Simple question about inertial frames of reference

[LBrandt]

Hello,

I know that this is going to sound stupid, so please forgive me, but could someone give me a brief and simple definition of what an inertial frame of reference is and what a non-inertial frame of reference is?

Thanks,
Louis

 

[bcrowell]

I know that this is going to sound stupid, so please forgive me, but could someone give me a brief and simple definition of what an inertial frame of reference is and what a non-inertial frame of reference is?

Not a stupid question at all. It's a very subtle concept.

In Newtonian mechanics, I would say that an inertial frame is one in which every force is caused by one material object acting on another material object. Once you've identified one inertial frame, all other frames moving at constant velocity relative to it are also inertial.

In relativity, an inertial frame is the local frame of reference defined by a free-falling observer.

For a longer discussion, see http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html#Section1.5 , section 1.5.4.

In Newtonian mechanics, the frame of reference of the dirt on the Earth's surface is approximately an inertial frame, and examples of noninertial frames are frames tied to an accelerating subway car and a rotating merry-go-round. In relativity, a frame tied to the International Space Station is inertial, but the frame of the dirt is not even approximately inertial.

-Ben

 

[ghwellsjr]

A brief, simple definition of an inertial frame of reference is one in which you are not stepping on the gas, not stepping on the brake, not turning, and not going over hills or into valleys, otherwise, it's non-inertial. In other words, you're coasting in a straight line. (We're pretending that there is no friction to slow you down.)

If you're in space, it means you're not firing any rockets.

 

[LBrandt]

Thanks. Is it permissible to say that a non-inertial frame is one in which a body is either accelerating or decelerating or changing direction, and an inertial frame is one in which a body is either at rest or moving in a straight line with constant velocity? (or am I way off base?)

Louis

 

[ghwellsjr]

You got it. That was even briefer and simpler than mine.

 

[LBrandt]

Thanks. Incidentally, this is a great forum.

Louis

 

[bcrowell]

Thanks. Is it permissible to say that a non-inertial frame is one in which a body is either accelerating or decelerating or changing direction, and an inertial frame is one in which a body is either at rest or moving in a straight line with constant velocity? (or am I way off base?)

In Newtonian mechanics, that is basically correct except that there's the subtle issue of what you define the acceleration relative to. Without defining that, you haven't really defined an inertial frame. That's why you need to do something more like what I said in #2.

In general relativity, your definition does not match up with what most people mean by an inertial frame. See the examples in #2.

 

[PeterDonis]

In Newtonian mechanics, the frame of reference of the dirt on the Earth's surface is approximately an inertial frame, and examples of noninertial frames are frames tied to an accelerating subway car and a rotating merry-go-round.

I agree this is correct as long as we limit ourselves to considering horizontal motion; but as far as vertical motion is concerned, wouldn't the dirt's frame be considered non-inertial even in Newtonian mechanics?

 

[bcrowell]

In Newtonian mechanics, the frame of reference of the dirt on the Earth's surface is approximately an inertial frame, and examples of noninertial frames are frames tied to an accelerating subway car and a rotating merry-go-round.

I agree this is correct as long as we limit ourselves to considering horizontal motion; but as far as vertical motion is concerned, wouldn't the dirt's frame be considered non-inertial even in Newtonian mechanics?

That might depend on what definition of "Newtonian inertial frame" you have in mind. If you accept my definition from #2 (an inertial frame is one in which every force is caused by one material object acting on another material object), then in the dirt's frame both the horizontal and vertical forces are caused by material objects acting on other material objects. In the dirt's frame, the vertical acceleration of a falling rock is caused by the force of the planet Earth acting on it.

 

[PeterDonis]

In the dirt's frame, the vertical acceleration of a falling rock is caused by the force of the planet Earth acting on it.

Ok, I see; you're using "inertial" as opposed to, for example, a rotating frame in which "fictitious" forces like centrifugal or Coriolis force are observed. (Technically, of course, the "dirt" frame *is* a rotating frame, but for many purposes the "fictitious" forces are small enough to be ignored.)

