Question about length contraction

[tom.capizzi]

When a moving observer views a stationary meter stick, it appears contracted. The stationary observer measures it at full length. Similarly, the stationary observer views a moving meter stick as contracted, and the moving observer views it as full length. This is conventional relativity. When a stationary observer views a point travel between two endpoints, is it not fair to deem this distance as contracted as well?
At this point I wish to make clear that we are talking about the path (which depends on the velocity) as opposed to the separation (which is determined by an observer who is stationary). The example from Thermodynamics of internal energy, E, will help clarify this distinction. Delta E is a state variable, and as such only depends on the beginning and end state of the system. The definition of delta E is heat minus work. In differentials, dE = dq - dw. The quantities of heat and work depend entirely on the path that the system takes to get from beginning to end. The only stipulation is that the difference is independent of path.
With respect to relativity, the actual path traveled depends on the velocity, while the separation between endpoints does not. If the actual path is expanded with velocity by the same factor gamma as the meter stick is contracted, then the contracted length of the path becomes equal to the separation. The significance of this interpretation is the actual velocity is also greater than the observed velocity by the same factor gamma. This results in greater momentum than expected by the same factor gamma, which is in agreement with experimental results. Einstein invented relativistic mass increase with velocity to account for this departure from Newtonian physics. This interpretation restores mass to the status of relativistic invariant. It also changes the speed of light into an asymptote that observable velocity approaches while actual velocity approaches infinity.
Does anyone have a good reason not to pursue this line of reasoning?

 

[ghwellsjr]

Yes, it is in violation of one of the rules--see the section entitled Overly Speculative Posts". You can post this on the "Independent Research Forum".

 

[tom.capizzi]

I disagree that it is overly speculative. After all, if the moving observer were carrying a stick gamma meters long, it would appear to be only 1 meter to the stationary observer. If the moving observer travels gamma meters, then the stationary observer sees him travel 1 meter. Same principle. Is that speculation? On the other hand, I have no objection to posting on the "Independent Research Forum". I just think it is appropriate here as well.

 

[JesseM]

When a moving observer views a stationary meter stick, it appears contracted. The stationary observer measures it at full length. Similarly, the stationary observer views a moving meter stick as contracted, and the moving observer views it as full length. This is conventional relativity. When a stationary observer views a point travel between two endpoints, is it not fair to deem this distance as contracted as well?

Are the endpoints actual physical objects with their own rest frame, or are they just the events of the point beginning and ending its journey? In the former case the distance between these objects in the frame where they're moving is just given by the regular length contraction equation, in the latter case I don't see what you are saying the distance is "contracted" relative to, although you can compare the distance between the events in different frames using the Lorentz transformation.

At this point I wish to make clear that we are talking about the path (which depends on the velocity) as opposed to the separation (which is determined by an observer who is stationary).

Not sure what you mean when you say the path "depends on the velocity". Can you give some sort of numerical example? For example, suppose there are two planets moving at 0.6c in my frame, and the distance between them is 10 light-years in my frame, and I see an object travel between them with a speed of 0.8c in my frame, so its closing speed relative to the front planet is only 0.8c - 0.6c = 0.2c meaning it takes 10/0.2c = 50 years to get from one planet to the other in my frame, during which time it travels 50*0.8c=40 light-years in my frame. Are you defining the path length as the 40 light-years it traveled in my frame, or the 10 light-years between the planets in my frame, or something else? If it's 40 light years, by "depends on the velocity" do you just mean it depends on the planets' speed of 0.8c and the object's speed of 0.6c.

The example from Thermodynamics of internal energy, E, will help clarify this distinction. Delta E is a state variable, and as such only depends on the beginning and end state of the system. The definition of delta E is heat minus work. In differentials, dE = dq - dw. The quantities of heat and work depend entirely on the path that the system takes to get from beginning to end. The only stipulation is that the difference is independent of path.

Here we have a specific quantity, energy, where we are measuring the difference at the endpoints. What specific quantity are you talking about in your relativity discussion? Distance? In the thermodynamic example the difference in energy is only independent of the path in a specific frame, the difference will not be the same in different frames since energy is frame-dependent (this is easiest to see with kinetic energy but I believe the same would be true of electromagnetic potential energy in different frames). Are you saying there is some quantity that should be frame-independent for a given path?

With respect to relativity, the actual path traveled depends on the velocity, while the separation between endpoints does not. If the actual path is expanded with velocity by the same factor gamma as the meter stick is contracted, then the contracted length of the path becomes equal to the separation.

That doesn't seem to make any sense, see my example above where the distance traveled by the object is 40 light-years in my frame whereas the separation between planets is only 10 light-years in my frame (and since they are moving at 0.6c in my frame, the distance between the planets in their own rest frame would be 12.5 light-years). Again it would really help if you gave some numbers to illustrate what you're saying, not just words. Also, we can only meaningfully talk about distances within the context of some coordinate system, are your statements supposed to make sense in the context of the inertial frames of SR given by the Lorentz transformation, or are you trying to suggest we use some different non-inertial form of coordinate system?

The significance of this interpretation is the actual velocity is also greater than the observed velocity by the same factor gamma.

What does "actual velocity" mean? Again "velocity" is a concept that only makes sense relative to some coordinate system, you have to pick a coordinate system and then the velocity will be (difference in position coordinate)/(difference in time coordinate) for the coordinates of the start and end of the journey (assuming the object was traveling at constant velocity).

Einstein invented relativistic mass increase with velocity to account for this departure from Newtonian physics.

I don't think Einstein introduced the notion of "relativistic mass", his 1905 papers only talk about the traditional rest mass, and most physicists don't like to use the notion of "relativistic mass" nowadays because it can be confusing to students and isn't really needed to do physics (see the discussion of the history at http://en.wikipedia.org/wiki/Mass_in_special_relativity#The_relativistic_mass_concept)

This interpretation restores mass to the status of relativistic invariant.

Rest mass already is an invariant, and as I said it's the only concept of "mass" that most physicists nowadays use.

