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From one book, I read that if we release a ball horizontally on earth, its velocity does change in very accurate observations. If we track its motion, we get full information of its motion (incl. acceleration respect to an true inertial frame).

And we can then compare it to the ideal linear uniform motion, then we get our acceleration to an true inertial frame (in which newtons 1st law is true), right? So with the developed formulas, we get all things we ever wanted in an exactly true inertial frame, can't we?

YiFei
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There is no absolute inertial frame because all motion is relative. The earth is a good approximation, and the frame of reference we use for most situations. However the earth is accelerating around the sun (i.e. towards the sun, that acceleration keeps the earth in orbit). The closest thing to an absolute inertial frame might be a space ship traveling with uniform velocity in a straight line -- however even this is relative it it's surroundings. What appears to us as a uniform velocity straight line may not be from other frames of reference. Furthermore keep in mind such a straight line would have to continue forever staying far away from any material bodies (such as planets and stars) which curve space and therefore re-define what is a straight line in their vicinity.

The failure of science to experimentally determine and absolute inertial frame (for example, the Michaelson-Morley experiment, and others like it) is what ultimately lead to the theory of relativity. When Lorentz first proposed his time-space transformations, many scientists still believed there would be found an absolute inertial frame, that space and time were constant, and that the transformations were merely a correction to account for some as yet undiscovered error in their calculations. Einstein came along and said, no, the speed of light is constant, and the Lorentz corrections are real. In other words, there is no absolute inertial frame. All frames of reference are relative. This is our current understanding of the nature of the universe.

To summarize, after many, many failures to find an absolute inertial frame, Einstein effectively proposed that there is no such thing. His theory explained all the experimental results so well (by assuming no such thing as an absolute inertial frame) that it has generally become accepted that the universe does not have an absolute inertial frame. Keep an open mind. Perhaps someday we will discover something new and realize that there is an absolute inertial frame, but for now all the physics we do works best assuming that there is none.

Daniel Goldfarb
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Consider this previous exchange on the Cosmic Microwave Background

The crucial bit is this:

However, the crucial assumption of Einstein's theory is not that there are no special frames, but that there are no special frames where the laws of physics are different. There clearly is a frame where the CMB is at rest, and so this is, in some sense, the rest frame of the Universe.

Community
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There is nothing called an absolute inertial frame however in an inertial frame obeying the laws of motion your calculations will hold true.

Dhritiman Banerjee
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