 

[bcrowell]

Ok, I see; you're using "inertial" as opposed to, for example, a rotating frame in which "fictitious" forces like centrifugal or Coriolis force are observed.

Exactly. The really mind-blowing thing that Einstein realized was that this distinction between fictitious and non-fictitious forces ultimately doesn't work. Once you accept that, you're forced to accept a revision of your notions of space and time that is much more radical than anything in special relativity.

 

[cesiumfrog]

Exactly. The really mind-blowing thing that Einstein realized was that this distinction between fictitious and non-fictitious forces ultimately doesn't work.

How so?

 

[bcrowell]

Exactly. The really mind-blowing thing that Einstein realized was that this distinction between fictitious and non-fictitious forces ultimately doesn't work.

How so?

Let's say LIGO observes a gravitational wave. Is that a fictitious force, or a non-fictitious force?

 

[cesiumfrog]

Is that your best example? Where is the force you're referring to?

Presumably there are pendula designed to minimise the real forces applied to the mirrors, so that they approximate unaccelerated geodesic motion (at least in the relevant dimensions). If you impose a cartesian frame of reference across the entire observatory, then wouldn't you say it is a fictitious force causing the relative acceleration of the mirrors in that frame?

Are you just trying to make the point that because global inertial frames do not exist in GR, the term "fictitious" may be less fitting (since inertial forces do have undeniable real consequences, such as communicating work), unlike in Newtonian physics (where there is always a choice of coordinates in which all inertial forces cleanly vanish)? Or are you actually disputing the possibility of distinction between inertial forces and all other (e.g., electromagnetic) forces?

 

[PeterDonis]

Is that your best example? Where is the force you're referring to?

I can't speak for bcrowell, but one way I could put the point I think he was trying to make is that, according to GR, gravity itself is kind of a "fictitious force"! That is, if an object is moving solely under the influence of what in Newtonian terms we call "gravity", that object will *feel* no force at all--it will be in free fall. GR uses this to define the notion of "local inertial frames"; they are the frames, at a given event, in which objects at rest are in free fall. A "real" force, according to GR, is a force that is actually *felt*, physically, by the object being subjected to it--in other words, it's a force that corresponds to a real, "proper" acceleration, one that can be measured with accelerometers, and which has an invariant geometric definition (the covariant derivative of the object's 4-velocity with respect to proper time is nonzero).

Part of the reason bcrowell may have used the term "mind-blowing" for this is that, under the above definition, a force like centrifugal force might actually be considered "real", not fictitious! That is, an object that in Newtonian terms we would say is subject to centrifugal force (for example, an object being thrown against the wall of a spinning cylinder, like the ones they have in amusement parks where you "stick" to the wall while the floor drops away) will actually *feel* a force--its proper acceleration will be nonzero. However, in GR, the force the object feels is not attributed to "centrifugal force", but to the straightforward mechanical force of some other object pushing it out of an inertial path (in the case just described, the force would be ascribed to the wall of the cylinder pushing on you). Similarly, the force we call "weight", that you feel while standing on Earth, is *not* due in GR to "gravity" pulling you down, but to the Earth pushing you up, out of the inertial path you would otherwise be following.

For the case of LIGO, as I understand it, the instrument detecting the gravitational wave would not feel any force--all its parts would be moving inertially the whole time. What the gravitational wave does is change what state of motion is "inertial" as it passes the instrument. So from GR's point of view, there is no "force" exerted by a gravitational wave on LIGO, in and of itself. However, it is quite possible that, as a result of a gravitational wave passing an object, internal stresses would be set up in the object itself, which would give rise to "real" forces between the object's parts (i.e., the parts would no longer be moving inertially, at least for a short period of time when the wave passed, even though the object as a whole, the "average" motion of all the parts, would be inertial).