It also changes the speed of light into an asymptote that observable velocity approaches while actual velocity approaches infinity.

Again, in what coordinate system?

 

[tom.capizzi]

In reply to JesseM, I refer to the quoted snippets by number in the interest of brevity:

1) I don't understand why you choose to distinguish between two actual objects and two events. In any case, I am not talking about the distance between two points in the frame in which they are moving. Nor am I talking about frames moving at different velocities that can be compared using the Lorentz transform. I am saying that the distance measured by the stationary observer is a contracted observation. The actual distance is greater than this by the traditional factor gamma, which depends on how fast the object is moving in the stationary frame.
2) In general terms, by path length depends on velocity I mean the distance actually traveled increases with velocity by the factor gamma. Specifically, per your numbers, it is neither 40 ly nor 10 ly. Without the complication of two moving planets, an object which appears to travel 40 ly at 0.8 c in the stationary frame is really going farther and faster than it appears. The gamma factor for 0.8 c is 1.67, so the actual path is 66.7 ly and the actual velocity is 1.33 c. Only the cosine projection of these quantities is measurable by the stationary observer, hence the appearance of 40 ly at 0.8 c.
3) The specific quantity I refer to is the separation between two points in the stationary frame. This is distinguished from the distance traveled, which varies with velocity. Perhaps I should have used a different analogy to illustrate the point. The separation compares to the straight line distance between two points on a map, while the distance traveled compares to the route chosen, which is completely arbitrary.
4) I am not suggesting that we use non-inertial coordinate systems.
5) I refer to "actual velocity" as the velocity parallel to the actual path, as opposed to the apparent velocity measured along the projection of the path which is parallel to the straight line between the endpoints.
6) Apparently someone else is responsible for the notion of mass increase with velocity.
7) We are in agreement about rest mass. I referred to the relativistic mass increase with velocity.
8) I refer to the coordinate system of the stationary observer.

I believe some of the confusion in terminology stems from the Euclidean concept that two points determine a unique line, with a unique distance between them. My point is that spacetime is non-Euclidean, and two points can determine many lines that connect them. Only one of these is the shortest (if I remember correctly). The others are followed by moving objects - the faster the motion the longer the path.

 

[ghwellsjr]

I hope you understand that when an object accelerates, its dimensions along the direction of acceleration actually, physically, change. It's not an artifact of the way it is measured. And when an object accelerates, its clocks and all other time related factors, actually and physically change their tick rates and aging rates. It's not an artifact of the way it is measured. These changes are true for all frames of reference from which the measurements are made. What is not true is that dimensions and time change just because they are viewed or analyzed from a different reference frame. So when you talk about "actual distance" or "actually travelled" or "actual velocity" compared to the measured versions of those items, you are making some assumptions that aren't warranted.

 

[tom.capizzi]

To ghwellsjr:

First, I am not referring to accelerated frames in any way. Special Relativity presumes constant velocity, which implies unaccelerated frames. Second, your description of distance and time changing (or not changing) depending on the frame of reference does not ring true according to relativity. Specifically, although there is no way to tell which observer is moving and which observer is stationary, a scenario could be constructed in which we know with certainty who is standing still. Both observers "see" the other meter stick as contracted, but only one stick is actually moving. Relative to each other, each sees the other as moving, but we know from the history of the system which is right. Only one meter stick is contracted due to its own actual motion. The other one appears contracted even though it is not moving, due to the motion of the observer's frame. Finally, when I use the term "actual distance", I refer to the integral of length along a path as opposed to the length of a straight line between two points. As I mentioned above, these two concepts are only synonymous in Euclidean geometry. In other geometries, more than one straight line can be determined by two points, and only one of these is the shortest. If "actual distance" is the length of one of these other lines, then it does not equal the measured distance (which I assume is the shortest).

 

[JesseM]

In reply to JesseM, I refer to the quoted snippets by number in the interest of brevity:

1) I don't understand why you choose to distinguish between two actual objects and two events.

Because events are localized to a single point in space and time, whereas objects are localized in space but not in time (they have extended worldlines with different positions at different times). Also, by convention the "distance between objects" in a given frame is always understood to mean their coordinate separation at a single moment of coordinate time, whereas we can talk about the spatial separation between events even if the events occurred at different times in the frame we're using.

In any case, I am not talking about the distance between two points in the frame in which they are moving.

For the reason above, it's better not to use the word "points" if you don't specify whether you're talking about point particles that exist for an extended period, or points in spacetime (i.e. events). But since you talk about the points "moving" I guess you must mean point particles.

In any case, I am not talking about the distance between two points in the frame in which they are moving.

So are you talking about the distance between two point particles in the frame where they are at rest?

I am saying that the distance measured by the stationary observer is a contracted observation.

By "distance measured by the stationary observer" do you just mean the spatial separation between the point particles at a single time-coordinate in the stationary frame? And again, are you defining "stationary frame" to mean the frame where the point particles are at rest?

The actual distance is greater than this by the traditional factor gamma, which depends on how fast the object is moving in the stationary frame.

But you still haven't defined what you mean by "actual distance" in any clear manner, you need to pick a frame, pick some physical events, and show that the coordinate separation between them in that frame corresponds to your notion of "actual distance". Also, what "object" is moving" in the stationary frame? In the above I thought you were talking about a pair of point particles at rest in the stationary frame. Are you talking about a third point particle traveling between the two stationary point particles?

2) In general terms, by path length depends on velocity I mean the distance actually traveled increases with velocity by the factor gamma.

And what is "distance actually traveled"? Is it just the difference between the position coordinate of the start of the journey and the position coordinate of the end of the journey?

Specifically, per your numbers, it is neither 40 ly nor 10 ly. Without the complication of two moving planets, an object which appears to travel 40 ly at 0.8 c in the stationary frame is really going farther and faster than it appears.

"Really" going faster in what coordinate system? It's meaningless to talk about speed unless you have a particular coordinate system in mind, and if an object has a speed of 0.8c in the stationary frame then by definition it is "really" moving at 0.8c in the stationary frame. Do you disagree with any of that?

