Special relativity and accelerated frames

[Dale]

Dale: Thanks, but the equations are ahead of my current understanding. I'm still a novice.

Sure, it took me several years to figure this stuff out.

The point to notice from what I posted is that the worldline for a uniformly accelerating object in SR is a hyperbola, not a parabola as it is in Newtonian physics. The coordinate acceleration is non-uniform, even though the proper acceleration is uniform.

 

[CKH]

Sure, it took me several years to figure this stuff out.

The point to notice from what I posted is that the worldline for a uniformly accelerating object in SR is a hyperbola, not a parabola as it is in Newtonian physics. The coordinate acceleration is non-uniform, even though the proper acceleration is uniform.

I see a little bit. The worldline cannot be a parabola since the coordinate acceleration leads to a limiting velocity, c. The uniform acceleration of a particle in it's own frame (proper acceleration) is constantly changing in an IRF because the relative velocity is changing. When we move from the particle's frame to the IRF, at each moment the particle is in a different ICRF that is moving faster. Now I understand why coordinate acceleration is relative (in magnitude as well as direction).

Good point and more food for thought. My Newtonian physics training tends to lead me to wrong conclusions. One always has to remember the transformations.

 

[CKH]

It doesn't even have a proper time - proper time is the space-time interval between two points, it's not defined for a single point.

Do you mean in the sense that there is no specific proper time defined for the clock? But given some chosen event at which the clock as some particular value, the value on the clock becomes a property of the particle that can be read by any observer. Without choosing such an event we can only talk about intervals of proper time.

An "event" is a point in the 4-dimensional spacetime of an IRF?

 

[Nugatory]

Do you mean in the sense that there is no specific proper time defined for the clock? But given some chosen event at which the clock as some particular value, the value on the clock becomes a property of the particle that can be read by any observer.

Yes, that's saying same thing that I am: proper time is the interval between two points, in this case your chosen event and the point we're talking about. Do not be confused by the fact that we can define the rate of change of proper time at a single point - that's done using differential calculus, where we select two points very near each other and see how the proper time between them varies as we move them ever closer (if you're familiar with the calculus notion of derivatives this will make sense to you - if not, you may have to take my word for it).

An "event" is a point in the 4-dimensional spacetime of an IRF?

The correction I made above is essential: an "event" is a point in four-dimensional spacetime independent of any frame, whether inertial or not. Take the simple case of ordinary flat 2-D Minkowski space, the kind that we represent as a sheet of paper with X and t axes drawn on it. Select a point, any point, on that sheet of paper; it's an event. I can draw other points and speak about their geometrical relationships: These three points form a triangle, these two lines are parallel, and so forth. I can even use a ruler to measure distances between them - I don't have any axes so I don't have any notion of $\Delta{x}$ or $\Delta{y}$, but the ruler measures the (Cartesian, not Minkowski, because that's how rulers on paper work) $\Delta{s}$ directly.

Now suppose I draw a pair of $x$ and $t$ axes on the paper. Nothing has changed about my points and lines distances and geometrical shapes; they still have the same relationships and they don't care whether I've drawn axes or not. But the moment I draw those axes, I have created for myself a convention for assigning $x$ and $t$ values to the various events that were already there. That's choosing a reference frame. Draw another set of axes, presumably with different angles, and I've set up another reference frame that will assign different $x$ and $t$ values but still respects all the geometrical relationships between the points and lines I've drawn on the paper.

The deeper you get into relativity, the more you will focus on the relationships described by frame-invariant quantities like the interval between two events or the proper time along a worldline between two events, and the more you will view the specific choice of frame and coordinates as a choice that make to simplify whatever calculations you have to do in a particular problem.

 

[CKH]

Yes, that's saying same thing that I am: proper time is the interval between two points, in this case your chosen event and the point we're talking about. Do not be confused by the fact that we can define the rate of change of proper time at a single point - that's done using differential calculus, where we select two points very near each other and see how the proper time between them varies as we move them ever closer (if you're familiar with the calculus notion of derivatives this will make sense to you - if not, you may have to take my word for it).

I understand. (I was a math major back in the stone age.)

