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analyst5

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- Thread starter analyst5
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In summary: This cannot be correct as you state it. Whether or not a given state of motion is inertial is invariant; it doesn't depend on the coordinates you choose. Inertial motion means zero proper acceleration; non-inertial motion means non-zero proper acceleration.

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analyst5

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A.T.

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It already differs significantly from inertial frames because of gravity. The rest frame of your house for example is highly non-inertial as it accelerates with 1g away from the the Earth's center. If the Earth was spinning faster, your house would be actually more inertial, than it is now.analyst5 said:if we could travel at greater speeds undergoing circular motion on the surface of Earth, what would happen, and how would it differ from the standard definition of these 3 effects in inertial frames?

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analyst5

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If we don't take gravitational effects into account, just circular motion, what are the differences?

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A.T.

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You mean just a rotating reference frame, or the restframe of a spinning disc. There are many threads on this here already. The key points:analyst5 said:If we don't take gravitational effects into account, just circular motion, what are the differences?

- You cannot synchronize clocks in that frame.

- You can synchronize clocks at equal radius, but not with the standard Einstein convention

- Light propagation is not isotropic

- The geometry measured with rulers at rest to the disc is non-Euclidean

Maybe you should first look into linearly accelerated frames, to see the differences to inertial frames there.

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Nugatory

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analyst5 said:

It's not "motion on Earth" that's not inertial, it's motion described using coordinates in which the surface of the Earth is at rest that's non-inertial. Seal me and my lab equipment into a box, then drop the box from an airplane... I can use coordinates in which my lab is at rest (until I hit the ground) and the experiments I do inside the box will register time dilation, length contraction, relativity of simultaneity, just as they do in the textbook thought experiments conducted in empty space. In this coordinate system the surface of the Earth is both rushing towards at 9.8 ##m/sec^2## and rotating sideways underneath me.

Or I could choose to use the coordinates that come naturally to an observer on the surface of the earth; in these coordinates the surface of the Earth is at rest and my lab is falling from the sky and moving sideways from Coriolis force. The calculation of the motions of objects in my lab will be much messier, but as long as I get the coordinate transformations right I'll get the exact same results - it has to come out that way because it's still the exact same experiment and the objects in my lab are all doing the exact same thing.

Many things are harder in the non-inertial coordinates: A.T. has provided a list of things that are hard or impossible using the coordinates in which the surface of the Earth is at rest. However, as long as we're aware of these difficulties, we can do the necessary calculations to predict how relativistic effects will appear on earth, and perform experiments that verify these predictions.

In practice and for realistic measurements we seldom have to worry about the non-inertial nature of the Earth's surface; the difference in relativistic results between a true inertial frame and a frame in which the surface of the Earth is at rest are generally negligible. Gravity is totally irrelevant for the motion of particles in a collider; the rotation of the Earth is pretty much negligible during the hundred or so microseconds that it takes a cosmic ray particle to make from upper atmosphere to ground; and so forth.

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PeterDonis

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Nugatory said:It's not "motion on Earth" that's not inertial, it's motion described using coordinates in which the surface of the Earth is at rest that's non-inertial.

This can't be correct as you state it. Whether or not a given state of motion is inertial is invariant; it doesn't depend on the coordinates you choose. Inertial motion means zero proper acceleration; non-inertial motion means non-zero proper acceleration. Proper acceleration is an invariant.

Nugatory said:Seal me and my lab equipment into a box, then drop the box from an airplane...

Which means, the box is in free fall (ignoring air resistance), therefore it has zero proper acceleration, therefore the motion is inertial. That's true regardless of the coordinates we use to describe it.

Nugatory said:I can use coordinates in which my lab is at rest (until I hit the ground) and the experiments I do inside the box will register time dilation, length contraction, relativity of simultaneity, just as they do in the textbook thought experiments conducted in empty space. In this coordinate system the surface of the Earth is both rushing towards at 9.8 ##m/sec^2## and rotating sideways underneath me.

Agreed.

Nugatory said:Or I could choose to use the coordinates that come naturally to an observer on the surface of the earth; in these coordinates the surface of the Earth is at rest and my lab is falling from the sky and moving sideways from Coriolis force. The calculation of the motions of objects in my lab will be much messier, but as long as I get the coordinate transformations right I'll get the exact same results - it has to come out that way because it's still the exact same experiment and the objects in my lab are all doing the exact same thing.

Right; and those experimental results will include a reading of zero on an accelerometer that is at rest in the free-falling lab, corresponding to a zero proper acceleration for the lab's worldline. So the lab's motion is inertial even though we are using a non-inertial frame to describe it.

On the other hand, if the lab was sitting on the surface of the Earth, an accelerometer at rest in the lab would *not* read zero; it would read 1 g, meaning the lab's motion would be non-inertial. This would change a number of other experimental results as well. And that would be true regardless of whether you chose coordinates in which the surface of the Earth was at rest, or coordinates in which an observer free-falling towards the Earth was at rest.

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Nugatory

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You're right. The point I was trying to make is that "on earth" (which is usually understood to mean "using coordinates in which the surface of the Earth is at rest"), has nothing to do with whether the motion is inertial or not; the motion is what it it is and it can be inertial or not on Earth as easily as anywhere else.PeterDonis said:This can't be correct as you state it. Whether or not a given state of motion is inertial is invariant; it doesn't depend on the coordinates you choose. Inertial motion means zero proper acceleration; non-inertial motion means non-zero proper acceleration. Proper acceleration is an invariant.

I'm hearing in OP's question, perhaps wrongly, some confusion between the invariant property (accelerometer does or does not read zero) and the coordinate accelerations that appear and complicate things when we choose coordinates in which the surface of the Earth is at rest.

[And for analyst5's benefit - PeterDonis and I are not arguing here. He's right and I'm trying to clean up my answer to our question so that it better describes physics about which we are in complete agreement while still responding to your question. Maybe instead I should have asked you to go back and look very carefully at the first two sentences of your original post where you slid almost imperceptibly from "inertial

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analyst5

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A.T.

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analyst5 said:Let's say a car is moving with some speed relative to Earth's surface.

A frame moving fast in a circle has locally the same SR effects as its instantaneous inertial rest frame. But additionally it has radial proper acceleration and thus (pseudo)gravitational time dilation along this direction.

Time dilation is a phenomenon that occurs when an object is moving at high speeds or in a strong gravitational field. It causes time to pass slower for the moving object compared to a stationary observer. On Earth, time dilation is very small and only becomes noticeable at extremely high speeds, such as in particle accelerators or with satellites orbiting the Earth.

Length contraction is the shortening of an object's length in the direction of its motion. This occurs when an object is moving at high speeds, relative to an observer. On Earth, length contraction is also very small and only becomes significant at extremely high speeds. This effect is often observed in particle accelerators, where particles appear to shrink as they approach the speed of light.

Yes, time dilation and length contraction affect all objects on Earth, but the effects are only noticeable at extremely high speeds. The faster an object is moving, the more significant these effects become. However, at everyday speeds, the effects of time dilation and length contraction are too small to be observed.

Relative simultaneity is the concept that events that appear simultaneous to one observer may not appear simultaneous to another observer, depending on their relative motion. This is due to the finite speed of light and the fact that the speed of light is constant for all observers. On Earth, this effect is very small and only becomes noticeable at extremely high speeds.

No, time dilation and length contraction are only noticeable at extremely high speeds, which are not achievable in everyday life on Earth. These effects are only significant in extreme conditions, such as in particle accelerators or space travel. In everyday life, the effects of time dilation and length contraction are too small to be observed.

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