3) The specific quantity I refer to is the separation between two points in the stationary frame. This is distinguished from the distance traveled, which varies with velocity. Perhaps I should have used a different analogy to illustrate the point. The separation compares to the straight line distance between two points on a map, while the distance traveled compares to the route chosen, which is completely arbitrary.

So if a ship travels in a straight line between two planets, would you say that in the rest frame of the planets the "separation between" the planets is equal to the "distance traveled" by the ship from one planet to another? If not I have no idea what you mean by "distance traveled".

5) I refer to "actual velocity" as the velocity parallel to the actual path, as opposed to the apparent velocity measured along the projection of the path which is parallel to the straight line between the endpoints.

No idea what you mean by that. It would help if you'd give a mathematical example--for example, if in our frame we have two planets, one at rest at x=0 light years and the other at rest at x=20 light-years, and we also have a ship traveling between them with worldline given by x(t) = 0.8c*t (so at t=0 it is at the first planet's position at x=0, then it travels towards the other planet with a coordinate velocity of 0.8c and reaches it at t=25 years), then can you show how to calculate both the "velocity parallel to the actual path' and the "apparent velocity measured along the projection of the path which is parallel to the straight line between the endpoints"? Please make sure not to just arbitrarily declare that one velocity is 0.8c multiplied by (or divided by) the Lorentz factor, actually show mathematically how notions like "velocity parallel to the actual path" and "the projection of the path which is parallel to the straight line between the endpoints" apply in this example, and how calculations involving these specific notions actually yield whatever final numbers you get for each type of velocity.

I believe some of the confusion in terminology stems from the Euclidean concept that two points determine a unique line, with a unique distance between them. My point is that spacetime is non-Euclidean, and two points can determine many lines that connect them. Only one of these is the shortest (if I remember correctly). The others are followed by moving objects - the faster the motion the longer the path.

Two events, i.e. two "points" in spacetime, do determine a unique straight line in spacetime between them, but two point particles are actually extended worldlines and one can pick different events on each worldline to draw a straight line in spacetime between them. Are you familiar with the notion of the spacetime interval? If the separation between the two events is spacelike, then the "straight line between them" will have a spatial distance, but no object could actually have this straight line as a worldline unless it traveled faster than light (though I suppose you could treat this spacelike path as a surface of simultaneity and figure out what velocity an object would need in order for that to be the surface of simultaneity in the object's rest frame). On the other hand, if the separation is timelike (as would be true for the events of a ship departing one planet and arriving at another) then the length of the "straight line between them" (the ship's worldline) will be measured in units of time, not distance. And if the separation is light-like, then the length of the "straight line between them" is zero, this is the only sense I can think of in which your statement "Only one of these is the shortest" would be correct.

 

[ghwellsjr]

To ghwellsjr:

First, I am not referring to accelerated frames in any way.

I never mentioned accelerated frames either so I'm glad we concur on that point.

Special Relativity presumes constant velocity, which implies unaccelerated frames.

SR, the way Einstein taught it and the way I learned it only discusses unaccelerated "inertial" frames but nowadays people like to talk about accelerated frames, for which you have to be a real expert at to not get into trouble. However, SR permits objects to accelerate in any way you like. So I think we concur here too.

Second, your description of distance and time changing (or not changing) depending on the frame of reference does not ring true according to relativity.

My only point with regard to viewing objects from different reference frames is to affirm that no real change occurs simply by changing to a different frame. I think we concur here.

Specifically, although there is no way to tell which observer is moving and which observer is stationary, a scenario could be constructed in which we know with certainty who is standing still.

This sure looks like a self-contradictory statement. I have no idea what you mean here.

Both observers "see" the other meter stick as contracted, but only one stick is actually moving. Relative to each other, each sees the other as moving, but we know from the history of the system which is right.

Again, I have no idea what you are saying here. I just can't figure out what you mean by the terms "actually" and "which is right". These are not terms used in SR and this is why I said your thread does not belong on this forum.

Only one meter stick is contracted due to its own actual motion. The other one appears contracted even though it is not moving, due to the motion of the observer's frame.

I just have trouble with your term "actual".

Finally, when I use the term "actual distance", I refer to the integral of length along a path as opposed to the length of a straight line between two points. As I mentioned above, these two concepts are only synonymous in Euclidean geometry. In other geometries, more than one straight line can be determined by two points, and only one of these is the shortest. If "actual distance" is the length of one of these other lines, then it does not equal the measured distance (which I assume is the shortest).

I just don't know what you are talking about here, sorry.

 

[Passionflower]

tom.capizzi is your topic perhaps about the difference between velocity and proper velocity or celerity?

The difference is that celerity takes the coordinate distance divided by proper time instead of proper distance divided by proper time. While velocity is bounded celerity is unbounded.

 

[tom.capizzi]

Reply to JesseM:

Thank you for the clarification. However, the usage of "point" is neither definition you offer. I simply refer to a grid point in a coordinate system. It is not a particle, nor is it an event, since it has duration for as long as I care to define the coordinate system. If I choose a particular instant of time, then I suppose it could be an event. I have assumed that a distance measurement is made according to the constraints of SR, that is at the same time at both endpoints.
As to "actual distance", please refer to my other recent post describing loxodrome geometry. This geometry shares enough mathematical properties with spacetime to illustrate what I mean. By the way, it also shows why I disagree that "... an object has a speed of 0.8c in the stationary frame ... is 'really' moving at 0.8c".

