# What are non-inertial frames of reference?

I am reading through a textbook on AP Physics, and I came across a few references to non-inertial frames of reference. It doesn't clearly say what a non-inertial frame of reference is. Based on the examples it gives, I assume that it is a frame of reference where the observer is experiencing acceleration.

Am I right?

QuantumQuest
Gold Member
A non-inertial frame of reference is one that is undergoing a non zero acceleration.

• General Scientist
A non-inertial frame of reference is one that is undergoing a non zero acceleration.
Thanks a lot!

• QuantumQuest
An example of a noninertial frame would be a coordinate system attached to and rotating with a spinning turntable.

• General Scientist
A non-inertial frame of reference is one that is undergoing a non zero acceleration.

I think one should be cautios here. As is pointed out in MTW "Gravitation" (6.3) the accelerated reference frame is a local thing - it does not cover the whole space. What does it mean? Well, according to my back-of-the-envelope calculation, if you were to stand at the front of the rocket of length ##L## which is accelerating with uniform acceleration ##a##, and your friend would stay at the very back of the same rocket, your clocks would tick differently. The relationship between your (##\tau_1##) and your friend's (##\tau_2##) proper times would be:

##\tau_2\approx(1-\frac{aL}{c^2})\tau_1## in the limit of ##a\to0##

Where ##c## is the speed of light. So as the space you include in your "accelerated reference frame" is small (##aL/c^2 \ll 1##), you can talk about a reference frame. Once you go beyond distance ##c^2/a##, the accelerated reference frame no longer makes sense.

QuantumQuest
Gold Member
I think one should be cautios here. As is pointed out in MTW "Gravitation" (6.3) the accelerated reference frame is a local thing - it does not cover the whole space. What does it mean? Well, according to my back-of-the-envelope calculation, if you were to stand at the front of the rocket of length ##L## which is accelerating with uniform acceleration ##a##, and your friend would stay at the very back of the same rocket, your clocks would tick differently. The relationship between your (##\tau_1##) and your friend's (##\tau_2##) proper times would be:

##\tau_2\approx(1-\frac{aL}{c^2})\tau_1## in the limit of ##a\to0##

Where ##c## is the speed of light. So as the space you include in your "accelerated reference frame" is small (##aL/c^2 \ll 1##), you can talk about a reference frame. Once you go beyond distance ##c^2/a##, the accelerated reference frame no longer makes sense.

What I really don't understand is why did you quote my post. The question of the OP was (quoting from post #1)

... It doesn't clearly say what a non-inertial frame of reference is. Based on the examples it gives, I assume that it is a frame of reference where the observer is experiencing acceleration.

Am I right?

Is a non-inertial frame of reference something different from what I wrote?

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Nugatory
Mentor
Is a non-inertial frame of reference something different from what I wrote?
You wrote "A non-inertial frame of reference is one that is undergoing a non zero acceleration."

I suggest that a more generally accepted definition is something along the lines of: An inertial frame is one in which a body with no net force acting on it does not accelerate. Conversely, if an object with no net force acting on it does accelerate in a given frame, then that frame is not inertial. Examples of each are left as an exercise for the reader - and everyone is reminded that this is the classical physics forum, so gravity is a considered a real force.

The problem with your definition is that it is not at all clear what it means to speak of a "frame undergoing acceleration". Objects can have acceleration, but a frame is just a convention for assigning time and space coordinates to events, and it's not clear how to accelerate a convention. That doesn't stop people from talking about "accelerated reference frames", but that's because natural language sacrifices precision for convenience.

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• QuantumQuest and dextercioby
Nugatory
Mentor
I think one should be cautious here. As is pointed out in MTW "Gravitation" (6.3) the accelerated reference frame is a local thing - it does not cover the whole space....
This is correct, and it points out one of the pitfalls in the "accelerated frame" concept..... But MTW (for the uninitiated, that's a graduate-level general relativity textbook, and one of the more demanding ones at that) would not be my go-to source for a B-level thread in the classical physics subforum.

• suremarc, dextercioby, vsv86 and 1 other person
Nugatory
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This thread is open again after removing a long digression.

I am reading through a textbook on AP Physics, and I came across a few references to non-inertial frames of reference. It doesn't clearly say what a non-inertial frame of reference is. Based on the examples it gives, I assume that it is a frame of reference where the observer is experiencing acceleration.