The correction I made above is essential: an "event" is a point in four-dimensional spacetime independent of any frame, whether inertial or not. Take the simple case of ordinary flat 2-D Minkowski space, the kind that we represent as a sheet of paper with X and t axes drawn on it. Select a point, any point, on that sheet of paper; it's an event. I can draw other points and speak about their geometrical relationships: These three points form a triangle, these two lines are parallel, and so forth. I can even use a ruler to measure distances between them - I don't have any axes so I don't have any notion of $\Delta{x}$ or $\Delta{y}$, but the ruler measures the (Cartesian, not Minkowski, because that's how rulers on paper work) $\Delta{s}$ directly.

The reason I added "spacetime of an IRF" is that in order to talk about a 4-dimensional space we require some coordinate system. A point (event) in that system also exists in other coordinate systems, but I don't see how we can abandon all coordinate systems and say anything about where an event is in space-time.

Now suppose Idraw a pair of $x$ and $t$ axes on the paper. Nothing has changed about my points and lines distances and geometrical shapes; they still have the same relationships and they don't care whether I've drawn axes or not. But the moment I draw those axes, I have created for myself a convention for assigning $x$ and $t$ values to the various events that were already there. That's choosing a reference frame. Draw another set of axes, presumably with different angles, and I've set up another reference frame that will assign different $x$ and $t$ values but still respects all the geometrical relationships between the points and lines I've drawn on the paper.

However, an arbitrary choice of axes is something you can do just once. That act is a choice that represents some particular IRF. Once you draw those axes (orthogonal or not), thereafter you must follow the rules (imposed by invariant c) when drawing axes for other IRFs.

You don't know how to draw another IRF in this spacetime, without knowing it's coordinates in this chosen IRF.

So here's what I meant by "spacetime for an IRF". On a piece of graph paper, you chose a point of origin, you use a horizontal line for x-axis and a vertical line for the t axis. That represents the coordinate system for a particular IRF. So I say by convention, this is the spacetime diagram for that IRF. With these coordinates we can assign a position to an event and draw axes for other IRFs. Without a base set of coordinates we don't how to draw other relative frames.

An "event" is a point in the 4-dimensional spacetime of an IRF?

To define an event we need to choose some coordinate system.

The deeper you get into relativity, the more you will focus on the relationships described by frame-invariant quantities like the interval between two events or the proper time along a worldline between two events, and the more you will view the specific choice of frame and coordinates as a choice that make to simplify whatever calculations you have to do in a particular problem.

Frame invariant quantities are nice.

You mentioned a geometry, independent of frames. I just did a simple example and I think distance measurements are invariant in IRFs. So you can measure distances on the graph (as defined by the base coordinate system/IRF) using a ruler and know that any other IRF will measure the same distance. If you measure t in seconds and x in light seconds, then distances have units of time (in seconds)? What is the meaning/interpretation of these distances?

Suppose I have one graph of spacetime in which the graph paper lines and axes correspond to an IRF S, And then I take another IRF S' moving to the right along the x axes, I can draw a new graph of spacetime in S'. Creating the S' graph requires shearing the original graph along both the x and t axes (by the same amount). In this S' graph, I think the geometric relationships of events as measured on the paper will be the same as before, but all points will be rotated in the new graph (and translated if the origins for S and S' are different). The possible rotation is limited to 45 degrees by c (using seconds and light seconds as units).

 

[Dale]

The reason I added "spacetime of an IRF" is that in order to talk about a 4-dimensional space we require some coordinate system. A point (event) in that system also exists in other coordinate systems, but I don't see how we can abandon all coordinate systems and say anything about where an event is in space-time

Nugatory is correct (he usually is).

In relativity spacetime is descibed by a Riemannian manifold, which consists of a topological manifold and a metric. The metric gives a geometrical structure to the manifold including notions of distance and angles. In the manifold you can define events, lines, volumes, etc. and even determine geometric properties like straightness, curvature, orthogonality, etc. All of this is done with the Riemannian manifold only.

A coordinate chart is an additional structure added on top of the manifold. The coordinates do not define the geometry, in particular, they do not define events, they merely label them.