 

[tom.capizzi]

To ghwellsjr:

Regarding your last post, the first point of disagreement concerns what you call a "self-contradictory statement."
Relativity asserts that there is no experiment or measurement that an observer in an isolated system moving at constant velocity can make that will reveal the absolute velocity.
So if two observers in such systems pass each other, neither one can tell who is moving and who is standing still. They can only agree on the relative velocity between them. My point was simply that if one frame is a laboratory on Earth and the other frame is a moving rocket ship, an observer outside of the two systems can easily distinguish which frame is moving even if not present at the instant the ship took off.
If we ask each observer to describe the other's meter stick, each will claim the other stick is contracted. However, only one of them is contracted because of its own motion. The other appears contracted because of the motion of the observer. Relativity says there is no way to distinguish between these two observations, but one is a physical contraction and the other is an artifact.
To clarify for you, with respect to the example above, "actually moving" refers to the rocket ship frame. Similarly, "which is right" refers to the lab observer who claims the other one is moving. I admit that this does not prove that the rocket ship is the moving frame, but it would be extremely absurd to think that the Earth would take off, leaving the rocket ship standing still, at the precise instant that the engines were ignited.
Finally, let me try to clarify "actual distance" with a specific example. Consider the geometric figure known as a loxodrome. Rather than elaborate all its properties here, I offer several keywords to google: loxodrome, rhumb line, gudermannian, and Mercator projection. It is a spiral on the surface of a sphere that intersects all longitude lines at the same tilt angle. The poles represent two stationary points in the discussion. The separation between them is the length of the diameter (2 * radius). The length of a loxodrome of zero tilt is the length of a longitude line, Pi * radius. A loxodrome of any other tilt angle also connects the two poles together, but the length of such a spiral is gamma * Pi * radius. I use gamma intentionally, because this factor is related to the tilt angle in the same way in Special Relativity, where beta is the sine of an angle, and gamma is the secant of the same angle and v = beta * c. In any case, the point is that even though it has the same two endpoints, the "actual length" is gamma times the length of a longitude line. Since the length of the spiral is linearly proportional to radius, two spirals with the same tilt and half the diameter placed end-to-end will have the same total arc length as one full size spiral. In the limit of an infinite number of infinitesimal spirals, the result is a line with no thickness, a length equal to the diameter, and an "actual" total length that depends on the secant of the tilt angle. The "actual distance" in our discussion would be 2 * radius of a sphere with a longitude arc of length = gamma * Pi * radius. If the diameter is 1 meter, then we have a meter stick that is "actually" gamma meters long. If it is "actually" 1 meter, then it appears to be contracted to 1/gamma meters, depending on the tilt angle of the spiral.

 

[tom.capizzi]

Reply to Passionflower:

The calculations produce similar results, but my question revolves around the possibility that a stationary observer cannot see the full extent of the distance traveled by a moving object. Therefore, the stationary observer cannot see the full extent of velocity also. Please refer to my recent reply describing loxodrome geometry as to how this could be.

 

[JesseM]

Reply to JesseM:

Thank you for the clarification. However, the usage of "point" is neither definition you offer. I simply refer to a grid point in a coordinate system. It is not a particle, nor is it an event, since it has duration for as long as I care to define the coordinate system.

OK, but then for all practical purposes it's just like a point particle, since we're just dealing with thought-experiments here so we're free to define a given point particle's position as being constant in some coordinate system, whether there's an actual particle there or it's just a "grid point" makes zero difference to the math. So one can still talk about the "worldline" of such a point, and then what I said at the end of my post about there being different straight-line paths between two worldlines still applies, it would help if you'd respond to that part at least (in particular tell me whether you understand what the spacetime interval is, and the difference between space-like, time-like and light-like intervals).

As to "actual distance", please refer to my other recent post describing loxodrome geometry. This geometry shares enough mathematical properties with spacetime to illustrate what I mean.

Doesn't help, sorry. There is nothing mysterious about what we mean when we talk about the "length" of a loxodrome, we are talking about a path length on a sphere, which might be defined using a coordinate system on the sphere along with the appropriate spherical metric that gives a line element which can be integrated along a path with known coordinates to give a coordinate-independent notion of the "length" of that path. Spacetime has a metric and line element which give a coordinate-independent notion of "spacetime distance" too, but clearly the notion of "distance" you are talking about does not correspond to this, since for inertial paths in flat spacetime the "spacetime distance" found by integrating the line element would be nothing more than the spacetime interval I mentioned before, which as I said gives an answer in units of time for timelike paths and an answer in units of distance for space-like paths (along with a distance of zero along light-like paths). Again, you really need to give some mathematical example of how you derive the "actual distance" in the context of special relativity, not just an analogy.

By the way, it also shows why I disagree that "... an object has a speed of 0.8c in the stationary frame ... is 'really' moving at 0.8c".

I didn't say "really moving at 0.8c", I said "really moving at 0.8c in the stationary frame", obviously all notions of speed are frame-dependent. Do you disagree that "speed" can only be defined relative to a given frame, and that speed in any frame is just (change in position)/(change in time) in the coordinates of that frame? For example, if in inertial frame A an object is moving inertially between two planets, and the event of its departing the first planet is at coordinates (x=0 light years, t=0 years) in frame A while the event of it reaching the second planet is at coordinates (x=8 light years, t=10 years), then by definition the object's speed in frame A is (8 - 0 light years)/(10 - 0 years) = 0.8c?

 

[Mentz114]

... but my question revolves around the possibility that a stationary observer cannot see the full extent of the distance traveled by a moving object. Therefore, the stationary observer cannot see the full extent of velocity also. Please refer to my recent reply describing loxodrome geometry as to how this could be.

Allowing you to nominate some inertial observer as your 'stationary' frame, I still can't make a lot of sense of what follows

... cannot see the full extent of the distance traveled

Distance traveled is defined how ? In a unit of time ? If the latter, on whose clock ? What does 'see' mean here ? Do you mean measure ?

It gets worse

the stationary observer cannot see the full extent of velocity

Velocity does not have an 'extent'. A material body may have 'extent'. Do you mean that this observer cannot measure the relative velocity, or that the measured relative velocity is not 'true' in some way ?

I have to admit that you have baffled me.

