# Length contraction viewed with Heaviside equations

• I
In summary, Richard Feynman, in his lectures on Physics, Vol I, chapter 28, provides an expression for calculating the E and B field of a moving point charge based on retarded radius. The equation, originally derived by Oliver Heavyside and re-discovered by Feynman, takes into account the knowledge of where the charge was at the time when the field was being formed. Using the Python programming language, a numerical simulation was set up to understand the equations, and it was found that the unit retarded radius vector and retarded radius value can be calculated using algebraic expressions. Feynman also points out that the third term of the equation is responsible for radiation due to acceleration of particles, leading to a linear contraction in
I have other posts to catch up on, but I'd like to comment on this point for I think it clarifies the special relativity issue / assumption that is causing me trouble.

Ibix said:
The only reason you need to accelerate at all is because there's no other way for you to meet your twin again in flat spacetime.

For future reference, if I make statements that something is "like" something else, I do not MEAN they are identical. Carefully note that they are likely Not the same in sense that identical "twins" are meant to be the same. For example, if I say "A duck is like an airplane" please do not jump to the conclusion that I think a duck IS an airplane or that an airpane must have feathers. etc.

In Special Relativity, the idea that light is accelerated by gravity is not yet accounted for. The purpose of using "twins" in special relativity experiments is to simplify the mathematics, for since they are "identical" in how they act, or measure time, etc. we are invoking the scientific method of investigation and attemption to limit what might interfere with a thought experiment; eg: I am trying to isolate what assumptions (false/true) we may be making that we don't realize we are making.

In the twin paradox I linked to by Dr. Don Lincoln, Femilab:

The experiment is broken into a "triplet's" paradox, in order to avoid acceleration. Please watch the video carefully. I think Doc does an admirable job.

The implicit assumption of the show, however, is that calling something an "inertial reference frame" is sufficient to solve the paradox. This is why I tried to call your attention to the doppler effect and measurements of acceleration; I wanted to try and isolate what "acceleration" is.

There are at least two ways I know to measure acceleration. I am not certain they are the same under the assumptions of Special Relativity.

One method is that if two "identical" lasers (twins) are in motion toward (or away) from each other, that they can measure a color change (doppler shift) of each other. Therefore, they can measure acceleration between the two lasers by measuing the time derivative of the laser's color change.

The second method of measuring acceleration is to use two masses (inertias) that are in torsion with each other. We attach one mass rigidly to a "reference" frame, and measure the shear force required to keep the other mass moving rigidly next to the first mass. When I accelerate the reference frame (eg: by shoving a cart with my hand that the accelerometer is attached to) the other mass is "dragged" along with the reference mass by a spring or other energy transfer device. We can therefore measure the amount of force the transfer device uses in order to measure/sense the "acceleration" of the cart.

An inertial accelerometer is the kind of accelerometer that a computer hard disk drive uses to detect when it is dropped or shoved suddenly, to protect the heads from "crashing." A doppler accelerometer is the kind of accelerometer used to measure whether a distant sun of the same spectra as our own, is moving (OR) accelerating toward or away from us.

However, I do not thnk an intertial accelerometer necessarily gives the same measurements in Special Relativity that a doppler shift accelerometer will give. The reason is that "free fall" can not be detected by an inertial accelerometer, but it can be detected by a doppler accelerometer.

When we did the "twin" paradox, using minkowski space, I suspect we are assuming that somehow we can magically tell the difference between inertial acceleration and gravitational acceleration in Special Relativity by the mere labeling of "frames of reference."

But it seems to me that we can always use gravitational acceleration to "hide" acceleration from an inertial accelerometer.

A special television show was aired last week about General Relativy's 100th birthday. At the end of the show they posed a Gedanken "Suppose the Sun disappeared from our solar system, never mind "HOW" it happened, what would be the effect on the earth?"

The television show's point was that it would take 8 minutes for the gravitational wave from the diappearing sun to reach the Earth, and for Earth to stop "curving" around the region where the sun used to be. The Earth would then move in a straight line flying off at a tangent...