Carroll has a great set of lecture notes, and Chapter 2 covers these ideas in detail.

 

[CKH]

Riemann's geometry is not in my list of skills.

I don't know how you can work without coordinates. Consider two events devoid of coordinates but suppose we know there is a causal relationship between E1 and E2. I plot these points arbitrarily on a coordinate-less surface with the only rule being that they are a certain distance apart (which is coordinate independent). You cannot then apply an arbitrary set of axes because E1 must precede E2 in all frames.

I'm not sure how using just the notions of distance and angle to construct a measurement system is any different from the usual way. But then again I know nothing about Riemann's geometry.

If it's not too complicated to explain, how does one define an event in a manifold without coordinates?

 

[pervect]

I understand. (I was a math major back in the stone age.)
The reason I added "spacetime of an IRF" is that in order to talk about a 4-dimensional space we require some coordinate system. A point (event) in that system also exists in other coordinate systems, but I don't see how we can abandon all coordinate systems and say anything about where an event is in space-time.

Coordinates are a convenience, but I don't see why you _have_ to have them. Is it your position that you need to supply coordinates for a space-time to exist? My own position (and I believe the position of the post your are responding to) is that a space-time is an abstract object that exists independently of the particular coordinates you may use to describe it.

However, an arbitrary choice of axes is something you can do just once. That act is a choice that represents some particular IRF. Once you draw those axes (orthogonal or not), thereafter you must follow the rules (imposed by invariant c) when drawing axes for other IRFs.

Why? My position would be that you can draw as many coordinate axes as you like, the coordinates are labels. Another way of saying this, a coordinate system is a map of a physical space-time, it's not the space-time itself. Having multiple coordinate systems is like having multiple maps of a physical territory. Much as you can use or discard a map at your convenience, you can use or discard a coordinate system. Nothing physical changes by doing this. Of course you may get confused if you're not careful.

Assuming for now that the coordinate system covers all of space-time (this requirement needs to be dropped if you wish to use coordinate systems that are accelerating, this makes the description a bit harder but doesn't change much important other than adding some qualifiers), then there is a 1:1 correspondence between the events in your abstract space-times, and each set of coordinates that you may use to describe the space-time. This one-one correspondence between events in the physical space-time, and the coordinates in the representation or map of space -time, implies that there is also some 1:1 map between the coordinates in every pairing of coordinate systems that describe that space-time. (If the coordinate systems cover different regions of space-time, the 1:1 correspondence still occurs, but only in the region that is covered by both coordinate systems).

You don't know how to draw another IRF in this spacetime, without knowing it's coordinates in this chosen IRF.

You don't know that, until you impose additional requirements. The details for how to do this are usually based on the abstract framework that I described above, though, with some aditional assumptions.

 

[PeterDonis]

I don't know how you can work without coordinates.

How do you get around on the surface of the Earth without using latitude and longitude? One way is to navigate by known landmarks and relative positions: "turn right at the Shell station, then take the third left, it's the second house on the right". You can do a similar thing in spacetime: label events by what happens at them, or by some relationship to a known event: "Event A is a lightning bolt striking a point on the track at the same instant that the front end of the train passes that point; Event B is a lightning bolt striking a point on the track at the same instant that the rear end of the train passes that point; Event C is the event at which the light beams emitted from those two lightning strikes cross."

Consider two events devoid of coordinates but suppose we know there is a causal relationship between E1 and E2. I plot these points arbitrarily on a coordinate-less surface with the only rule being that they are a certain distance apart (which is coordinate independent).

Distance in space, or "distance" in spacetime? "Interval" would be a better term if you mean the latter (which I think you do).

You cannot then apply an arbitrary set of axes because E1 must precede E2 in all frames.

Yes, but that's not a restriction on the events themselves; it's only a restriction on how you can assign coordinates to them. You can talk about the events and their relationship without using coordinates at all; you just did so in your post.

 

[Dale]

I don't know how you can work without coordinates. Consider two events devoid of coordinates but suppose we know there is a causal relationship between E1 and E2. I plot these points arbitrarily on a coordinate-less surface with the only rule being that they are a certain distance apart (which is coordinate independent). You cannot then apply an arbitrary set of axes because E1 must precede E2 in all frames.