 

[ghwellsjr]

To ghwellsjr:

Regarding your last post, the first point of disagreement concerns what you call a "self-contradictory statement."
Relativity asserts that there is no experiment or measurement that an observer in an isolated system moving at constant velocity can make that will reveal the absolute velocity.
So if two observers in such systems pass each other, neither one can tell who is moving and who is standing still. They can only agree on the relative velocity between them. My point was simply that if one frame is a laboratory on Earth and the other frame is a moving rocket ship, an observer outside of the two systems can easily distinguish which frame is moving even if not present at the instant the ship took off.
If we ask each observer to describe the other's meter stick, each will claim the other stick is contracted. However, only one of them is contracted because of its own motion. The other appears contracted because of the motion of the observer. Relativity says there is no way to distinguish between these two observations, but one is a physical contraction and the other is an artifact.
To clarify for you, with respect to the example above, "actually moving" refers to the rocket ship frame. Similarly, "which is right" refers to the lab observer who claims the other one is moving. I admit that this does not prove that the rocket ship is the moving frame, but it would be extremely absurd to think that the Earth would take off, leaving the rocket ship standing still, at the precise instant that the engines were ignited.
Finally, let me try to clarify "actual distance" with a specific example. Consider the geometric figure known as a loxodrome. Rather than elaborate all its properties here, I offer several keywords to google: loxodrome, rhumb line, gudermannian, and Mercator projection. It is a spiral on the surface of a sphere that intersects all longitude lines at the same tilt angle. The poles represent two stationary points in the discussion. The separation between them is the length of the diameter (2 * radius). The length of a loxodrome of zero tilt is the length of a longitude line, Pi * radius. A loxodrome of any other tilt angle also connects the two poles together, but the length of such a spiral is gamma * Pi * radius. I use gamma intentionally, because this factor is related to the tilt angle in the same way in Special Relativity, where beta is the sine of an angle, and gamma is the secant of the same angle and v = beta * c. In any case, the point is that even though it has the same two endpoints, the "actual length" is gamma times the length of a longitude line. Since the length of the spiral is linearly proportional to radius, two spirals with the same tilt and half the diameter placed end-to-end will have the same total arc length as one full size spiral. In the limit of an infinite number of infinitesimal spirals, the result is a line with no thickness, a length equal to the diameter, and an "actual" total length that depends on the secant of the tilt angle. The "actual distance" in our discussion would be 2 * radius of a sphere with a longitude arc of length = gamma * Pi * radius. If the diameter is 1 meter, then we have a meter stick that is "actually" gamma meters long. If it is "actually" 1 meter, then it appears to be contracted to 1/gamma meters, depending on the tilt angle of the spiral.

Tom, you understand, don't you, that your so-called laboratory frame on the surface of the Earth is in motion and not even in inertial motion? It is constantly accelerating, changing it speed and direction of motion. It is not an inertial frame.

In fact the Michelson-Morley experiment was in such a laboratory when they were trying to measure this change in motion on a daily and yearly basis. If it were already an inertial frame, they were wasting their time because obviously, that would explain the null result. And everybody would be drawing a false conclusion that the MMX was such a dramatic experiment.

And your spaceship is also not an inertial frame, at least not while it is firing its rockets.

So your statement:

"My point was simply that if one frame is a laboratory on Earth and the other frame is a moving rocket ship, an observer outside of the two systems can easily distinguish which frame is moving even if not present at the instant the ship took off."

begs the question. In SR, we consider any inertial (non-accelerating) frame to be stationary but we consider only one frame at a time. Your observer can analyze both the rocket ship and the laboratory from his frame of reference as long as he is not accelerating.

And your statement:

"However, only one of them is contracted because of its own motion. The other appears contracted because of the motion of the observer. Relativity says there is no way to distinguish between these two observations, but one is a physical contraction and the other is an artifact."

is wishful thinking. Whatever you want to say about length contraction of a moving object as viewed from any frame with regard to your terms "physical" and "artifact", it is the same for all objects as viewed from all frames. You cannot give priority to one frame's view over another frame's view. And as I pointed out earlier, whenever any object accelerates, it experiences a real, actual, physical change in its dimensions along the direction of the acceleration, it is not an artifact or a result of merely the way in which it is viewed.

 

[tom.capizzi]

It seems that my usage of the English language as opposed to Special Relativese has caused the focus of this thread to drift away from my original question. It also seems that I can never be precise enough with terminology to satisfy everyone. With so many tangents to follow it allows the substance to be ignored. When I suggested the Earth as the lab frame it was pointed out that the Earth rotates, etc. When I described the loxodrome, it was admitted that the length could be measured, but the fact that the length varies with tilt angle while the diameter remains constant was ignored. Or that a string of infinitesimal spirals could be indistinguishable from a straight line, hence "actually" be longer than the line itself.
So, can we consider one specific point? First, can we agree that an inertial frame exists? If so, let's call it the lab frame, and place it anywhere in the universe that you choose. A bunch of observers are in the lab with identical meter sticks. None of these observers are moving relative to each other, so they all see each other's meter stick as uncontracted. Each observer is paired with another observer who takes off into space with a meter stick, each moving at a different velocity. At some time later, all moving observers pass by the lab at the same time. Each pair of observers sees the partner's meter stick contracted according to SR. The question is, are the meter sticks in the lab actually physically contracted? If so, how much? Remember, all the lab observers agree that the lab meter sticks are the same.
Or do they just look that way? And if SR says that we can't tell the difference between the pairs of meter sticks, how do we know that the moving stick is physically contracted?

 

[ghwellsjr]

First, we can agree that an infinite number of inertial frames exist (because they are in our minds, they are not physical) and they all extend physically to infinity in all directions and they all encompass time from eternity past to eternity future and they all contain all the objects in the universe, or if we are talking about a thought problem, they all contain all the objects, labs, spaceships, rulers, clocks, observers, and anything else we want to consider.

So now we arbitrarily pick anyone inertial reference frame. You can call it the lab frame because in it you place a lab that is at rest in the lab reference frame. Then in that lab you place any number of stationary observers with any number of meter sticks that are defined to be the same length. Then if you accelerate any number of those meter sticks, they will each contract physically along the direction of acceleration, in the lab reference frame that you have defined. Now if you have an observer that is stationary in the lab reference frame, he will observe those moving meter sticks to be shortened. However, an observer with his own meter stick traveling with another traveling meter stick will discover that they are the same length and will not realize that they are both shortened in the lab reference frame. If a traveler observes a meter stick that is stationary in the lab reference frame, he will determine that the stationary one is shorter than his moving one.