The issue I see is that the Earth is normally in "Free" fall around the sun (in Special Relativity) and an inertial accelerometer can not detect the acceleration of the Earth by the sun (accurately). Whether the sun is there (or not), an inertial accelerometer will measure approximately 1g because of the accelerometer's close proximity to Earth's surface. I am unsure what the accelerometer would meausre if it was some-how moved magically to the center of the Earth ... but I suspect it would measure zero.

Both before and after the "sun" went out, the Earth has the same mass , and therefore an inertial accelerometer will still measure 1g if oriented parallel to the Earth's gravitational field at the surface of the Earth, or probably zero if at the center. The inertial accelerometer is not going to measure the same amount of acceleration as a doppler accelerometer will measure because an inertial accelerometer doesnt' "know" the total mass of the entire universe. (Especially if the sun suddenly just "disappeared.).

That leads me back to the assumption I'm trying to explore:
Whenever we do measurments in Minkowski space, for special relativity, we always measure with time of travel via the length "ct", to make the measurement co-variant. But this automatically implies that light is some-how involved in the measurement and acceleration is inherently a doppler measurement.

The second issue is that Minkowski space uses the generalized Pythagorean theorem, which is the same as taking the square root of an absolute value. In every case where a square root is taken, there are mathematically TWO solutions. It's not clear to me that one can't use Newtonian gravity in Special Relativity, to artificially manipulate which soultion applies to path length, or time length, etc.

For example, in the Twin paradox, at the 8 year mark ... what happens if two massive (but tiny) black holes moving in opposite directions were to move approximately perpendicular to both A and B at the "midway" point between them? The inertial accelerometers on A and B are not going to measure much change (if any!) ... But the doppler shift accelerometer is going to suddenly blue shift on both A and B very distinctively. This change will happen under the assumptions of Special Relativity, as well.

Therefore, I think that "inertial reference frames" are a necessary condition to claim that something is not accelrating, but merely being in an inertial reference frame is not a *sufficient* condition to guarantee that measurable acceleration is not happening in Special Relativity.

I realize that General Relativity may come to a different conclusion, but I think General Relativity uses different assumptions.

I have completed my calculations for the orbit of the planet-satellite system based on F=ma. One very important point is that the effective mass is different in different directions for 3 vectors. It goes as gamma in directions perpendicular to the direction of motion, and gamma cubed in the direction of motion. Therefore, the direction of the force vector is only the same as the direction of the acceleration vector when going exactly perpendicular or exactly parallel to the direction of motion. In all other cases, the direction of acceleration is not exactly the same as the direction of the applied force.

One method is that if two "identical" lasers (twins) are in motion toward (or away) from each other, that they can measure a color change (doppler shift) of each other. Therefore, they can measure acceleration between the two lasers by measuing the time derivative of the laser's color change.

More precisely, these will measure relative acceleration between two different objects.

The second method of measuring acceleration is to use two masses (inertias) that are in torsion with each other. We attach one mass rigidly to a "reference" frame, and measure the shear force required to keep the other mass moving rigidly next to the first mass. When I accelerate the reference frame (eg: by shoving a cart with my hand that the accelerometer is attached to) the other mass is "dragged" along with the reference mass by a spring or other energy transfer device. We can therefore measure the amount of force the transfer device uses in order to measure/sense the "acceleration" of the cart.

This is one possible implementation of an accelerometer, which measures proper acceleration of a single object. There are multiple ways of making an accelerometer:

https://en.wikipedia.org/wiki/Accelerometer
The key point is that an accelerometer is a measurement on a single object, whereas the Doppler shift measurement you describe is a measurement involving two objects, the light source and the receiver. So they're not measuring the same thing.

I do not thnk an intertial accelerometer necessarily gives the same measurements in Special Relativity that a doppler shift accelerometer will give. The reason is that "free fall" can not be detected by an inertial accelerometer, but it can be detected by a doppler accelerometer.

No, you have it backwards. The inertial accelerometer does detect free fall: you're in free fall if and only if your inertial accelerometer reads zero.

