How do accelerometers measure acceleration without using relative motion?

[PatrickPowers]

In Newtonian physics gravity accelerates things, so the distance at which they work in a GR space containiny mass is null, because there is no distance in GR that gravity accelerates things?

Whoops, you got me there. I was thinking of Newton's laws of motion, not his theory of gravitation.

Do you mean as I change perspective? It seems that from a coordinate system fixed to my hand, it could only measure ONE vector of acceleration at a time

Hmm. You can have a string and weight attached to your hand, but you can't have a coordinate system attached to your hand. You could have several strings and weights attached to your hand and notice that they are pulling in almost all different directions at the same time. The net force would depend on many things. It would be complicated and doesn't seem helpful.

The way I see it, Albert found out that once you establish a universal coordinate system then you have already lost. It can't be done. His theory helps you to do the best you can for your purposes.

As for proper acceleration, I don't know and others who do know have already answered, it seems to me.

 

[Zula110100100]

Hmm. You can have a string and weight attached to your hand, but you can't have a coordinate system attached to your hand. You could have several strings and weights attached to your hand and notice that they are pulling in almost all different directions at the same time. The net force would depend on many things. It would be complicated and doesn't seem helpful.

You can't have a coordinate system "attached" to you hand, but you can choose a coordinate system in which you hand is at rest, this is what I meant. The pull on all these string would cancel other than the net pull of gravity(wording issues, I know) These multiple string would complicate things, so to be simple I am using a unidirectional accelerometer with one weight and one string. It should measure the net acceleration I could be wrong here. To simplify further, let me add I am talking about a universe containing one large massed body and no other significant sources of gravity.

The way I see it, Albert found out that once you establish a universal coordinate system then you have already lost. It can't be done. His theory helps you to do the best you can for your purposes.

I am not establishing a universal coordinate system, but for anyone problem I must choose at least one coordinate system to work with(agree?), for this problem I am first choosing a cooridinate system in which my hand is always at the origin.

And I do appreciate that it helps, but to further out understanding of reality we must test each theory we have with every way we can think of, it's what Einstein did with Newtonian mechanics and to imply it need not be done is to say you are content with the level of understanding we currently have.

As for proper acceleration, I don't know and others who do know have already answered, it seems to me.

I will re-read the posts but I do not believe it has been satisfactorily answered in regards to this specific question.

 

[PAllen]

Just a few clarifying points:

1) Proper acceleration is a strictly local measurement. This follows immediately from its mathematical definition, which is second derivative with respect to proper time along a world line. You can't get more local than infinitesimal. (Note, in the formal definition, proper acceleration is a 4-vector; its norm (magnitude) is also frequently called proper acceleration when the context makes clear a scalar quantity is referenced. The vector is contravariant, the magnitude is a scalar invariant). Thus, an accelerometer to measure proper acceleration must be small.

2) Given the above, a weight on a long string is not considered an accelerometer (at least in the GR context under discussion). This type of experiment, instead, measures:

a) Newtonian terminology: in addition to how a small accelerometer would work, gravitational potential difference and tidal gravity, in a mixture depending on orientation.

b) GR terminology: a nonlinear combination of proper acceleration plus a measure of degree of curvature spanned by the long string.

3) The GR way of looking at things, I explained many posts ago, is motivated by replacing two laws with one (instead of a gravitational force law and a law of inertia, you have only a law of inertia). Given this foundation of GR, you cannot discuss or try to understand gravity in GR as a force. [Caveat: this is true in classical GR, classically interpreted. There are alternative viewpoints motivated in part by making GR look more like QFT forces. I prefer to avoid bringing these into discussion until the basics are understood.] Any time you want to say: "what about a force acting on everything the same way proportional to mass", that is exactly what caused Einstein to say: "What sort of force is that? It seems to affect all matter the same, acting on exactly the same quantity as inertia. Ah, it is actually inertia in disguise.".

 

[Zula110100100]

All you need to measure magnitude and direction of acceleration from your point of view (you being stationary relative to yourself) is a standard mass on a spring.

