# What is an inertial reference frame?

Tempest

## Main Question or Discussion Point

I am not really sure I have the concept of an inertial reference frame down, can anyone help me?

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I am going to give an answer that's probably more difficult than necessary. Classicaly, two frames are inertial if observer in each frame can agree on Newton's Laws (law of inertia, f=ma, action/reaction). Turns out that they must be moving at constant speed relative to each other. If you are in an accelerated frame, then newton's laws won't hold. The first one, for example, would break down, because object would have acceleration without force (there is an ambient force due to the fact that the observer is accelerating).

The problem is how do we jump from one frame to another so that measurements in each frame agree. By simple assumption of homogeneity (no preferred origin) and isotropic (no preferred direction) of space one can prove that there are three kinds of transformation possible: one with a speed minimum, one with a speed maximum (Lorentz), and one with neither (Galileon). The first kind turns out to violate causality. And the third kinds turns out to be invalid because some other law of physics is not invariant under it (Maxwell's Eqn). So one conclude that the second one to be correct and the third one to be a limiting case of it (limit as the speed limit goes to infinity). And for all law (included EM) to be correct, this speed limit must be the speed at which EM disturbance travel in vaccum, i.e. the speed of light. Arriving at this conclusion requires 5 assumptions: 1) all Law of physics is invariant when you go from one inertial frame to another. 2) the transformation from one frame to another is linear 3) spacetime is homogeneous 4) spacetime is isotropic 5) Causality is preserved. I like this derivation better because the speed limit turn out to be a result, not an assumption. And the 5 assumptions above are more "obvious", "trivial", and "intuitive".

In short, inertial frame are frames that moves at constant speed with respect to each other. And by assuming the 5 assumptions above, and using some group theory and linear algebra. One arrive at the Lorentz transformation which has stand the tests of experiments.

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• fireflies
HallsofIvy
Homework Helper
More simply, an inertial frame is one that is not accelerating.

Velocity is "relative"- always measured with respect to something that is taken to be stationary. Acceleration is not relative so asserting that a frame of reference is or is not accelerating can be done independently of other frames.

Of course, in general relativity it is shown that such tests of acceleration depend on effects that could be attributed to forces such as gravity so the idea of an inertial frame becomes blurred in general relativity.

Tempest
Originally posted by HallsofIvy
More simply, an inertial frame is one that is not accelerating.

Velocity is "relative"- always measured with respect to something that is taken to be stationary. Acceleration is not relative so asserting that a frame of reference is or is not accelerating can be done independently of other frames.

Of course, in general relativity it is shown that such tests of acceleration depend on effects that could be attributed to forces such as gravity so the idea of an inertial frame becomes blurred in general relativity.
Ok I have a few questions. First you say that an inertial reference frame is one that is not accelerating. Is that a necessary and sufficient condition for a reference frame to be inertial, or is that just a necessary condition.

Next you go on to say that "acceleration is not relative". But acceleration is relative. Consider a falling apple. IN a reference frame fixed at the center of mass of the apple, the center of mass of the earth accelerates towards the center of mass of the apple. In a reference frame fixed at the center of mass of the earth, the center of mass of the apple accelerates towards the earth. Hence, acceleration is relative.

Consider a free floating body in space, say a spaceship. The engines are off, and you release your apple. The apple floats. From that simple observation, you know the net force on the ship is zero, from which you conclude that the ship is not accelerating. Now consider another person in another ship which has its engines on, and is accelerating as it passes by you. You see it moving faster and faster, and suppose somehow you measure its rate of acceleration as 9.8 meters per square seconds. Now, in a reference frame whose origin is the center of mass of the rocket, you are acclerating at a rate of 9.8 meters per second. Hence, acceleration is relative. What isn't relative is force.

You are inside your ship, and you dont feel any force, but for observers in the other ship, they do feel forces. You see, acceleration is a kinematic variable, and is relative, for the same reason that velocity is relative. It is only when you multiply the accleration of an object by its inertial mass, that you come up with the quantity known as force.

And it is here that we see the concept of an inertial reference frame begin to emerge. For occupants in the ship that don't feel any forces, those people can say they are in an inertial reference frame, and for people on the rocket with its engines on, those people are not in an inertial reference frame. It is easy to see why the rocket with its engines on isn't an inertial reference frame. Suppose you let the apple go. You will see the apple fall to the floor, as if gravity is pulling it down. But there is no action reaction PAIR for this apple, hence Newton's third law isn't satisfied. So in other words, there is no force acting upon the apple as soon as it is released. If you were inside the apple the moment it was let go, you would instantaneously be in an inertial reference frame, while the rest of the rocket would be a non-inertial reference frame.

And from these simple thoughts, we eventually arrive at Einsteins general theory of relativity, which is that there is no way for an observer inside a reference frame accelerating uniformly, to tell that his ship isn't at rest in a gravitational field.

Thus, einstein asserts that the frames are equivalent, and this is his principle of equivalence.

The error in the general theory of relativity is exposed when one considers electrodynamics inside the ship. Einstien only thought of dropping neutrally charged objects to the floor. And if that is the only kind of object you are permitted to drop, then indeed the two frames are equivalent. However, consider Maxwellian electrodynamics. When an electrically charged object experiences a force, it radiates an EM wave. OR in more modern terms, if an electron experiences a force, it radiates a photon. Thus, suppose that we are in our ship, and that ship is on the surface of a planet. Now, instead of dropping a neutrally charged object, let us drop an electron. What really will happen is that the electron will radiate photons. So if you perform your experiment in the dark, and you see light hitting you after you release the electron, you can conclude that you are in a gravitational field, rather than uniformly accelerating through empty space.