First, in GR it is permitted to have a timelike coordinate which assigns E2 a smaller value than E1. Coordinates just label events, and as long as the labeling is smooth and one-to-one all of the math works out.

Second, why do you need to apply any set of axes? Don't think in terms of coordinates or axes, think in terms of geometry, lines, and distances. For example, I can tell you "my house is at lattitude X and longitude Y" or I can tell you "my house is on DaleSpam road 3.4 km past the intersection with PhysicsForums street". As Peter Donis mentioned, this is actually the more common way to think of navigation in your daily life.

If it's not too complicated to explain, how does one define an event in a manifold without coordinates?

I would strongly recommend reading ch 2 here: http://arxiv.org/abs/gr-qc/9712019

 

[CKH]

Maybe there is a misunderstanding. Yes things of course can exist without coordinates. It's just that you need to measure things in a proper way to apply physical theory. For example in the Lorentz transformations there are quantities that must be measurable in a coordinate system with axes that represent space and time.

(You can do what Peter suggests and follow instructions to a treasure. But that is arbitrary method of measurement as well. Perhaps with a complete set of instructions that take you any point, you can then assign some coordinate system that has the same number of coordinate values as the dimension of the space.

So I'm not sure how there is a fundamental difference, but maybe there is. Like I said, I'm not familiar with Riemann's geometry. Maybe coordinates have a special meaning that is different from a "metric" and angles.)

Why? My position would be that you can draw as many coordinate axes as you like, the coordinates are labels. Another way of saying this, a coordinate system is a map of a physical space-time, it's not the space-time itself. Having multiple coordinate systems is like having multiple maps of a physical territory. Much as you can use or discard a map at your convenience, you can use or discard a coordinate system. Nothing physical changes by doing this. Of course you may get confused if you're not careful.

You can even assign "wiggly" coordinates if you like (e.g. where the axes look like sine waves) or polar coordinates. But these coordinates aren't useful in applying the Lorentz transformations. You are forced to map such coordinates to linear ones that represent actual measurements of time and space to use the LTs (as canonically written).

I was referring in particular to assigning a coordinate system that measures space and time for some IRF (IRFs are the basis for the theory). Given a spacetime (a 4D space, reduced to 2D in this argument), suppose you choose an arbitrary pair of x and t axes and state that these represent space and time. In those coordinates, an object moving at the speed of light along x is a ray which is a bisector (C) of the angle between the x and t axes (using seconds and light seconds). Any other x' and t' axes pair (for any other IRF) is now constrained such that the ray C must be parallel to the to the bisector of x' and t' because in spacetime the speed of light is invariant. I believe the constraint is a bit more extensive to keep the object from moving backward in time, etc. Moreover you cannot have two sets of space and time coordinates using the same units in which a unit on one space/time axis has a different length than on the other space/time axis. It just doesn't make sense when you are measuring the specific quantities we call space and time.

Suppose someone gives me a spacetime in which they have plotted a ray which they say is the inertial trajectory of a massive particle. Such things may exist in any spacetime and they constrain the interpretation of that spacetime using x and t coordinates (space and time). That ray must have positive direction for any time axis for any IRF you care to draw. You cannot draw x and t axes for an IRF such that the ray represents a speed faster than light.

You can of course assign an arbitrary linear coordinate system to a spacetime with axes r and q and another arbitrary system r' and q', but what meaning do they have? How can I apply the LTs to transform from (r,q) to (r',q')? I cannot do SR physics using such coordinates until someone defines at least one pair of linear x and t axes. Such axes are not completely arbitrary for given spacetime. They define what that spacetime means. For example, they define which ordered pairs of events may have a causal relationship and which may not. They define what rays are possible trajectories of massive objects.They define the measurements of space and time needed to apply the theory.

A given spacetime "diagram" in which no space and time axes are defined is physically meaningless. If the things plotted in that spacetime represent reality you are not free to assign any arbitrary space and time axes. Those axis must be consistent with the reality that the diagram is suppose to represent. That does not mean that the choice of x and t axes is fixed, but just that they must physically represent an IRF in that spacetime diagram.