If we transform the entire scenario from your previously defined lab frame to another frame, say one in which an observer is now traveling at a high speed, his meter stick will now be considered to be the correct length and the ones in the lab are now short, but this is not a physical change in any lengths. So we cannot conclude absolutely whose meter sticks are absolutely the correct lengths but we can conclude that every meter stick that is accelerating will be physically changing its length because that is what happens no matter which reference frame we chose to analyze any situation.

 

[Mentz114]

The question is, are the meter sticks in the lab actually physically contracted? If so, how much? Remember, all the lab observers agree that the lab meter sticks are the same.
Or do they just look that way? And if SR says that we can't tell the difference between the pairs of meter sticks, how do we know that the moving stick is physically contracted?

No, they are not actually contracted. The meter stick in question would have many 'actual' lengths if this were the case. Any meter stick has one proper length only - as measured in a comoving frame. The fact that an observer can see a meter stick contracted in no way affects the meter stick.

"Relativity is something that happens to other people"

 

[tom.capizzi]

ghwellsjr said:

"... Then if you accelerate any number of those meter sticks, they will each contract physically along the direction of acceleration,..."

and

"... his meter stick will now be considered to be the correct length and the ones in the lab are now short, but this is not a physical change in any lengths. ..."

It seems we are in agreement after all. However, you have not answered this question.
If we can't distinguish between a physical contraction in the direction of acceleration and the shortening of the lab meter sticks as viewed by the moving observer, how do we know that the contraction of the accelerating stick is actually physical?

 

[tom.capizzi]

reply to Mentz114:

Agreed. However, you did not answer the rest of the question. First, do you agree with ghwellsjr that the moving meter sticks are actually physically contracted? Personally, I'm not sure I believe that anymore. So I ask again, how do we know that the moving sticks are physically contracted?

 

[JesseM]

If we can't distinguish between a physical contraction in the direction of acceleration and the shortening of the lab meter sticks as viewed by the moving observer, how do we know that the contraction of the accelerating stick is actually physical?

Can you define what you mean by "actually physical"? If you're asking whether there's a coordinate-invariant sense in which the length is shorter, for example, then the answer is no, any measurement of the "length" of a physical object like a ruler depends on your choice of simultaneity convention (though you can define a coordinate-invariant notion of 'length' along a space-like worldline, akin to the coordinate-invariant notion of 'proper time' along time-like worldlines)

 

[Mentz114]

reply to Mentz114:

Agreed. However, you did not answer the rest of the question. First, do you agree with ghwellsjr that the moving meter sticks are actually physically contracted? Personally, I'm not sure I believe that anymore. So I ask again, how do we know that the moving sticks are physically contracted?

I thought my post was unambiguous - nothing is affected in it's rest frame by moving observers' observations. A meter stick only has one length.

I don't know what ghwellsjr means when says that lengths are contracted and clocks slowed down by acceleration.

 

[tom.capizzi]

Can you define what you mean by "actually physical"? ...

I was referring to a statement made by ghwellsjr that "... those meter sticks ... will each contract physically along the direction of acceleration,..."

You may delete the word "actually" if it confuses the issue. From the rest of your reply, are you saying that meter sticks do not contract physically?

 

[JesseM]

Can you define what you mean by "actually physical"?

I was referring to a statement made by ghwellsjr that "... those meter sticks ... will each contract physically along the direction of acceleration,..."

I don't understand what ghwellsjr means by "physically" in that sentence. What is your definition of "physical"? Can a coordinate-dependent fact, like the statement that object A has a greater speed than object B in some frame, or that object A has a greater x-coordinate than object B at time t, be "physical"?

 

[tom.capizzi]

reply to Mentz114:

Your post was unambiguous, but incomplete. Disregard the question about ghwellsjr. The other issue still stands. Do the moving meter sticks physically contract when they move? We are not in the frame of the moving object this time, and whether an observer is present or not is irrelevant.

 

[Mentz114]

Do the moving meter sticks physically contract when they move? We are not in the frame of the moving object this time, and whether an observer is present or not is irrelevant.

If we are talking about a set of inertial frames ( I prefer to call then inertial observers) then there is no way to decide who is 'moving'. (If you use the history of the observers to make such a decision, it is no longer SR.) So the question 'does the moving meter stick contract' has no meaning. In their own frame, not one of them is moving, but viewed from another frame, they might be.

If I happen to see a meter stick whizzing by I might decide it was shorter than 1m, but my state-of-mind will not affect the meter stick one bit.

If an observer accelerates, their velocity relative to other observers will change, and objects may experience stresses, but any relativistic effects are purely the result of instantaneous relative velocity.

I endorse what JesseM says above. What do you mean by 'physical' ?

 

[tom.capizzi]

I looked it up on the internet.

According to Einstein, length contraction of a moving body is a real, physical, actual phenomenon, not merely a visual artifact. He proposed a somewhat complex experiment involving a train and multiple stationary observers to illustrate this claim. It is rather difficult to confirm by direct experiment, as opposed to time dilation. Although two observers in relative motion can each claim the other one's clock is running slow, the moving clock actually loses time. When the moving clock is returned to the frame of the stationary clock, it still shows lost time (the twin paradox?). The details are still a little fuzzy, but if the speed of light is invariant, and time dilation is real, then length contraction is physically required to maintain correct velocity. Still, without knowledge of the history of the system, neither observer can determine which one is actually moving (assuming that one of them was not moving in the first place).

I'm not yet sure how this affects my original question. I like to associate velocity with a rotation that projects part of an object's length out of normal spacetime, resulting in contraction. Since the rotation angle back to the stationary frame is equal and opposite, the moving observer gets the same view of the stationary object. Even though the moving object may measure as contracted, the total length never changes, as is apparent to the moving observer, who is at zero degrees rotation relative to the moving object. But if a moving object/frame/observer has a phase angle attached to it, all of the observations from the (relatively) stationary frame are projections, including path length. Hence, my original question. However, at this point it seems unlikely to be resolved here.

