Can someone explain non-inertial reference frames

[wbandersonjr]

What is a non-inertial reference frame? how is it defined?

 

[G01]

Simply, a non-inertial reference frame is a reference frame that is accelerating.

Example:

1. You are in a car traveling at a constant 70mph, doing experiments of some sort. This reference frame is inertial.

2. If the car was accelerating, the reference frame would be non-inertial.

3. If the car was traveling around a curve at a constant speed the reference frame would also be non-inertial, because there will be a centripetal acceleration.

 

[BruceW]

Any inertial frame is traveling at a constant speed with respect to any other inertial frame. This is the definition.

 

[vanhees71]

First of all one has to understand the issue of inertial frames. This is not so easy as it might seem.

In Newtonian physics and also on the level of the Special Theory of Relativity (STR) it's simply postulated that there exists a class of reference frames, where the principle of inertiality holds true. That means that anybody moves with constant velocity if there is no force that accelerates it.

This has brought up the question, how to determine physically one inertial frame and then with it all inertial frames since two inertial frames must be in uniform motion with respect to each other. This is achieved by either a Galilei boost (in Newtonian physics) or a Lorentz boost (in STR). It is indeed not so easy to define such an inertial reference frame on purely physical grounds. The physicist Ernst Mach came up with the idea that the reference frame, which is at rest with respect to the fixed stars around our galaxy determines such an inertial frame, but that's as much a postulate as is the more abstract formulation about the existence of an inertial frame, but at least it's an assumption, one can check experimentally in principle.

The best model of space-time we have today is Einstein's General Theory of Relativity (GTR), which grew out of the question, how to describe gravity within relativistic physics. As Einstein figured out quite quickly, a very natural description of gravity is by what's called "geometrization of the force" in modern language. The specialty of gravity compared to the other fundamental forces is the validity of the equivalence principle. In its weak form it says that any body, no matter of which material and of which mass will follow the same trajectory as any other body as long as the only acting force is gravity.

Now, as one can easily figure out in both Newtonian and STR mechanics, in an non-inertial frame, i.e., in a reference frame which is accelerated against any inertial frame, one finds forces that are proportional to the mass of the body. In Newton's 2nd Law thus the (inertial) mass on the left-hand side of the equation cancels against the mass in the gravitational-force Law on the right, and that's why the weak equivalence principle holds. But know consider a physicist in a black box making experiments. Then he cannot distinguish the situation wheter his box determines an inertial frame of reference, and he has some gravitational field around him (say the one of the earth) which accelerates all bodies in the same way, independent of their mass and material or whether there's no gravitational field, and his box is accelerated against a inertial frame and thus what he observes are not gravitational fields but simply the inertial forces originating from the non-inertiality of his reference frame. Thus, within Newtonian mechanics, at least locally a gravitational field (observed with respect to an inertial frame) cannot be distinguished from the motion of bodies in a non-inertial frame without gravitational field.

Now, Einstein analyzed this aspect of the connection between non-inertial frames and inertial forces with gravity within STR. What comes out in the case of STR is that a non-inertial frame not only leads to the appearence of inertial forces, but for an observer at rest relative to a non-inertial frame the geometry of space becomes non-Euclidean, i.e., if he measures the three angles in a triangle, he doesn't find its some to be 180 degrees but some different value. Also the ratio between the circumference of a circle and its diameter is no longer $\pi$ but some other value, and in general it will not be a constant independent of the size of the circle. That led Einstein to the idea that gravitation is equivalent to the non-Euclidicity of space, and that the motion of a body in a gravitational field can be described as moving along the straightest possible lines (the socalled geodesics) of a non-Euclidean space.

Then, after a long struggle over about 10 years, Einstein came to the conclusion that not only three-dimensional space is non-Euclidean at presence of a gravitational field but four-dimensional space-time as a whole! This lead to the final theory, namely the General Theory of Relativity (GTR).

Now the issue of inertial frames is pretty clear! In general you cannot define a global inertial reference frame (except in an empty universe, where nothing can cause gravity), but only in a small four-dimensional space-time interval. Any body that is free falling, i.e., for which no other forces than gravity act, defines a local inertial frame. E.g., the International Space Station (ISS) is to a good approximation freely falling in the gravitational field of the Earth, the Sun etc. That's why within this quite small environment inside the ISS, one feels no gravity, and thus the interior of the ISS is (to a good approximation) a local inertial frame.