The Doppler shift accelerometer does not detect free fall, because knowing that your velocity relative to something else is changing, which is what the Doppler shift accelerometer is measuring, does not tell you that either your or the something else are or are not in free fall. In flat spacetime, if your Doppler shift accelerometer reads nonzero, then either you or the something else must not be in free fall, and it's possible that you both are not. In curved spacetime (i.e., when gravitating masses are present), your Doppler shift accelerometer can read nonzero even if both you and the something else are in free fall (this is a manifestation of tidal gravity, i.e., spacetime curvature).

When we did the "twin" paradox, using minkowski space, I suspect we are assuming that somehow we can magically tell the difference between inertial acceleration and gravitational acceleration in Special Relativity by the mere labeling of "frames of reference."

No. The twin paradox can be resolved without even using reference frames at all, and does not require any violation of the equivalence principle.

it seems to me that we can always use gravitational acceleration to "hide" acceleration from an inertial accelerometer.

No. An inertial accelerometer measures proper acceleration directly; there's no way to "hide" it.

At the end of the show they posed a Gedanken "Suppose the Sun disappeared from our solar system, never mind "HOW" it happened, what would be the effect on the earth?"

And this is an excellent example of why you should not use pop science shows to discuss science. This scenario violates stress-energy conservation so it is inconsistent with the laws of physics. You can't validly conclude anything from an inconsistent scenario.

the Earth is normally in "Free" fall around the sun (in Special Relativity) and an inertial accelerometer can not detect the acceleration of the Earth by the sun (accurately).

Of course not, since free fall means zero proper acceleration.

Whether the sun is there (or not), an inertial accelerometer will measure approximately 1g because of the accelerometer's close proximity to Earth's surface.

It will if it is sitting at rest on the Earth's surface, yes. This is due to the Earth's surface pushing up on the accelerometer and preventing it from freely falling.

But if you drop the inertial accelerometer off a cliff (or, better, from a platform at the top of the inside of a tall tower which has been evacuated to remove air resistance), it will read zero while it is falling.

Whenever we do measurments in Minkowski space, for special relativity, we always measure with time of travel via the length "ct", to make the measurement co-variant. But this automatically implies that light is some-how involved in the measurement

It implies no such thing. The speed of light is a universal constant, which really means it's just a unit conversion factor. Using units in which ##c = 1##, which is what you're describing, works just fine for measurements that have nothing to do with light.

The second issue is that Minkowski space uses the generalized Pythagorean theorem, which is the same as taking the square root of an absolute value. In every case where a square root is taken, there are mathematically TWO solutions.

And since we are dealing with lengths in time and space, which are obviously positive, the positive square root is the only one that is relevant.

what happens if two massive (but tiny) black holes moving in opposite directions were to move approximately perpendicular to both A and B at the "midway" point between them?

Nothing much. See below.

The inertial accelerometers on A and B are not going to measure much change (if any!)

Yes.

But the doppler shift accelerometer is going to suddenly blue shift on both A and B very distinctively.

No, it won't. The light going between A and B might be bent by the black holes, but its frequency as seen by A or B won't change; it falls into the gravity well and then climbs back out again, and the two frequency shifts cancel.

This change will happen under the assumptions of Special Relativity, as well.

No, it won't, because you can't use SR to analyze a scenario that includes black holes, since SR only works if spacetime is flat and black holes require curved spacetime.

I think that "inertial reference frames" are a necessary condition to claim that something is not accelrating

Neither inertial acceleration nor Doppler acceleration have anything to do with reference frames. They are direct measurements and will be the same regardless of any choice of frame, or even if you make no choice of frame at all.

merely being in an inertial reference frame is not a *sufficient* condition to guarantee that measurable acceleration is not happening in Special Relativity

If by "being in an inertial reference frame", you mean "being in free fall", then this is exactly the sufficient (and necessary) condition for zero inertial acceleration, and it tells you nothing by itself about Doppler acceleration. See above.

I realize that General Relativity may come to a different conclusion, but I think General Relativity uses different assumptions.

I have no idea what you mean by this.

I thought you were going to stay out of the thread and watch IBEX and others answer my questions. Couldn't stand not to inject again?

PeterDonis said:
The key point is that an accelerometer is a measurement on a single object, whereas the Doppler shift measurement you describe is a measurement involving two objects, the light source and the receiver. So they're not measuring the same thing.