Thank you all for your comments.

 

[JesseM]

I looked it up on the internet.

Where, specifically?

According to Einstein, length contraction of a moving body is a real, physical, actual phenomenon, not merely a visual artifact.

Did Einstein use the words "real", "physical", or "actual"? If so, where? Length contraction isn't a "visual artifact" (in fact moving objects don't always appear contracted visually, sometimes they appear expanded depending on the direction relative to the viewer), but it is dependent on your choice of coordinate system, or at least your choice of simultaneity convention.

He proposed a somewhat complex experiment involving a train and multiple stationary observers to illustrate this claim.

Again, where? Any experiment to measure length contraction will require each observer to have a definition of simultaneity, and if they are using the same definition as the one used in their inertial rest frame, then observers in relative motion will disagree about simultaneity.

I'm not yet sure how this affects my original question. I like to associate velocity with a rotation that projects part of an object's length out of normal spacetime, resulting in contraction. Since the rotation angle back to the stationary frame is equal and opposite, the moving observer gets the same view of the stationary object.

Again, it would really help if you would give some math so the meaning of odd phrases like "projects part of an object's length out of normal spacetime" would be made clear, no one is going to know what you're talking about otherwise, and if you can't define this stuff mathematically (or at least show what it means on a spacetime diagram) I would say that you probably don't have any well-defined idea in mind and are just expressing some vague intuitions based on analogies like the one with the loxodrome.

 

[ghwellsjr]

I don't know what ghwellsjr means when says that lengths are contracted and clocks slowed down by acceleration.

I never said that.

What I do say, is that objects change their length whenever they are accelerated along the direction of acceleration and clocks (and all time related features of other objects) change their tick rates (or aging rates) during accelerations. I have made it very clear that there is no absolute sense of which clock is running "slower" or which lengths are shorter.

Tom gave a very abbreviated and truncated quote of mine here:

ghwellsjr said:

"... Then if you accelerate any number of those meter sticks, they will each contract physically along the direction of acceleration,..."

Here's the full post of what I said:

First, we can agree that an infinite number of inertial frames exist (because they are in our minds, they are not physical) and they all extend physically to infinity in all directions and they all encompass time from eternity past to eternity future and they all contain all the objects in the universe, or if we are talking about a thought problem, they all contain all the objects, labs, spaceships, rulers, clocks, observers, and anything else we want to consider.

So now we arbitrarily pick anyone inertial reference frame. You can call it the lab frame because in it you place a lab that is at rest in the lab reference frame. Then in that lab you place any number of stationary observers with any number of meter sticks that are defined to be the same length. Then if you accelerate any number of those meter sticks, they will each contract physically along the direction of acceleration, in the lab reference frame that you have defined. Now if you have an observer that is stationary in the lab reference frame, he will observe those moving meter sticks to be shortened. However, an observer with his own meter stick traveling with another traveling meter stick will discover that they are the same length and will not realize that they are both shortened in the lab reference frame. If a traveler observes a meter stick that is stationary in the lab reference frame, he will determine that the stationary one is shorter than his moving one.

If we transform the entire scenario from your previously defined lab frame to another frame, say one in which an observer is now traveling at a high speed, his meter stick will now be considered to be the correct length and the ones in the lab are now short, but this is not a physical change in any lengths. So we cannot conclude absolutely whose meter sticks are absolutely the correct lengths but we can conclude that every meter stick that is accelerating will be physically changing its length because that is what happens no matter which reference frame we chose to analyze any situation.

Now if you want to claim that physical, actual, real, changes in lengths along the direction of acceleration do not happen then you should consider the following:

The relativity FAQ page claims that it is real, except they actually call it length contraction.

Lorentz's explanation for the null result of MMX prior to Einstein was physical length contraction along the direction of the ether wind.

Of course, no one can observe length contraction of objects that are stationary to themselves, not even MMX, it is an explanation that explains other things that are measureable.

So for example, the means by which all observers measure the same value of c for the round trip speed of light is that those that are moving with respect to the defined frame of reference have distances that are contracted along the direction of motion (and they experience time dilation) so that their calculation of the speed of light is always c. This has nothing to do with Einstein's second postulate, it is what we always measure in any experiment.

(Einstein's second postulate is talking about the one-way speed of light which cannot be measured but is assumed to also be c for both halves of the round trip measurement of the speed of light.)

 

[tom.capizzi]

reply to JesseM:

I looked at over a dozen sites that claimed to have an answer until I found one that reminded me of something I read in one of Einstein's own books - the train experiment. So don't believe me. Go to a real, physical library and lookup Einstein.

By the way, your comment:

"... (in fact moving objects don't always appear contracted visually, sometimes they appear expanded depending on the direction relative to the viewer) ..."

puzzled me. Maybe some math would help.

As to the issue of simultaneity, reread the post. I said "multiple stationary observers". Perhaps I should add "stationary relative to each other." So, there is no problem with simultaneity. In any case, it is Einstein's experiment not mine. Read his book.

I find it inconsistent that someone proficient in science would have difficulty understanding projective geometry references. Read "Flatland", or if you have already, read it again. A line segment only appears full length when viewed at 90 degrees. As the line is rotated (or the observer rotates) the line vanishes (contracts) to a single point. A 3-D observer off the plane can clearly see that the line segment never changes actual length, but merely rotates away from the axis that is visible towards the one that is not. The rotated line will also "fit" between two parallel lines that are separated by the contracted length, even though it is really not any shorter.
Maybe that isn't enough math for you, but you should know that you can't prove anything with any number of examples. Plus, a logical argument doesn't require specific numbers.

 

[JesseM]

reply to JesseM:

I looked at over a dozen sites that claimed to have an answer until I found one that reminded me of something I read in one of Einstein's own books - the train experiment. So don't believe me. Go to a real, physical library and lookup Einstein.

