Is Acceleration an Invariant? (Taylor & Wheeler)

[Freixas]

I've been reading Spacetime Physics by Taylor and Wheeler at @PeterDonis's suggestion. In chapter 3, they say:

Apply force to a moving object: Its velocity changes; it accelerates. Acceleration is the signal that force is being applied. Two events are enough to reveal velocity; three reveal change in velocity, therefore acceleration, therefore force. The laboratory observer reckons velocity between the first and second events, then he reckons velocity between the second and third events. Subtracting, he obtains the change in velocity. From this change he figures the force applied to the object.

The rocket observer also measures the motion; velocity between the first and second events, velocity between second and third events; from these the change in velocity; from this the force acting on the object. But the rocket-observed velocities are not equal to the corresponding laboratory-observed velocities. The change in velocity also differs in the two frames; therefore the computed force on the object is different for rocket observer and laboratory observer. The Principle of Relativity does not deny that the force acting on an object is different as reckoned in two frames in relative motion.

Ok, so force is not an invariant. Fine.

Then I go to http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html, where the term $a$ appears in various equations and is defined as acceleration. Is it an invariant?

The authors didn't state how they computed the force. I would assume they used the standard $F = ma$. If $F$ is not invariant, then either $m$ or $a$ is invariant, or both, correct?

My mental model was of someone on a rocket accelerating at 1g. When they step on a scale calibrated for the Earth's surface, a 1 kg object would generate a 1 kg reading, which I thought was a measure of force. The accelerated rocket has an instantaneous moving frame, one that changes each instant, but the 1 kg object would continue to generate a 1 kg reading.

As I'm not in school and can't ask the teacher a question, I thought I'd ask the group here to straighten me out. Thanks.

 

[PeroK]

I've been reading Spacetime Physics by Taylor and Wheeler at @PeterDonis's suggestion. In chapter 3, they say:

Ok, so force is not an invariant. Fine.

Then I go to http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html, where the term $a$ appears in various equations and is defined as acceleration. Is it an invariant?

The magnitude of the proper acceleration is an invariant.

The authors didn't state how they computed the force. I would assume they used the standard $F = ma$.

That's Newton's second law, which breaks down at relativistic speeds.

Although, it can be replaced by $\mathbf f = m \mathbf a$, where $\mathbf f$ and $\mathbf a$ are the four-force (aka Minkowski force) and four-acceleration.

 

[Orodruin]

Then I go to http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html, where the term a appears in various equations and is defined as acceleration. Is it an invariant?

Yes, it is invariant, but it is not the coordinate acceleration, which you may be thinking of. This $a$ is the proper acceleration, which is what an accelerometer would measure. It is equal to the coordinate acceleration only in the instantaneous rest frame of the rocket.

The authors didn't state how they computed the force. I would assume they used the standard F=ma. If F is not invariant, then either m or a is invariant, or both, correct?

No. Force is the time derivative of momentum. This makes things more complicated in relativity where momentum is given by $\vec p = m \vec v \gamma(v)$.

 

[PeterDonis]

Then I go to http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html, where the term $a$ appears in various equations and is defined as acceleration. Is it an invariant?

Yes, because that article is considering the very simple special case of acceleration in a straight line. Their $a$ is the magnitude of the proper acceleration, which is an invariant, and since they are only considering motion along a single straight line, no issues of changing direction enter into anything so the proper acceleration is entirely described by its magnitude $a$.

The authors didn't state how they computed the force.

That's because they didn't compute force at all. They used conservation of energy and momentum to compute things like the mass ratio that could not be computed simply from the equations for hyperbolic motion in Minkowski spacetime (i.e., for motion with constant proper acceleration in a straight line).

My mental model was of someone on a rocket accelerating at 1g. When they step on a scale calibrated for the Earth's surface, a 1 kg object would generate a 1 kg reading, which I thought was a measure of force.

A typical scale measures force indirectly by the compression of a spring or something similar yes. (Note that scales calibrated in kg are really measuring force in units of 9.8 Newtons, i.e., the force required to hold a 1 kg object at rest in the gravitational field at the Earth's surface.) However, you don't need to compute this force or anything like it in order to compute any of the things computed in the article.

 

[Freixas]

That's because they didn't compute force at all. They used conservation of energy and momentum to compute things like the mass ratio...

There are two source I refer to: the Spacetime Physics book you recommended and the rocket web site. I'm not sure the two of us are referring to the same things in the quotes above.

When I said, "The authors didn't state how they computed the force.", I was referring to the Spacetime Physics book authors. They talk about the laboratory and rocket observers calculating the force by measuring the motion; they don't tell us how. I don't see them talking about conservation of energy and momentum or computing mass ratios, so I'm guessing you were talking about the rocket web site.

I myself was unclear in that I was limiting my references to formulas 1-8 of the rocket web site. I forgot there was a lot more stuff on that page.

 

[Freixas]

Thanks to @PeroK, @Orodruin, and @PeterDonis.

Some of the concepts you introduced to answer my question are concepts that I have yet to learn about, however, what I did understand was that term used by the rocket equations, the magnitude of the proper acceleration, is an invariant, and this is what I wanted to know.

 

[PeterDonis]

When I said, "The authors didn't state how they computed the force.", I was referring to the Spacetime Physics book authors.

Ah, ok.

I don't see them talking about conservation of energy and momentum or computing mass ratios, so I'm guessing you were talking about the rocket web site.

Yes. Sorry for the confusion on my part.

 

[cianfa72]

The authors didn't state how they computed the force. I would assume they used the standard $F = ma$. If $F$ is not invariant, then either $m$ or $a$ is invariant, or both, correct?

They say to compute the force applied w.r.t. each frame from the change in object's velocity evaluated in that frame. Note that even though $F$ is not invariant it does not invalidate at all the Principle of Relativity.