The inertial accelerometer example that I gave was attached to a cart. It was accelerating with respect to my hand pushing it. The measurement was not a measurement on one object but on two masses that are connected by a spring relative to 'me' who pushed the cart.

I have never seen an example where anyone can truly make measurement (even in a gedanken) with a single object. There is always a reference of some kind used as a standard, whether it be a point in space, a clock, or something !

So, I have no idea what you are talking about.

PeterDonis said:
No, you have it backwards. The inertial accelerometer does detect free fall: you're in free fall if and only if your inertial accelerometer reads zero.

Fair enough, but you've missed something as usual because there were two objects being considered...

The accelerometer reads NON zero, on Earth while both the accelerometer and Earth freely fall toward the Sun. There is no string, or material object, keeping the accelerometer positioned with respect to the sun ... The meter is still not detecting Free fall even though Free fall is going on.

If you removed the sun, there would still be the gravity of Earth being measured by the inertial accelerometer. The sensitivity of the accelerometer is ovewhelmed by the Earth's gravity regardless of whether the sun exists or not; because the accelerometer is closer to the earth.

If you put the accelerometer exactly at the center of gravity between the sun and earth, ONLY then it will read zero but it will be not be falling (in any intelligible sense of the english words) toward EITHER mass, especially when measured relative to the stars.

I'm talking only about special relativity, but since you bring up the point:

In special relativity, if there aren't "black holes" according to you, then substitude "very large mass" for what I said. The point is that because of Newtonian gravity, when a large mass is (temporarily) placed between two free floating objects (lasers) that are a fixed distance apart; the gravity will cause those objects to accelerate toward each other. Therefore, a doppler shift will occur even though the inertial accelerometers do not detect the relative acceleration.

The doppler shift has nothing to do with a gravity well itself, as you pointed out ... So I have no idea why you brought it up.measure zer

weirdoguy and Dale
I thought you were going to stay out of the thread and watch IBEX and others answer my questions. Couldn't stand not to inject again?

The thread has been dormant for a few weeks, and the items I responded to in this latest post of yours are very different from the original topic of the thread anyway.

The measurement was not a measurement on one object but on two masses that are connected by a spring relative to 'me' who pushed the cart.

Yes, but one mass is held fixed to the cart, and you say you're measuring the force necessary to keep the second mass rigidly attached to the first mass, and "rigidly attached" means "not moving relative to", so you're not measuring the relative motion of the masses (which is what a Doppler accelerometer measures), you're measuring the proper acceleration of the cart. Which in the case you describe is caused by your hand pushing the cart: but you could just as easily attach your apparatus to a rocket in empty space and measure the proper acceleration of the rocket when its engine fires.

I have never seen an example where anyone can truly make measurement (even in a gedanken) with a single object.

It depends on what you mean by "a single object". The point is that an accelerometer (an inertial accelerometer, in your terminology) measures proper acceleration locally, without regard to any external reference. For example, in the case of the rocket in empty space that I just described, it measures the proper acceleration of the rocket without reference to anything outside the rocket (and the apparatus, which is attached to the rocket). This is a common and well understood measurement, for example in inertial navigation systems.

The accelerometer reads NON zero, on Earth while both the accelerometer and Earth freely fall toward the Sun.

It reads nonzero sitting at rest on the Earth's surface because, as I said, the Earth's surface is pushing up on it. Take that away by dropping the accelerometer off a cliff (or, as I said, inside an evacuated tunnel to eliminate air resistance) and it will read zero: everything is in free fall, so there is no need to exert any force on the second mass to keep it at the same distance from the first mass. The spring will sit in its unstressed equilibrium position and the masses will simply fall towards the Earth. The fact that the Earth is free falling in orbit about the Sun doesn't change that. Or you could put the masses in a spaceship well away from the Earth in a free-fall orbit, and the whole apparatus would just float along with the ship reading zero.

If you removed the sun

This is inconsistent with the laws of physics, so there is no way to even discuss what would happen. It's like asking what would happen if two plus two were five.

In special relativity, if there aren't "black holes" according to you, then substitude "very large mass" for what I said.