Am I right?

How about we try to define a reference frame? I will not try to be rigrous here, but merely gather the things I think are necessary for this concept.

Nothwithstanding my previous digression (I agree it was a digression - my bad) to relativity, in Newtonian mechanics we can define a reference frame as an origin ##O## and a set of unit vectors, ##\hat{\vec{a}}, \hat{\vec{b}}, \hat{\vec{c}}##, such that any vector ##\vec{v}## in space can be written as a linear combination of this set ##\vec{v}=v_a\hat{\vec{a}}+v_b\hat{\vec{b}}+v_c\hat{\vec{c}}##.

Next we can define an "observer in the reference frame" as observer who's position is (relative to ##O##): ##\vec{u}=u_a\hat{\vec{a}}+u_b\hat{\vec{b}}+u_c\hat{\vec{c}}##, where ##u_{a,b,c}## are constant in time.

Now the difference between the intertial reference frame and a non-intertial one is simply that the basis vectors depend on time (t): ##\hat{\vec{a}}=\hat{\vec{a}}(t)## and same for others. More precisely, for a non-inertial reference frame the dependence of the basis vectors on time, should be at least quadratic (so that second derivative should not vanish).

We can now easily deduce that the observer in the non-intertial reference will be accelerating.

This all may seem quite trivial, but the TC complained about the lack of definition of the reference frame. I think my little sketch above contains sufficient flexibility to cover all exercises with non-inertial reference frames I have seen thusfar.

PS: Of course on must also allow for unit-vectors to depend on position to have the full generality of treatment.

Dale
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2020 Award
I came across a few references to non-inertial frames of reference. It doesn't clearly say what a non-inertial frame of reference is.
I would tend to take the approach outlined here:

http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node9.html

Basically, the idea is to understand Newton’s first law as a definition of an inertial frame. Then a non inertial frame is simply one that violates Newton’s first law such that force free objects do not travel in a straight line with constant speed.

• QuantumQuest, vsv86, suremarc and 2 others
QuantumQuest
Gold Member
I want to apologize for digressing in this thread. It was really not my intention to do so.

You wrote "A non-inertial frame of reference is one that is undergoing a non zero acceleration."

I suggest that a more generally accepted definition is something along the lines of: An inertial frame is one in which a body with no net force acting on it does not accelerate. Conversely, if an object with no net force acting on it does accelerate in a given frame, then that frame is not inertial. Examples of each are left as an exercise for the reader - and everyone is reminded that this is the classical physics forum, so gravity is a considered a real force.

The problem with your definition is that it is not at all clear what it means to speak of a "frame undergoing acceleration". Objects can have acceleration, but a frame is just a convention for assigning time and space coordinates to events, and it's not clear how to accelerate a convention. That doesn't stop people from talking about "accelerated reference frames", but that's because natural language sacrifices precision for convenience.

Yes, you're absolutely right. I thought that it was sufficient to write it like I did i.e. that it would convey the meaning I intended to, but yes, I think that I should phrase it in a better way so I apologize for this too.

A non-inertial frame of reference is one that is undergoing a non zero acceleration.

Don't you need a second system to define the acceleration of the first, which only raises the question if the second system is inertial?

Now you could argue that there exist accelerometers. But they use Newton's laws to work, which already require an inertial frame in their assumptions. This seems like a circular argument.

robphy
Homework Helper
Gold Member
Possibly useful:
Frames of Reference (1960) .. go to 13m27s

QuantumQuest
Gold Member
Don't you need a second system to define the acceleration of the first, which only raises the question if the second system is inertial?

Now you could argue that there exist accelerometers. But they use Newton's laws to work, which already require an inertial frame in their assumptions. This seems like a circular argument.

Yes, all motion is relative but in a non-inertial frame of reference an object (having no force acting on it) does accelerate while in an inertial frame of reference it does not.

• Mason Smith
Dale
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2020 Award
Now you could argue that there exist accelerometers. But they use Newton's laws to work, which already require an inertial frame in their assumptions. This seems like a circular argument.
Accelerometers may use Newton’s laws to work, but that needn’t be part of their definition. They can be defined simply by instructions to build one. Similarly with clocks, rulers, and force plates. That eliminates the circularity.

However, a little bit of circularity is not in and of itself a problem, as long as everything is self consistent and subject to experimental verification and falsification.