According to Robert Resnick's book "Introduction to Special Relativity", a line states the following as the definition of an inertial frame of reference: "We define an inertial system as a frame of reference in which the law of inertia - Newton's first law - holds. In such a system, which we may also describe as an unaccelerated system, a body that is acted on by zero net external force will move with a constant velocity."
Therefore, a frame of reference, with respect to which, objects move in a straight line with uniform velocity in the absence of any unbalanced forces. Now my problem with Resnick's definition arises from the above line: "...In such a system, which we may also describe as an unaccelerated system...". How can an observer, occupying a particular frame of reference, realize that he is part of an unaccelerated system. He can only state, that with respect to other frames of reference, there is a uniform relative motion in a straight line. The definition of an inertial frame of reference is restricted only to comparisons between frames of reference. If a frame of reference is to be considered an inertial one, the condition that its relative motion with respect to other frames of reference should be uniform motion in a straight line, is to be fulfilled. Here is where my confusion arises with relation to the above quoted statement: If, for instance,the relative motion observed between two frames of reference is that of uniform acceleration, how can we determine which frame is the unaccelerated system? It is obviously not possible. But according to the statement made above, Resnick states that the frame of reference he occupies is an unaccelerated one. With respect to what? If accelerated motion were to be observed with respect to other frames of reference, how are we to determine that we occupy an inertial frame of reference at all?
Similarly, another statement made by Resnick in his book, which is related to the above question is as follows: "The objects whose motions we study may be accelerating with respect to such frames but the frames themselves are unaccelerated."
He states that inertial frames of reference are still to be considered frames of reference if other frames of reference are accelerating with respect to the occupied frame of reference. My simple question is this: How can we define an inertial frame of reference as an unaccelerated frame of reference unless and until we observe this particular frame of reference from another frame of reference such that the relative motion between these frames of reference is uniform motion along a straight line as per Newton's first law. Another part of this very question is also: How can we call the occupied frame of reference as being inertial regardless of whether other frames of reference are accelerating with respect to the occupied frame of reference? Please resolve these questions as best as you can without any ambiguity, as you know, specificity is very important in conveying ideas regarding to relativity.
Answer
You have said: If,for instance,the relative motion observed between two frames of reference is that of uniform acceleration, how can we determine which frame is the unaccelerated system? It is obviously not possible. and
Another part of this very question is also: How can we call the occupied frame of reference as being inertial regardless of whether other frames of reference are accelerating with respect to the occupied frame of reference?
Both these questions have been answered below.
Why would it not be possible? If you are in a reference frame which is accelerating at all, then you will experience pseudo-forces(forces whose source is not determined in that frame). That will tell you that your frame is accelerating. Moreover,if the relative motion between two frames is that of uniform acceleration,then both are accelerating! You do not have to determine WHICH is accelerating! The presence of acceleration(uniform or not) for any reference frame, guarantees that you will experience pseudo-force if you are in it. for example, if you throw a ball from a height,it seems to hit the ground after travelling a path perpendicular to ground. but the actual trajectory is not so. as the ball falls it is deflected due to Coriolis force,which is a pseudo-force. so technically the earth is not an inertial frame of reference in any way since we can never point to a source who caused this Coriolis force!
You have said: Resnick states that the frame of reference he occupies is an unaccelerated one. With respect to what? If accelerated motion were to be observed with respect to other frames of reference, how are we to determine that we occupy an inertial frame of reference at all?
According to Resnick he occupies an inertial frame that means, in his frame, Newton's first law holds true. obviously you need a reference object. when we say a car travels at 75m/s then we actualy mean it travels 75m/s with respect to, say,a stationary tree. but it would travel at 50m/s with respect to another car travelling with 25m/s. so you need a reference object.
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