Doesn't matter as far as using SR is concerned. SR is only valid if spacetime is flat. If any mass (and therefore gravity) is present, spacetime is not flat. You can certainly set up such a scenario, but you can't analyze it using SR; you need GR. However, for the particular question you're interested in, the GR answer is pretty simple and is basically what you say; see below.

when a large mass is (temporarily) placed between two free floating objects (lasers) that are a fixed distance apart; the gravity will cause those objects to accelerate toward each other.

Yes, that's true. I was confused because you talked about the twin paradox, and there's no need to bring in the twin paradox for a scenario like this. Just have two free floating lasers and have a large mass fly by between them. Yes, in that case the lasers will start falling towards each other and a Doppler accelerometer attached to each one will read nonzero.

Therefore, a doppler shift will occur even though the inertial accelerometers do not detect the relative acceleration.

Yes, in the case just described, inertial accelerometers attached to each laser would read zero, since both lasers are in free fall. The nonzero Doppler acceleration is due to the spacetime geometry.

One method is that if two "identical" lasers (twins) are in motion toward (or away) from each other, that they can measure a color change (doppler shift) of each other.
This is most definitely not a measurement of proper acceleration. This measurement could be zero when the proper acceleration is nonzero or it could be nonzero when the proper acceleration is zero.

I recommend reading about proper acceleration to get a better understanding of its meaning and importance. Accelerometers, by definition, measure proper acceleration. Your Doppler measurement does not measure proper acceleration. Therefore it is not an accelerometer.

The second method of measuring acceleration is to use two masses (inertias) that are in torsion with each other. We attach one mass rigidly to a "reference" frame, and measure the shear force required to keep the other mass moving rigidly next to the first mass.
This is essentially correct. A few details are off, but the basic concept is right. It doesn’t need to be torsion, usually it is compression or bending. In either case you usually measure the strain rather than the force, but of course they are related.

However, I do not thnk an intertial accelerometer necessarily gives the same measurements in Special Relativity that a doppler shift accelerometer will give.
That is correct because your Doppler shift device is not an accelerometer.

When we did the "twin" paradox, using minkowski space, I suspect we are assuming that somehow we can magically tell the difference between inertial acceleration and gravitational acceleration in Special Relativity by the mere labeling of "frames of reference."
There is no gravitational acceleration in special relativity. If you have gravitational acceleration then you need to use general relativity, not special relativity.

Therefore, I think that "inertial reference frames" are a necessary condition to claim that something is not accelrating, but merely being in an inertial reference frame is not a *sufficient* condition to guarantee that measurable acceleration is not happening in Special Relativity
Remaining at rest in an inertial frame in SR is both a necessary and a sufficient condition for having zero proper acceleration. In GR this still holds except that inertial frames are only local rather than global as they are in SR

I'm talking only about special relativity ... In special relativity, if there aren't "black holes" according to you, then substitude "very large mass" for what I said. The point is that because of Newtonian gravity
Newtonian gravity is inconsistent with talking only about special relativity. That is the whole reason that general relativity was developed. If you are talking about gravity (which you keep doing) then you are not taking about SR.

You are contradicting yourself every time you talk about gravity and then claim you are only talking about SR. If you want others to stop talking about GR then you need to stop bringing it up by discussing gravity. You cannot discuss gravity at all if you want to limit the conversation to SR.

Last edited:
Dale said:
If you have gravitational acceleration then you need to use general relativity

And strictly speaking, "gravitational acceleration" is a poor choice of terminology since a body moving solely under the influence of gravity is in free fall, with zero proper acceleration.