 

[bcrowell]

PeterDonis has done a great job of explaining what I had in mind :-)

 

[JesseM]

Part of the reason bcrowell may have used the term "mind-blowing" for this is that, under the above definition, a force like centrifugal force might actually be considered "real", not fictitious! That is, an object that in Newtonian terms we would say is subject to centrifugal force (for example, an object being thrown against the wall of a spinning cylinder, like the ones they have in amusement parks where you "stick" to the wall while the floor drops away) will actually *feel* a force--its proper acceleration will be nonzero.

But the reason it feels a force is that there's a real electromagnetic force from the wall acting as a centripetal force, I don't see how the idea that what we call gravity is a "fictitious force" makes the "centrifugal force" any more "real".

 

[bcrowell]

But the reason it feels a force is that there's a real electromagnetic force from the wall acting as a centripetal force, I don't see how the idea that what we call gravity is a "fictitious force" makes the "centrifugal force" any more "real".

Are you saying that there is a clearly definable distinction between gravity as a nonfictitous force and a centrifugal force as a fictitious force? In that case I disagree with you.

Or are you saying that there is no clearly definable distinction between gravity as a nonfictitous force and a centrifugal force as a fictitious force, and that this is obvious or trivial? In that case I disagree with you as well, because it wasn't obvious to anyone before Einstein.

Another thing I don't understand about your post: there is no frame of reference in which, as you seem to claim, the normal force from the wall is the reason for the centrifugal force.

In the nonrotating frame, the centrifugal force doesn't exist.

In the rotating frame, the normal force from the wall is not the reason for the centrifugal force, because objects that are not subject to the wall's normal force are still subject to the centrifugal force.

 

[PeterDonis]

I don't see how the idea that what we call gravity is a "fictitious force" makes the "centrifugal force" any more "real".

If pressed, I admit I would have to back off from the precise wording of that statement; I didn't stop to think about its implications if taken literally when I wrote it. The key point, which I think you and bcrowell would both agree with, is that the GR definition of "force" as that which produces a proper acceleration draws a better distinction, physically, than the Newtonian distinction between "real" and "fictitious" forces, with gravity being "real" and centrifugal force being "fictitious".

 

[JesseM]

Are you saying that there is a clearly definable distinction between gravity as a nonfictitous force and a centrifugal force as a fictitious force? In that case I disagree with you.

Or are you saying that there is no clearly definable distinction between gravity as a nonfictitous force and a centrifugal force as a fictitious force, and that this is obvious or trivial? In that case I disagree with you as well, because it wasn't obvious to anyone before Einstein.

No, I'm agreeing that gravity can be seen as a fictitious force (not sure why both the options you give in the two paragraphs suppose I was saying gravity is a 'nonfictitious force'), I'm just saying I don't see how reclassifying gravity as fictitious justifies PeterDonis' statement 'a force like centrifugal force might actually be considered "real", not fictitious!' It seems to me that with this reclassification, we preserve a clear line between fictitious forces (gravity, centrifugal) and real forces (electromagnetic, strong, and weak). Of course if some version of Kaluza-Klein is true (and I think it's incorporated into string theory somehow) then the line would be blurred again, since in that other "forces" besides gravity could all have something to do with spacetime curvature and thus be "fictitious" as well.

Another thing I don't understand about your post: there is no frame of reference in which, as you seem to claim, the normal force from the wall is the reason for the centrifugal force.

Conventional inter-atomic forces like the normal force can explain all the physical observations that give us a sense of being pushed outwards, like our body being squished and our handy pocket accelerometer showing a nonzero reading. So in that sense I would say these conventional forces are the "reason for the centrifugal force".

 

[JesseM]

If pressed, I admit I would have to back off from the precise wording of that statement; I didn't stop to think about its implications if taken literally when I wrote it. The key point, which I think you and bcrowell would both agree with, is that the GR definition of "force" as that which produces a proper acceleration draws a better distinction, physically, than the Newtonian distinction between "real" and "fictitious" forces, with gravity being "real" and centrifugal force being "fictitious".

Yes, I agree.

