# Inertial Frames of Reference - Inertial vs. Accelerated Frames

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.

• You really should cut down the length of this post and ask your question in as short a way as possible. – joshphysics Mar 15 '13 at 16:19
• @joshphysics. My question is specific, therefore in order to convey the context in which I have understood the concept as above, I need to provide a specific description of my line of thought. The length of this post is completely justifiable and does not need to be cut down. – Ram Sidharth Mar 15 '13 at 16:53
• All right; it's too bad though because if it were shorter, more people would be likely to read the question and venture a response (and or upvote the question)... – joshphysics Mar 15 '13 at 23:12
• @joshphysics. Yes, I understand, I appreciate your concern. If you could make suggestions on how to edit it, it may help, because, I really don't know how to better convey my question. – Ram Sidharth Mar 16 '13 at 21:50
• There is a nontrivial issue here. Relativity requires that inertial frames be redefined compared to the Newtonian definition. In relativity, an inertial frame is a free-falling frame, which would not have been considered inertial in Newtonian mechanics. +1 for a nontrivial question whose answer is nontrivial. – user4552 May 20 '13 at 21:00

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.

• I had thought about pseudo-forces as well, but only after I had posted this question. Nonetheless, you have answered all of my questions with great specificity. Thank you very much. – Ram Sidharth Mar 16 '13 at 21:54

Let me assume that we are talking about global inertial frames as opposed to local inertial frames. I encourage you to research the difference!

It seems to me that your confusion essentially comes from the following statement that you make:

The definition of an inertial frame of reference is restricted only to comparisons between frames of reference.

This is not true. Suppose we use the definition of inertial frame given by Resnik which I paraphrase here

Definition. We call a frame globally inertial provided any particle that is isolated from interactions with all other particles moves with a constant velocity.*

Notice that this definition makes no comparison between frames. The definition of inertial frame is being made via measurements that can be performed in the frame itself. However, it is interesting to note that if two frames satisfy this definition of being inertial, and if you accept that Poincare transformations describe the change of coordinates between inertial frames, then you can prove that any two globally inertial frames move with constant velocity relative to one another.

What is the meaning of making a statement that an inertial frame of reference is an unaccelerated one?

We can define an accelerated reference frame as a frame that is accelerating relative to some inertial reference frame (where inertial reference frame is defined above). Now we can ask the following question:

"Is every accelerated reference frame non-inertial given our definition of inertial above?"

The answer to this is yes because a person making measurements in an accelerating frame as we have defined it will find that even in the absence of interactions with other objects, objects in his frame will not travel with constant velocity. Instead, such objects will be measured to accelerate as if there are "forces" acting on them; these forces are commonly called fictitious or non-inertial forces for this very reason.

• My question is simple. What is the meaning of making a statement that an inertial frame of reference is an unaccelerated one? Further, what is the meaning of the statement that frames that are accelerating with respect to the occupied frame of reference do not determine whether the occupied frame is accelerated or not? Are these observations made from another frame of reference? – Ram Sidharth Mar 15 '13 at 16:50
• @RamSidharth Let me know if the Addendum makes things more clear. – joshphysics Mar 15 '13 at 23:11

An inertial frame is defined as: a frame in which Newton's laws hold. By testing whether Newton's laws hold in your reference frame, you can decide whether your reference frame is an inertial one.

• Could you elaborate a bit?! This is just restating the quoted lines from Resnick's book you can find in the first paragraph of the question, so apparently this is unclear for Ram – Michiel Mar 15 '13 at 18:32
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.

As others have pointed out: to identify an inertial frames of reference you observe motion, and then the laws of motion are the criterium: it's an inertial frame of reference if the laws of motion hold good.

The crucial point: there is a mutual dependence. We have the equivalence class of inertial frames of referencem, and we have the laws of motion. Each depends on the other.

For the laws of motion the concept of inertial frame of reference is the very reference. At the same time, the only way to identify an inertial frame of reference is to invoke the laws of motion as criterium.

That raises the question: is this circular reasoning? Each concept needs the other concept in order to be defined.

It's not circular reasoning, of course. Circular reasoning is it's own world, with no relation to the real world. We all know that the laws of motion are very much real properties of the world.

I emphasize this mutual dependence character because of your repeated assertions that to define some frame of reference all one has is comparison with other frames of reference.

You don't have to look at other frames of reference to find out if yours is an inertial frame. With respect to your frame of reference, that is using your cartesian coordinate axes and your clock, if you determine that Newton's first law is valid then yours is an inertial frame of reference. Let's consider two examples. (1) You are sitting inside a windowless railway wagon, moving uniformly on a smooth, straight track. You place a ball on the floor. Its weight is balance by the floor's normal reaction and the net force on it is zero. You also observe it to be at rest. Therefore, you are in an inertial frame of reference. (2) If at a later time, the train slows, you will observe the ball begin to move. There is no force other than gravity and normal reaction (balancing each other) and yet the ball accelerated (in a direction perpendicular to both). Newton's first law is violated and therefore you are no longer in an inertial frame of reference.

Note that, in neither of the cases did you have to refer to other frames of reference. You were in a windowless wagon !!.

You can apply similar logic to the frame of reference attached to a ball tied to a string and rotating uniformly to conclude that it is non-inertial.

@Ram Sidharth - I read your question to mean; what is it about mass that exhibits inertia which is the same for any mass regardless of relative velocity. Good question, from my readings it would seem that at present we have only an operational definition of inertia. John R. Cox