Dale
The implicit assumption of the show, however, is that calling something an "inertial reference frame" is sufficient to solve the paradox.
No frames are needed at all. The only reason you need to introduce frames explicitly is to explain how the "paradox" arises from using frames implicitly in a careless manner.
There are at least two ways I know to measure acceleration. I am not certain they are the same under the assumptions of Special Relativity.
You specify a mass-on-a-spring accelerometer and a Doppler accelerometer. These do, indeed, measure two different things, usually called proper acceleration and coordinate acceleration (assuming the coordinates in question treat the laser accelerometer as at rest). The distinction is present in Newtonian physics as well, although we don't usually make a point of it.
However, I do not thnk an intertial accelerometer necessarily gives the same measurements in Special Relativity that a doppler shift accelerometer will give. The reason is that "free fall" can not be detected by an inertial accelerometer, but it can be detected by a doppler accelerometer.
They won't give the same readings in general, no. For example, fit two carts with an inertial and a laser accelerometer each and point the laser at the other cart. Push one cart. The inertial accelerometer on that cart will kick, but the Doppler accelerometer on both carts will kick.
When we did the "twin" paradox, using minkowski space, I suspect we are assuming that somehow we can magically tell the difference between inertial acceleration and gravitational acceleration in Special Relativity by the mere labeling of "frames of reference."
If we are working in Minkowski spacetime then there is no gravity by hypothesis. If there is gravity then it's not Minkowski spacetime.
But it seems to me that we can always use gravitational acceleration to "hide" acceleration from an inertial accelerometer
This is backwards. General relativity does not treat gravity as a force so does not regard it as causing acceleration. It simply modifies the concept of "moving in a straight line".
they posed a Gedanken "Suppose the Sun disappeared from our solar system, never mind "HOW" it happened, what would be the effect on the earth?"
As Peter points out, this violates the local conservation of energy and is therefore not a situation that can be described in general relativity, since local conservation of energy follows directly from the Einstein Field Equations. Attempting to reason from this point is, therefore, nothing to do with GR.
Whenever we do measurments in Minkowski space, for special relativity, we always measure with time of travel via the length "ct", to make the measurement co-variant. But this automatically implies that light is some-how involved in the measurement and acceleration is inherently a doppler measurement
The constant c in special relativity does not have to be the speed of light. It's perfectly possible to model photons as having non-zero mass, in which case light does not travel at c (or indeed, any unique speed). There is no effect on relativity. I believe we currently place the photon mass at less than 10-54kg, and we usually assume it to be zero, but we do not actually know that. In modern treatments of relativity you derive the existence of a constant with dimensions of velocity directly from symmetry requirements. Only by experiment do you then discover that it is the same as the speed of light.
The second issue is that Minkowski space uses the generalized Pythagorean theorem, which is the same as taking the square root of an absolute value. In every case where a square root is taken, there are mathematically TWO solutions. It's not clear to me that one can't use Newtonian gravity in Special Relativity, to artificially manipulate which soultion applies to path length, or time length, etc.
I don't understand how you think your last sentence relates to the first two, but Newtonian gravity is fundamentally incompatible with relativity. It requires a notion of global simultaneity that isn't present.

Last edited:
I have never seen an example where anyone can truly make measurement (even in a gedanken) with a single object. There is always a reference of some kind used as a standard, whether it be a point in space, a clock, or something !
I agree to a point. However, the point about the inertial accelerometer is that it is a "closed room" experiment. You can build one in a sealed opaque box and it will work. Not so your laser accelerometer (unless you use it on a free-floating mass inside the box, which is simply using the device as another form of inertial accelerometer).
The accelerometer reads NON zero, on Earth while both the accelerometer and Earth freely fall toward the Sun.
The accelerometer is not in free fall. Free fall is exactly what it says - falling freely. Sitting on the floor, it is not falling. Thus it shows the acceleration due to the force exerted on it by the floor.

And can I suggest you leave off the personal attacks? It just makes you look foolish, especially when you are incorrect as in this case.
I'm talking only about special relativity,
...yet you keep bringing up gravity. Special relativity only works where gravity can be neglected.

Last edited:
weirdoguy

• Special and General Relativity
Replies
54
Views
1K
• Special and General Relativity
Replies
6
Views
2K
• Special and General Relativity
Replies
11
Views
2K
• Special and General Relativity
Replies
12
Views
1K
• Special and General Relativity
Replies
33
Views
2K
• Special and General Relativity
Replies
40
Views
3K
• Special and General Relativity
Replies
12
Views
1K
• Special and General Relativity
Replies
27
Views
2K
• Special and General Relativity
Replies
4
Views
1K
• Special and General Relativity
Replies
23
Views
3K