 

[PeterDonis]

Of course if some version of Kaluza-Klein is true (and I think it's incorporated into string theory somehow) then the line would be blurred again, since in that other "forces" besides gravity could all have something to do with spacetime curvature and thus be "fictitious" as well.

Yes, but only if you allow "spacetime" to have extra dimensions--how many depends on what other forces you want to include--and to have those extra dimensions be curled up and knotted in a variety of ways that I don't fully understand (I've read a bit about Calabi-Yau manifolds, which are, last I checked, the favorites for the structure of the extra dimensions in which the Standard Model forces--everything except gravity--"live", and it pretty much made my brain explode). Also, even with that caveat, I believe that the other forces would not obey the equivalence principle the way gravity does--different objects could "fall" at different rates in response to them, even in the higher-dimensional spacetime.

 

[TrickyDicky]

Going back to the inertial frame definition, I have some doubts, for instance a planet in a elliptical orbit is supposed to follow an inertial geodesic, and yet it's evident that it doesn't have a constant velocity, it goes much faster in some parts of the orbit (by the keplerian notion of covering equal areas of the ellipse in equal time). So it accelerates and decelerates relative to the sun, so it should be a non-inertial frame?
I think this might be due to the fact that a planet is not a point test particle and due to its size and mass, only its center point could be considered inertial. But is that the reason the orbit is elliptical? would it be perfectly circular if it was test particle with negligible mass and dimensions?

 

[JesseM]

Going back to the inertial frame definition, I have some doubts, for instance a planet in a elliptical orbit is supposed to follow an inertial geodesic, and yet it's evident that it doesn't have a constant velocity

Well, the center does have zero proper acceleration throughout the orbit, even if it has nonzero coordinate acceleration in some global coordinate system covering its entire orbit. In curved spacetime no global coordinate system can really be an "inertial" one, you can only talk about local inertial frames arbitrarily small patches of spacetime, and in such a patch which included a small bit of the worldline of the planet's center, the center would be moving at constant velocity in a local inertial frame on that patch.

 

[TrickyDicky]

Well, the center does have zero proper acceleration throughout the orbit, even if it has nonzero coordinate acceleration in some global coordinate system covering its entire orbit. In curved spacetime no global coordinate system can really be an "inertial" one, you can only talk about local inertial frames arbitrarily small patches of spacetime, and in such a patch which included a small bit of the worldline of the planet's center, the center would be moving at constant velocity in a local inertial frame on that patch.

Ok, thanks, that is moreless what I had in mind but I needed someone to word it so I could understand it better.
Am I right in the final questions about a circular orbit if the test particle was arbitrarily small and only subject to the gravity of the central mass? (and if the central mass was a perfect sphere)

 

[bcrowell]

Conventional inter-atomic forces like the normal force can explain all the physical observations that give us a sense of being pushed outwards, like our body being squished and our handy pocket accelerometer showing a nonzero reading. So in that sense I would say these conventional forces are the "reason for the centrifugal force".

I think what this really shows is that those methods of measurement don't work as operational definitions of force in Newtonian mechanics. They give zero for objects in free fall, and in Newtonian mechanics an object in free fall is considered to have a force acting on it.

 

[bcrowell]

Going back to the inertial frame definition, I have some doubts, for instance a planet in a elliptical orbit is supposed to follow an inertial geodesic, and yet it's evident that it doesn't have a constant velocity, it goes much faster in some parts of the orbit (by the keplerian notion of covering equal areas of the ellipse in equal time). So it accelerates and decelerates relative to the sun, so it should be a non-inertial frame?

Newtonian mechanics and GR have different definitions of inertial frames. To see the distinction, you can just use a planet in a circular orbit. A frame tied to a planet in a circular orbit is noninertial in the Newtonian sense (because its velocity vector isn't constant) but inertial in the GR sense.

 

[JesseM]

Am I right in the final questions about a circular orbit if the test particle was arbitrarily small and only subject to the gravity of the central mass? (and if the central mass was a perfect sphere)

No, a point particle could still have an elliptical orbit. But regardless of whether the orbit was circular or elliptical, the point particle would still have zero proper acceleration, and it would still be moving at constant velocity in a locally inertial frame on a tiny patch of spacetime that included a bit of the orbit.