On the other hand, suppose that you are uniformly accelerating through empty space, and you release the same electron. The moment you release the electron, the electron will be in an inertial frame, and it wont be experiencing any forces, and so it wont radiate, and so if your lights are off, you will not see the electron emit light.

Now, in either case the electron "falls to the spaceship floor" but it is the laws of electromagnetism themselves, that demand that a uniformly acclerating reference frame be NON equivalent to a reference frame at a constant position in a gravitational field well.

However you slice it, general relativity contains an error. Einstein went to all the trouble to demand that the laws of EM hold in SR, so its ironic that he totally forgot about them when he created GR.

One other thing I would point out, is that after Einstein made his special theory public, I think he began to think about ways that SR could be proved false. I think it occurred to him, that if a photon could accelerate in an inertial reference frame (IRF), then SR is wrong. I think he then stumbled onto the idea of a gravitational field pulling on a photon, and thus accelerating it. So what I am saying is that Einstein himself tried to falsify SR, and a result hit upon GR. This is pure conjecture of course, but it makes a bit of sense. He wanted to hit upon the idea of GR, before anyone else came up with the idea of a photon that can accelerate in an IRF.

As for the idea of an IRF becoming blurred in relativity, that should be impossible. Once you define an IRF, you are done. There is no changing the definition later to suit your needs. If you want something new, you define something new.

Now, consider a reference frame centered at the earth. If we release a coin at the CM of earth, and that coin doesnt move, then this an IRF. Now, raise an apple into the air and hold it there. Locate the center of mass of the earth apple system (its roughly where the CM of earth was before but raising the apple changed things a bit). Now switch to the CM of the earth apple system, and release the apple.
The apple exerts force F on earth, and vice versa by newtons third law, and from the CM frame, newtons third law is satisfied, and so the event was witnessed in a IRF. At any rate, the general theory of relativity ignores a fact about how real electrons behave. Again, either when you let that electron go it emits light or it does not emit light, and depending on which you can tell whether your are uniformly accelerating through deep space, or at rest in a gravitationaly field. The principle of equivalence is false, but you must take into account EM.

Now if you really want to confuse yourself, consider the fact that when an electron experiences a force, it must radiate a photon. Now think of the Bohr model of a hydrogen atom.

He starts out with the idea that the force of attraction between an electron and proton is given by the coloumb formula:

F = K Qq/r^2

He then assumes the mass of the proton far exceeds the mass of the electron, so that he can assume the CM of the system is located where the proton is. Thus, the proton is at the origin of the frame he is thinking of. He then imagines the electron orbiting around the proton, and so it thus has centripetal acceleration v^2/r, and the centripetal force is the electron mass times that. He then equates the two:

mv^2/r = KQq/r^2

From which it follows that

mv^2 = KQq/r

Since the proton and electron have equal electric charges, Q=q, hence he wrote:

mv^2 = KQ^2/r

Now, without going on, by EM this electron experiences a force, and so it should radiate light, but then a hydrogen atom is unstable, and the time it would take for an electron to spiral inwards and strike the proton is less than one second. But of course, the hydrogen atom is stable, and so he fixes the electron orbits by assuming that the electrons angular momentum mvr is equal to an integral multiple of planck's constant. This helped usher in the age of Quantum Mechanics.

At any rate Bohr's model of a hydrogen atom makes you wonder if there is something wrong with EM theory. EM theory predicts that hydrogen atoms should be unstable, but they are stable as quantum mechanics was designed to say. And well, if there is an error in EM then the theory of special relativity contains an error, and simultaneity is absolute and not relative. Who knows.

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• Tempest: Consider a free floating body in space, say a spaceship. The engines are off, and you release your apple. The apple floats. From that simple observation, you know the net force on the ship is zero, from which you conclude that the ship is not accelerating. Now consider another person in another ship which has its engines on, and is accelerating as it passes by you. You see it moving faster and faster, and suppose somehow you measure its rate of acceleration as 9.8 meters per square seconds. Now, in a reference frame whose origin is the center of mass of the rocket, you are accelerating at a rate of 9.8 meters per second. Hence, acceleration is relative. What isn't relative is force.
I disagree. You can always tell that you are accelerating no matter how something else is moving relative to you. Acceleration can be defined dynamically. (a=f/m) If I am moving at a constant V and I see a ship accelerating toward me, I can't say it's relative because I know I'm not accelerating.

Consider two ships who are accelerating toward each other at 9.8 m/s^2. Would it be correct to say that one ship is accelerating toward the other at 18.6 m/s^2? No. You might say that kinematically that is correct but you can still tell exactly what your acceleration is without even looking at the other ship. Knowing your acceleration, you can easily figure out the other ship's acceleration.

Tempest: I am not really sure I have the concept of an inertial reference frame down, can anyone help me?
How is it that you don't fully understand the concept of an inertial reference frame and yet you are able to find errors in General Relativity?

Oh wait, I get it. That's what they call trolling, right?

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In the context of Newtonian Physics, an inertial reference frame is a reference frame in which "every body remains in a state of rest or uniform motion (constant velocity) unless it is acted upon by an external unbalanced force".

So, our first Newton´s law gets transformed into: There exists at least one inertial reference frame. MaribuS.

JesseM
Hi MaribuS, just a reminder, when searching through old threads make sure to check the dates on the upper left of each post before adding a new post of your own...in this case no one had posted on this thread since 2003! Better to start a new thread if you want to talk about the subject.

Sorry, folks... I entered the forum via Google, while I was looking for something else (Ludwig Lange on Inertial Reference Frames)... Well... I could't resist posting something.

By the way, one problem in "my" above definition (and law) is that time is a "pre-concept", something that has to be known before using it (like force), not a post-concept "derived" from the first law itself... Just remember how in Special Relativity nature provides us with a fundamental clock (light). MaribuS.