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I have doubts on whether it is true or false that an object has zero coordinate acceleration if and only if it has zero proper acceleration.

In special relativity, I think it is true because coordinate acceleration $a$ and proper acceleration $\alpha$ is related by the formula $a=\alpha/\gamma^3$. So if $\alpha=0$, then $a=0$ and vice versa.

But I am not sure if the same formula $a=\alpha/\gamma^3$ holds in general relativity because 4-acceleration now contains an extra term $A_\text{GR}^\lambda$ from spacetime geometry:

$$A^\lambda=A_\text{SR}^\lambda+A_\text{GR}^\lambda=\frac{dU^\lambda}{d\tau}+\Gamma_{\mu\nu}^\lambda U^\mu U^\nu$$

So we cannot be certain that $\alpha=0$ implies $a=0$ and vice versa, right?

There is one thing has been bothering me for a while: I think in general relativity (even in special relativity), it is impossible to tell whether an object feels zero proper acceleration or not. Suppose we stand on some reference frame and see an object moving with nonzero coordinate acceleration, since coordinate acceleration is relative, I don't think we can tell whether a reference frame we are standing on is accelerating with respect to the object or the object is accelerating with respect to our reference frame, let alone whether an object feels zero proper acceleration or not. Even if we observe the object is at rest, how can we tell whether it has zero proper acceleration or not?

Qmechanic
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Jimmy Yang
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2 Answers2

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I have doubts on whether it is true or false that an object has zero coordinate acceleration if and only if it has zero proper acceleration.

Why do you start from such a premise at all? It's wrong!

Perhaps you are confusing it with a different statement: an inertial frame is one where objects with zero proper acceleration, also have zero coordinate acceleration. This in turn implies, that a non-inertial frame is one where objects with zero proper acceleration may have non-zero coordinate acceleration.

In this answer of mine to one of your previous questions, that is exactly the scenario I took as an example. The observer on the earth that describes the motion of a free falling body, is in a non-inertial frame. The earth is non-inertial because the ground beneath our feet keeps resisting the inertial geodesic motion that we would have had in its absence, towards the center of the earth.

Consequently, in this earthbound coordinate system, the freely falling body has a coordinate acceleration. Near the surface of the earth, we denote that coordinate acceleration usually by $g$ where $g\approx 9.8\text{ m/s}^2$.

In this answer I also showed, that the proper acceleration still computes to zero in the earth's frame (as it will in any other) because, to use your present notation: $$ A_\text{SR}^\lambda = -A_\text{GR}^\lambda $$

So with regards to the first part of your question, I am really quite perplexed because I was sure this was addressed in this previous question.

Now, with regards the second part:

Suppose we stand on some reference frame and see an object moving with nonzero coordinate acceleration, since coordinate acceleration is relative, I don't think we can tell whether a reference frame we are standing on is accelerating with respect to the object or the object is accelerating with respect to our reference frame, let alone whether an object feels zero proper acceleration or not. Even if we observe the object is at rest, how can we tell whether it has zero proper acceleration or not?

This raises a separate topic, that's in fact just as relevant to ask about in the context of Newtonian physics as it is in the context of relativity. That is, you're basically asking whether acceleration is completely relative, similar to uniform velocity, or is it in some sense absolute and detectable from within the accelerating frame. The latter is the correct option.

There are many questions of a similar nature on this site, this one is a rather good example. But to summarize this in a few points one can say:

  1. Proper acceleration is an absolute property of a frame. It can be detected from within it by an inside observer.
  2. The laws of physics do not take their simplest form in an accelerating frame. Hang a pendulum so its string is perpendicular to the direction of acceleration in an accelerating frame. You will measure what seems to be a violation of Newton's first law: The pendulum will, for some "mysterious" reason stray from the vertical alignment when at equilibrium.
  3. Simplest and most concrete of all: accelerometers are physically realizable instruments, that we can and do build in the real world. Proper acceleration is what accelerometers measure.
Amit
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I have doubts on whether it is true or false that an object has zero coordinate acceleration if and only if it has zero proper acceleration.

It is false. Either acceleration may be zero while the other is non zero.

In special relativity, I think it is true because coordinate acceleration an and proper acceleration α is related by the formula a=α/γ3. So if α=0, then a=0 and vice versa.

That expression only holds in inertial frames in SR. It does not hold in non-inertial frames in SR.

I think in general relativity (even in special relativity), it is impossible to tell whether an object feels zero proper acceleration or not.

Proper acceleration can be directly measured with an accelerometer in both SR and GR.

Even if we observe the object is at rest, how can we tell whether it has zero proper acceleration or not?

Proper acceleration is covariant and is measured by an accelerometer attached to the object. All frames agree on it.

Dale
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