform motion. In an accelerating frame is inside an accelerating vehicle.

 

Inertial frames are frames in uniform motion. The tricky part is that, for uniform motion, any other "vehicle" moving at the same speed and in the same direction as the original vehicle is in the same inertial frame. All objects moving at that speed and direction are in that inertial frame.

 

See if the following link helps:

 

www.youtube.com/watch?v=ZSmz2XAjl1E

 

So you are describing a frame in uniform motion as inside a vehicle in uniform motion. But not everything inside a vehicle is necessarily going the same speed or direction...things can be in motion, both uniform or otherwise with respect to this frame.

 

So is everything inside the vehicle still in this frame? The driver? The wheels? The passenger in the back that is switching seats?

[IM Egdall]

So you are describing a frame in uniform motion as inside a vehicle in uniform motion. But not everything inside a vehicle is necessarily going the same speed or direction...things can be in motion, both uniform or otherwise with respect to this frame.

 

So is everything inside the vehicle still in this frame? The driver? The wheels? The passenger in the back that is switching seats?

 

YA, good point. I should have been more rigorous. I am assuming that everything inside the vehicle is at rest with respect to everything else inside the vehicle. So all things inside the vehicle are moving along with the vehicle at the same speed and in the same direction as the vehicle. So with this assumption, all things inside the vehicle are in the same frame.

 

Oh, and I don't think of the wheels as "inside" the vehicle. In my mental construct, they are outside the vehicle in contact with the road.