 

[jason12345]

Hello,

I know that this is going to sound stupid, so please forgive me, but could someone give me a brief and simple definition of what an inertial frame of reference is and what a non-inertial frame of reference is?

Thanks,
Louis

It's a coordinate system where Newton's law of inertia holds:

A body shall remain at rest or in constant uniform motion unless acted upon by a net force.

 

[bcrowell]

It's a coordinate system where Newton's law of inertia holds:

A body shall remain at rest or in constant uniform motion unless acted upon by a net force.

This actually doesn't quite work, at least without some significant elaboration. For example, an astronaut aboard the International Space Station can do experiments, and those experiments all show that Newton's first law holds, if motion is measured relative to the walls of the Space Station. Nevertheless, the ISS isn't considered to be an inertial frame as defined in Newtonian mechanics. Essentially Newton likes the frame of the solar system's barycenter better than he likes the frame of the ISS, because to him it's a matter of a majority vote; there is a lot more matter in the sun and planets than in the ISS.

You really can't give a satisfactory definition of a Newtonian inertial frame without going through some kind of "census" of what objects exist in the universe and could be making gravitational forces within your laboratory. You can widen and widen the census, and the hope is that as you go to larger and larger scales, the calculated net force on a given object in your lab will settle down to a sufficiently a well-defined value sufficiently rapidly. Newton believed that the universe was infinite and homogeneous on large scales, so that once you got up to the scale of homogeneity, this limiting process would very rapidly settle down to a very well defined result. We don't live in that kind of universe. What Einstein figured out was that the limiting process was a fool's errand, and the Newtonian concept of an inertial frame was ultimately not logically well founded.

[EDIT] Deleted the following, which didn't make sense: "One problem is that you can't define uniform motion unless you first decide what you're measuring motion relative to."

 

[D H]

For example, an astronaut aboard the International Space Station can do experiments, and those experiments all show that Newton's first law holds, if motion is measured relative to the walls of the Space Station.

Your thinking is too local and is not sufficiently omniscient. You've been working with general relativity for too long. Astronauts on the ISS know they are orbiting the Earth. An object floating around in the ISS is subject to one external force, that of gravity. Yet the object does not fall toward the Earth. The ISS is not an inertial frame in Newtonian mechanics.

 

[bcrowell]

Your thinking is too local and is not sufficiently omniscient. You've been working with general relativity for too long. Astronauts on the ISS know they are orbiting the Earth. An object floating around in the ISS is subject to one external force, that of gravity. Yet the object does not fall toward the Earth. The ISS is not an inertial frame in Newtonian mechanics.

Yes, the Newtonian definition of an inertial frame requires omniscience. My point is that omnisicience isn't possible, so unless the presumed omniscience can be redefined as a limiting process, the definition fails. I claim that there is no successful way of redefining it in terms of a limiting process (at least not in a universe that isn't infinite and homogeneous, as Newton assumed).

 

[yuiop]

Hello,

I know that this is going to sound stupid, so please forgive me, but could someone give me a brief and simple definition of what an inertial frame of reference is and what a non-inertial frame of reference is?

Thanks,
Louis

The simplest and shortest definition of inertial observer is one that does not feel any proper acceleration (as measured by an accelerometer attached to the observer). By this definition, an observer free falling in a vacuum is an inertial observer. An observer standing on the surface of the Earth is a non-inertial observer because an accelerometer attached to them will show a non zero acceleration.

An inertial frame is a grid of inertial observers each with there own rulers and clocks that are stationary with respect to each other. The clocks of all these observers will be running at exactly the same rate as each other. Normally to define the inertial frame as a frame of reference, all the clocks should be synchronised with each other so that they always show the same elapsed time as each other after allowing for light travel times. In a non inertial frame, two clocks that are spatially separated and experiencing different proper accelerations, will appear to be running at different rates even if their spatial separation appears constant from the point of view of observers at rest with the clocks.