Can you be specific about the book/page number instead of just making these vague claims, and if you have the book in hand perhaps give a direct quote? For example, Einstein discusses a train thought-experiment in section 8 and section 9 of the book Relativity: The Special and General Theory which is available online, but first of all this thought-experiment is about the relativity of simultaneity rather than length contraction, and second I don't see him using words like "physical" or "real" or "actual" in his description, which was what I was asking about.

By the way, your comment:

"... (in fact moving objects don't always appear contracted visually, sometimes they appear expanded depending on the direction relative to the viewer) ..."

puzzled me. Maybe some math would help.

I was referring here to the the Terrell-Penrose effect which depends on the fact that the light rays from different points on a moving object which reach my eyes at the same instant were actually emitted at different time-coordinates in my frame. For example, one simple case of the Terrell-Penrose effect is that for an object traveling straight towards me, if a light ray from the back of the object reaches my eyes at the same moment of a light ray from the front, the light ray from the back must actually have been emitted at an earlier time-coordinate in my frame, which causes the object to visually appear stretched to a length greater than its coordinate length in my frame.

For example, suppose a ship with a rest length of 10 light-seconds is traveling towards me at 0.8c, traveling in the +x direction. If the front reaches my own position x=0 at t=100, then the front has x(t) = 0.8c*t - 80 (when you plug in t=100 you get x=0). Since the ship is length-contracted to a length of 6 light-seconds in my frame, that suggest that at t=100 the back must have been at position x=-6, so the back must have x(t) = 0.8c*t - 86. Now, suppose at t=50 the front emits a light signal from position x=0.8*50 - 80 = -40, which reaches my eyes 40 seconds later at t=90. Also suppose at t=20 the back end emits a light signal from position x=0.8*20 - 86 = 16 - 86 = -70, which reaches my eyes 70 seconds later, also at t=90. That means that at t=90, I am seeing the front end lined up with the x=-40 mark on my ruler, and also seeing the back end lined up with the x=-70 mark, so visually the object appears to be 30 light-seconds long at that moment, five times greater than its "true" length of 6 light-seconds in my frame.

See here for more on the difference between visual changes in length and "actual" length contraction of an object which is moving relative to your rest frame:

http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html

And this page has a good discussion of differences in apparent visual length for objects moving towards you or away from you (showing how objects moving towards you appear stretched visually, not contracted, as shown by my example above):

http://www.spacetimetravel.org/bewegung/bewegung3.html

As to the issue of simultaneity, reread the post. I said "multiple stationary observers". Perhaps I should add "stationary relative to each other." So, there is no problem with simultaneity. In any case, it is Einstein's experiment not mine. Read his book.

What book? What page? The train thought-experiment in the book above is all about observers in relative motion with different definitions of simultaneity.

I find it inconsistent that someone proficient in science would have difficulty understanding projective geometry references.

I understand geometrical projections just fine, but you are being very vague about what specific thing (a worldline of some sort?) you want to project onto what surface (a surface of simultaneity? Something else?) so your comments are impossible to interpret. I don't want analogies to Euclidean geometry, I want to know what you are actually talking about in the context of spacetime, so that's why I'm asking for either math or a spacetime diagram.

Read "Flatland", or if you have already, read it again. A line segment only appears full length when viewed at 90 degrees.

Sure. This analogy doesn't help one whit in understanding what you are talking about with your comments about "actual distance" vs. "apparent distance" in spacetime though.

As the line is rotated (or the observer rotates) the line vanishes (contracts) to a single point. A 3-D observer off the plane can clearly see that the line segment never changes actual length, but merely rotates away from the axis that is visible towards the one that is not. The rotated line will also "fit" between two parallel lines that are separated by the contracted length, even though it is really not any shorter.

Again, completely obvious and also completely unhelpful. I'm beginning to think all you have is homespun analogies to Euclidean geometry, that you have no clear non-analogical idea about what your comments about actual vs. apparent distances and velocities actually mean in spacetime.

Maybe that isn't enough math for you, but you should know that you can't prove anything with any number of examples. Plus, a logical argument doesn't require specific numbers.

You sure are trying hard to rationalize not giving either any math or even a spacetime diagram. If you don't know how to describe your ideas in those terms, why not just admit that?

 

[tom.capizzi]

Einstein wrote more than one book. I read the paperback edition when you were in grade school and I no longer have my copy. It had two different train experiments in it, and at the time I was more interested in the one you refer to (relativity of simultaneity). The book is in an actual library. Online sources tend to be abridged versions anyway.

 

[JesseM]

I read the paperback edition when you were in grade school and I no longer have my copy.

If you read it decades ago, I doubt your memory of it is clear enough that you would specifically remember him using words like "physical" or "real" or "actual" to describe length contraction, which is the only thing my question was about. Just because a thought-experiment deals with physical objects like trains doesn't mean that the thing it's deriving is frame-independent (for example, he uses a train thought-experiment to show how simultaneity varies between frames), and personally I wouldn't use words like "physical" or "real" to refer to quantities that depend on one's choice of reference frame.

In any case, the question of whether you were justified in claiming "According to Einstein, length contraction of a moving body is a real, physical, actual phenomenon" is a bit of a sidetrack. Are you going to address my main question about whether you are able to what your notions of "actual" vs. "apparent" distance and velocity in terms of spacetime (preferably using a mathematical example or a spacetime diagram), not just using homey analogies to Euclidean geometry where it's completely unclear what specific elements in spacetime the geometrical objects are supposed to be analogous to?

 

[tom.capizzi]

Here's a URL for a demonstration of the train experiment in question. It is not the relativity of simultaneity experiment. It is designed for a college lecture, but contains very little math so that it can be understood by students of varying degrees of familiarity with the subject. Click on the link to visit the site.

http://www.fas.harvard.edu/~scdiroff/lds/QuantumRelativity/RelativityTrain/RelativityTrain.html

By the way, although I read the book many years ago, I aced relativity when I finished my college years in 2000.