So that's what I meant about coordinate system constraints for space and time coordinates. I wasn't talking about arbitrary units of measurement but specifically about units of space and time.

BTW, I'm not talking about GR which may be more complicated to talk about.

 

[Dale]

I'm not familiar with Riemann's geometry. Maybe coordinates have a special meaning that is different from a "metric" and angles.

It is all explained in that chapter I linked to. I think that the things that you are thinking about as being physically important actually belong to the metric rather than to the coordinates. In particular, the metric is what relates to measurements, not the coordinates.

 

[CKH]

I would strongly recommend reading ch 2 here: http://arxiv.org/abs/gr-qc/9712019

Thanks, I think I can follow that chapter, but it will take some work. The first chapter is a little over my head, the notation is unfamiliar, although I can guess what it means.

But before digesting foundations for GR, I'd like to get SR straight.

 

[Dale]

This is just math, you can apply it to SR or GR. The difference between the two doesn't start until chapter 3 where he introduces curvature. But SR is still a 4D pseudo Riemannian manifold, just a flat one.

GR does not "own" the mathematical techniques.

In any case, that is the formal version of what Nugatory was referring to.

 

[Nugatory]

You can even assign "wiggly" coordinates if you like (e.g. where the axes look like sine waves) or polar coordinates. But these coordinates aren't useful in applying the Lorentz transformations. You are forced to map such coordinates to linear ones that represent actual measurements of time and space to use the LTs (as canonically written).

That's true, but I think you're drawing the wrong conclusion from it. You should instead be concluding that there must be a class of problems, namely all those in which you cannot find such a map, that are not addressed by the Lorentz transformations of special relativity. Indeed, that's why we call it "special" relativity - it works only for the special case of flat spacetime.

It's worth noting that latitude and longitude on the surface of the Earth are "wiggly" coordinates that cannot be mapped to the Minkowski x,y,z coordinates in any way that preserves geometrical relationships. There are a lot of problems like that.

 

[pervect]

Thanks, I think I can follow that chapter, but it will take some work. The first chapter is a little over my head, the notation is unfamiliar, although I can guess what it means.

But before digesting foundations for GR, I'd like to get SR straight.

A good basic plan, I'd recommend understanding the situation in non-wiggly coordinates first, the way it is presented in SR. This implies, though, not too much focus on acceleration, because the natural coordinate system of an accelerated observer is "wiggly". These coordinates not really wiggly in the spatial dimensions, but rather "wiggle" the time dimension. To thouroghly justify that statement I'd have to talk about the metric, which is one of the things I think you're trying to avoid at this point, so I won't clarify the vague general statement and hope it enlightens more than it confuses.

What you can do with the SR approach easily is to describe the coordinates of an accelerating object in an inertial frame, and calculate some basic coordinate independent observations such as proper time and proper acceleration on the accelerated clock, maybe even calculate the proper time of emission and reception of a few radar signals. What you'll be lacking is the "viewpoint" of the accelerated observer, the most direct route to that viewpoint is to learn about metrics and some of the (non-tensor) basics of how they can be used to deal with "wiggly" coordinate systems, but that can wait until after you learn about the non-wiggly coordinate approach in SR.

 

[CKH]

Yes. I would prefer to master the flat cases before think about others. My basic questions about acceleration in SR have been answered. After acquiring a sound understander of SR. It would be interesting to attempt to analyze the case of acceleration and see why these metrics are needed.

Dale,

This is tangential but I'm curious. Manifolds can be thought of as surfaces in a higher dimensional space. Can all manifolds be represented in a higher dimensional flat space? I saw a comment in a wiki article that suggested that a Klein bottle (a 2D manifold) cannot be represented in a 3D space but a 4D space does the trick. I suppose this is because the surface intersection in a Klein bottle in 3D leads to an ambiguous tangent plane.

 

[Dale]

Manifolds can be thought of as surfaces in a higher dimensional space. Can all manifolds be represented in a higher dimensional flat space?

Yes, there are several "embedding theorems" about how many dimensions are needed.