So you imply your method counts as an accelerometer, so long as the spring is infinitesimal in length, in a strong enough gravitational gradient all you would need is 1mm to measure a noticable pull on the spring... As your sensitivity to the measurement increases it becomes smaller and small that is needed.

Proper acceleration is a strictly local measurement. This follows immediately from its mathematical definition, which is second derivative with respect to proper time along a world line. You can't get more local than infinitesimal. (Note, in the formal definition, proper acceleration is a 4-vector; its norm (magnitude) is also frequently called proper acceleration when the context makes clear a scalar quantity is referenced. The vector is contravariant, the magnitude is a scalar invariant). Thus, an accelerometer to measure proper acceleration must be small.

I am neither a scientist or a mathematician so I apologize that my calculus terminology is lacking...

In a free-falling frame my acceleration I never change velocity, since A change of no velocity per proper time equals no acceleration, If I fire a rocket to escape(the not pull of) gravity, and use a frame that accelerates thus, then I still measure no change in velocity per proper time...so what is an example of proper acceleration please?

Now the object that is 50m from the surface does change its velocity(in my free-falling coordinate system 100m from surface) per proper time. So does an object 150m from surface, seems to slowly accelerate away? I am probably misusing proper acceleration here, but can you help clear that up?

 

[D H]

In a free-falling frame my acceleration I never change velocity

Velocity with respect to what? You are still appear to have the mistaken concept of absolute space, and you apparently do not understand the difference between proper acceleration and coordinate acceleration.

Keep in mind that the prefix "proper" does not mean "everything else is incorrect". A better prefix would be "eigen" or "characterestic".

 

[PAllen]

So you imply your method counts as an accelerometer, so long as the spring is infinitesimal in length, in a strong enough gravitational gradient all you would need is 1mm to measure a noticable pull on the spring... As your sensitivity to the measurement increases it becomes smaller and small that is needed.

Correct. The more extreme the curvature (GR equivalent of Newtonian gradient), the smaller the accelerometer must be to measure proper acceleration.

I am neither a scientist or a mathematician so I apologize that my calculus terminology is lacking...

In a free-falling frame my acceleration I never change velocity, since A change of no velocity per proper time equals no acceleration, If I fire a rocket to escape(the not pull of) gravity, and use a frame that accelerates thus, then I still measure no change in velocity per proper time...so what is an example of proper acceleration please?

Now the object that is 50m from the surface does change its velocity(in my free-falling coordinate system 100m from surface) per proper time. So does an object 150m from surface, seems to slowly accelerate away? I am probably misusing proper acceleration here, but can you help clear that up?

Here you are being limited by your math back ground, and I'm not sure I can help. I'll try.

A world line is a path through space and time. It represents, for example, the complete history of you, sitting down, running, flying in a plane or a rocket. Even if use coordinates where your world line is always of the form (t,0,0,0) (that is, you are always at the origin of your coordinate system, the only change in your coordinates being your wristwatch time), this does not make either your 4-velocity or 4-acceleration zero. The reason is that the derivative I referred to is really the covariant derivative (this involves a term called the connection as well as ordinary derivatives). One qualitative statement I can make is that at any point on your world line, there is a unique geodesic tangent to it. This geodesic would be the free fall path of an object you let go of at that point. To the extent that object moves away from you, you have proper acceleration.

 

[Zula110100100]

DH: With respect to my free-falling frame, at no point in this am I saying there is one correct coordinate system.

(that is, you are always at the origin of your coordinate system, the only change in your coordinates being your wristwatch time), this does not make either your 4-velocity or 4-acceleration zero

Right, so my 4-velocity would be a constant (ct, 0, 0, 0)?

 

[PAllen]

DH: With respect to my free-falling frame, at no point in this am I saying there is one correct coordinate system.



Right, so my 4-velocity would be a constant (ct, 0, 0, 0)?

Yes, you could construct such coordinates that your 4-velocity would be always be (1,0,0,0) in units of c=1. However, your proper acceleration vector would not, therefore, be zero unless an object you let go of stayed with you.

 

[Zula110100100]

What are the dr and dt in the swartzchild equation relative to, the center of the body M?