All the above assumes "natural" clocks. There are some weird coordinate systems that use artificially sped up clocks so that the clocks in a non-inertial frame are artificially made to keep synchronised time with each other.

A perhaps even simpler definition of an inertial observer is to imagine the observer holding a ball. If he releases the ball from his hand and the ball stays where it was put then he is an inertial observer. If the ball moves then he is a non inertial observer.

 

[LBrandt]

The simplest and shortest definition of inertial observer is one that does not feel any proper acceleration (as measured by an accelerometer attached to the observer). By this definition, an observer free falling in a vacuum is an inertial observer. An observer standing on the surface of the Earth is a non-inertial observer because an accelerometer attached to them will show a non zero acceleration.

It appears to me that you've contradicted the definitions in the above statement, but maybe I'm just missing something. In the first sentence, you stated that it is an INERTIAL observer who feels no acceleration. In the third sentence, you stated that it is a NON-Inertial observer who feels no acceleration.

Louis

 

[JesseM]

In the third sentence, you stated that it is a NON-Inertial observer who feels no acceleration.

I think you misread it, the third sentence says that the observer on the surface of the Earth is non-inertial because they feel "non zero" acceleration, i.e. they do feel acceleration.

 

[LBrandt]

You are correct, I misread your statement. Sorry. But in that case, I'm a little confused. I was under the impression that a stationary observer would be considered to be in an INERTIAL frame of reference. I realize that the Earth is moving, but at it is moving at constant velocity. My original understanding was that an inertial frame of reference would be one in which there is either no movement or movement at constant velocity, whereas a non-inertial frame of reference would be one in which there is either acceleration, deceleration or change of direction.

Louis

 

[JesseM]

You are correct, I misread your statement. Sorry. But in that case, I'm a little confused. I was under the impression that a stationary observer would be considered to be in an INERTIAL frame of reference. I realize that the Earth is moving, but at it is moving at constant velocity. My original understanding was that an inertial frame of reference would be one in which there is either no movement or movement at constant velocity, whereas a non-inertial frame of reference would be one in which there is either acceleration, deceleration or change of direction.

The problem is that statements like "no movement or movement at constant velocity" are meaningless unless you specify a coordinate system in which to define "movement" in terms of change in position coordinate / change in time coordinate...no matter what object you are considering, you can always define some coordinate system where the object's coordinate position as a function of coordinate time is either unchanging or changing at a constant rate. An inertial observer is only one who is at rest or moving at constant speed relative to inertial coordinate systems, so that definition is of little use to you unless you've already come up with a definition of what it means for a coordinate system to qualify as "inertial". In Newtonian physics gravity is treated as a force so it is possible for someone to feel G-forces and still be at rest in an inertial frame, but in general relativity things work differently. In GR gravity is instead understood in terms of curved spacetime, and no coordinate system covering a large region of curved spacetime (say, a coordinate system covering an entire orbit of the Earth) is considered "inertial", but according to the equivalence principle there is the concept of a "local inertial frame" in a region of spacetime small enough that the effects of spacetime curvature (tidal forces) can be considered negligible and the laws of physics look the same as they do in flat SR spacetime, and such a local inertial frame is always the rest frame of an observer in free-fall who feels no G-forces.

 

[LBrandt]

Thanks again for the quick reply. I certainly learn more from this forum than I contribute, and that's because I'm still basically a novice when it comes to special and general relativity. However, I thought that one of the principles that is usually mentioned when discussing the twin paradox is the notion that the stay-at-home twin represents the inertial frame member of the paradox and the traveling twin (the accelerating one), the non-inertial member.
Again, I'm just learning.

Louis

 

[JesseM]

However, I thought that one of the principles that is usually mentioned when discussing the twin paradox is the notion that the stay-at-home twin represents the inertial frame member of the paradox and the traveling twin (the accelerating one), the non-inertial member.