 

[DickL]

Vanhees, thanks, that is a good explanation of inertial and non-inertial frames, and more interesting for me is the development from STR to GTR that was included. It has helped me fill in some context between these two theories.

 

[olivermsun]

In classical physics, it's kind of circular in the sense that an inertial frame is one in which Newton's First Law (inertia) more-or-less holds.

The accelerating car as mentioned above is a common example of a non-inertial frame. The surface of the rotating Earth is another good example, and one where noninertial coordinates are often used.

 

[fluidistic]

The physicist Ernst Mach came up with the idea that the reference frame, which is at rest with respect to the fixed stars around our galaxy determines such an inertial frame, but that's as much a postulate as is the more abstract formulation about the existence of an inertial frame, but at least it's an assumption, one can check experimentally in principle.

Does Mach mean he can notice a motion of the stars when he's accelerating inside his car?
Or does he mean that as long as you don't see stars moving then you can consider yourself over an inertial frame? Or neither of these sentences and I'm not understanding what Mach said.

 

[RedX]

Does Mach mean he can notice a motion of the stars when he's accelerating inside his car?
Or does he mean that as long as you don't see stars moving then you can consider yourself over an inertial frame? Or neither of these sentences and I'm not understanding what Mach said.

I was wondering about that too. You can see the fixed stars move because the Earth is rotating. But if the Earth were not rotating but accelerating through space recti-linearly, then the fixed stars should not move. But clearly such a frame is not inertial.

Now the issue of inertial frames is pretty clear! In general you cannot define a global inertial reference frame (except in an empty universe, where nothing can cause gravity), but only in a small four-dimensional space-time interval. Any body that is free falling, i.e., for which no other forces than gravity act, defines a local inertial frame. E.g., the International Space Station (ISS) is to a good approximation freely falling in the gravitational field of the Earth, the Sun etc. That's why within this quite small environment inside the ISS, one feels no gravity, and thus the interior of the ISS is (to a good approximation) a local inertial frame.

Doesn't a tetrad define a local inertial reference frame for each point in spacetime? Tetrad indices are pure Minkowski, and the relationship between the tetrad and the metric is given by the vielbein, allowing two equivalent formulations of general relativity, one based on local inertial frames, and the other based on the metric function. So can't local inertial frames be defined at each spacetime point, and not just in space-time intervals?

 

[Hurkyl]

So can't local inertial frames be defined at each spacetime point,

But they can't be sewn together to form a Minkowski coordinate chart on an extended region of space-time.

 

[RCP]

Can't believe this dialogue finished here. So much I would like to learn about irf's but am reluctant to spell it all out on a necromanced thread like this. Perhaps all my questions would have been answered had humber not been banned here as a crack-pot. IMO such posters should be refuted-as Dalespam was attempting-rather than dismissed. Now a year later I'm still confused about reference frames. (in the name of full disclosure I'll state the obvious; I'm not very smart. :()

 

[iVenky]

Then, after a long struggle over about 10 years, Einstein came to the conclusion that not only three-dimensional space is non-Euclidean at presence of a gravitational field but four-dimensional space-time as a whole! This lead to the final theory, namely the General Theory of Relativity (GTR).

Now the issue of inertial frames is pretty clear! In general you cannot define a global inertial reference frame (except in an empty universe, where nothing can cause gravity), but only in a small four-dimensional space-time interval. Any body that is free falling, i.e., for which no other forces than gravity act, defines a local inertial frame. E.g., the International Space Station (ISS) is to a good approximation freely falling in the gravitational field of the Earth, the Sun etc. That's why within this quite small environment inside the ISS, one feels no gravity, and thus the interior of the ISS is (to a good approximation) a local inertial frame.

Thanks a lot for your explanation. I was searching for this. I have few doubts. You say "Three-dimensional space is non-Euclidean at presence of a gravitational field". Do you mean whenever we have gravitational field we have non-inertial frame? Also at the same time you are saying "Any body that is free falling i.e., for which no other forces other than gravity act, defines a local inertial frame". Does that mean that everything is Euclidean whenever you are in local inertial frame though they experience gravitational field?

Thank you all