# Inertial Frames of Reference - Description of an Inertial Frame of Reference

An inertial frame of reference is described as being a frame of reference in which the first law of Newton (the law of inertia) holds. This means that all events as described with respect to this frame of reference must have a zero net force acting on it and therefore traces a straight line with a uniform non-translatory motion. But, I have read in some books, especially "Introduction to Special Relativity" by the well-known Robert Resnick, wherein his definition of an inertial frame of reference also refers to such a frame of reference as being an unaccelerated system. This is where I am confused.

How can we describe a frame of reference as being unaccelerated if we occupy the frame of reference itself? No mechanical experiment conducted solely confined to a single frame of reference can determine the absolute motion of the frame of reference relative to another frame of reference. All that can be understood is that there is a certain uniform relative motion between frames of reference and no more. Is Robert Resnick saying that the inertial frame of reference is unaccelerated with reference to another frame of reference?

all events as described with respect to this frame of reference must have a zero net force acting on it

This is wrong: there may exist nonzero forces in an inertial frame of reference. These have to obey Newton's laws. If the forces do not obey Newton's laws in your frame of reference, then it is not an inertial frame.

Newton's laws are correct only in an inertial frame of reference. What is an inertial frame of reference? One in which Newton's laws are correct.

• Your answer is much appreciated. – Ram Sidharth Mar 20 '13 at 0:45

Yes, the (any) inertial frame is unaccelerated relatively to any other inertial frame. The previous sentence, if it were the only thing we could say, would be a circular definition of the inertial systems of a sort. But given one inertial frame, it would still be enough to find all the other inertial frames.

You should view the situation as follows: Newton's theory or, analogously, Einstein's special theory of relativity postulates that there exist inertial frames. We may also say that it's those in which all the objects that are unexposed to any forces remain in a uniform motion in the same direction. One may see that the previous sentence implies that there exist infinitely many inertial frames; and they're in uniform unaccelerated motion with respect to each other.

All other frames, those that are not in uniform unaccelerated motion in the same direction relatively to an inertial frame, are non-inertial and according to these frames, objects may move along curved or accelerated paths even if no forces act upon them.

• Your answer appears to partially resolve my question, but could you be a little more specific? – Ram Sidharth Mar 15 '13 at 12:35
• Specific about what? I guess that the answer is No because I don't understand what you're dissatisfied with. – Luboš Motl Mar 15 '13 at 17:15

Although you cannot do any experiment to tell if you're in motion at constant velocity, you CAN determine if you are accelerating. In such cases, in order to make sense of what you see, you need to add some "fake" forces to make Newton's 2nd law work out correctly.

An example of this is how your body seems "forced" to one side as your moving car makes a sharp turn, or is "forced" forward as your car brakes. Your accelerating (=velocity changing) car is an accelerating frame of reference. If you make the incorrect assumption that your joyriding frame of reference isn't accelerating, then in order to make Newton's 2nd law work, you're going to have to insist that there are some real forces making your body move sideways around a turn, or making your body lean forward when the car is braking. What's even more difficult is to explain why such forces seem to depend on the velocities of objects outside the car, since you've assumed you're at constant velocity or at rest.

An inertial frame of reference is described as being a frame of reference in which the first law of Newton (the law of inertia) holds. This means that all events as described with respect to this frame of reference must have a zero net force acting on it and therefore traces a straight line with a uniform non-translatory motion.

Your explanation of the bolded statement is a misrepresentation.First and foremost all inertial reference frames move at a relative velocity with respect to each other.You can never tell if your inertial reference frame is moving, you can only observe that all other reference frames are moving at defined velocities relative to your reference frame.So the book you cited most probably means that. Nonetheless, non-zero forces can act on bodies in an inertial reference frame.However Newton's first law of motion is invariant in any inertial reference frame.This means that if a body accelerates in an inertial reference frame with an acceleration, $a$, in all other inertial reference frames, the body will have the same acceleration. Proof:Let $r$ be the position vector of a body in motion in an inertial reference frame S, let $r'$ be the position vector of the same body in another inertial reference frame T, and R be the position vector of T relative to S. Then by vector addition,$$r= r' + R$$ $$\frac{dr}{dt}= \frac{dr'}{dt} + \frac{dR}{dt}$$ $$\frac{d^2r}{dt^2}= \frac{d^2r'}{dt^2}$$ $$a=a'$$ Notice that $\frac{d^2R}{dt^2}=0,$since the bodies are not accelerating with respect to each other.So you the body's acceleration does not change from one inertial reference frame to another. I would have provided a diagram to represent the coordinate systems of the two inertial reference frames.It was a little tedious, so i hope you can follow the equations.