The twin paradox is usually understood to be a problem occurring in the flat spacetime of special relativity, with no gravity. You could have a variant of the twin paradox in curved spacetime, but then no coordinate system covering the entire path of each twin from beginning to end would be an inertial one. In general relativity, even if one twin is moving inertially in a local sense (i.e. at each point on his worldline, he is at rest in a locally inertial frame in the neighborhood of that point, and thus is experiencing no G-forces) that doesn't necessarily mean he'll have aged more than a different twin who moves non-inertially between their two meetings.

 

[LBrandt]

I'll accept that. Thanks.

 

[yuiop]

In GR gravity is instead understood in terms of curved spacetime, and no coordinate system covering a large region of curved spacetime (say, a coordinate system covering an entire orbit of the Earth) is considered "inertial", but according to the equivalence principle there is the concept of a "local inertial frame" in a region of spacetime small enough that the effects of spacetime curvature (tidal forces) can be considered negligible and the laws of physics look the same as they do in flat SR spacetime, and such a local inertial frame is always the rest frame of an observer in free-fall who feels no G-forces.

Louis, please avert your eyes, as what I am going to say next might be confusing at this stage!

I just thought I would clarify that even an accelerating observer experiencing proper acceleration can consider themselves to be in flat SR spacetime for a sufficiently local region and the time dilation of nearby passing clocks traveling at relative velocities that are near the speed of light will be just the regular SR time dilation factor. If you are still reading Louis (and I know you are ) the important thing is that measurements by an observer in a gravitational field of distant events require more than just SR to analyse. In the twins example, we tend to ignore the fact that the stay at home twin on the Earth is not really inertial and treat the effect of the Earth's gravitational field on time dilation as negligible relative to the time dilation due to the high speeds involved.

 

[JesseM]

I just thought I would clarify that even an accelerating observer experiencing proper acceleration can consider themselves to be in flat SR spacetime for a sufficiently local region and the time dilation of nearby passing clocks traveling at relative velocities that are near the speed of light will be just the regular SR time dilation factor.

Wouldn't the accelerating observer be locally equivalent to an accelerating observer in flat SR spacetime, not an inertial one, though? Are you saying that in some sense even an accelerating observer in SR can make local measurements of passing clocks and find that they are running slow by the normal SR time dilation factor seen in inertial frames? If so, it seems to me this would depend on exactly how you define the local measurements of an accelerating observer (what kind of ruler they use to measure distances, how they synchronize clocks, etc.)

 

[yuiop]

Are you saying that in some sense even an accelerating observer in SR can make local measurements of passing clocks and find that they are running slow by the normal SR time dilation factor seen in inertial frames?

Yes.

If so, it seems to me this would depend on exactly how you define the local measurements of an accelerating observer (what kind of ruler they use to measure distances, how they synchronize clocks, etc.)

There are a number of ways to do this. One is to assume that a sufficiently nearby clock is running at approximately the same rate and ignore the slight error. Another is to speed up or slow down a nearby clock so that it remains synchronised and allow for the adjustment after carrying out the experiment. Finally this last method should give fairly accurate results.

Measurement method in an accelerating frame using a single clock:

Place one mirror a short distance above the clock at and measure its radar distance using the single clock. (The clock and mirror are both co accelerating and at rest relative to each other). Let's say the the radar distance is one pico-light second. Place another (co-accelerating) mirror below the clock and adjust its distance so that the radar distance is also one pico-light second. The distance to the two mirrors can be calibrated precisely using an interferometer. Arrange things so that when a test particle passes the upper mirror, a light signal is sent to the central clock to start it and when the particle passes the lower mirror a second light signal is sent to stop the central clock. The elapsed time on the central clock is a good representation of the time it took the particle to transfer from the first mirror to the second mirror because the stop/start signal delays of one pico second cancel each other out. You now have a method to measure the velocity of passing particles in an accelerating reference frame. If the velocity is measured to be v, then the time dilation factor should just be sqrt(1-v^2) to a reasonable degree of accuracy.