# Definition of inertial frame

1. Nov 3, 2008

### calculus_jy

einstein: a set of frames which move without acceleration to one another and that the laws of physics hold in the simplest-is the a definition or that just a assumption that are existence of such frame,

i have beeen told about using acclerometer, but the problem is that even acceleration and force is also relative -what is acclerometer measuring( a force(relative to what) but how do you conclude it is fictiticious ie with no actor of force, such that newton's 1st law does not hold)
and is it not possible that newtons law is always valid... because if you observe acceleration then you must conclude acted on by a force??(and dont say the actor of force cannot be found so the force is fictiticious...)

the concept is still a bit vague... can some help me find logic to this seemingly circular definition?

Last edited: Nov 3, 2008
2. Nov 3, 2008

### atyy

It is a definition and an assumption. To figure out your frame, you have to do experiments and see if the results match the "simple laws". If they do, then you've found an inertial frame. If they don't, but you manage to find some other laws, then the assumption says that you will be able to make those laws simple by some coordinate transformation.

3. Nov 3, 2008

### matheinste

Hello calculus jy

Acceleration is not relative.

Matheinste.

4. Nov 3, 2008

### HallsofIvy

In Special Relativity acceleration is not relative. In General Relativity, acceleration is equivalent to an outside force (acceleration cannot be distinguished from a "gravitational force") so an "inertial frame" is defined as one without outside forces.

5. Nov 3, 2008

### DrGreg

A very simple and crude accelerometer would consist of a small mass attached by a short spring to the observer. If the observer accelerates relative to an inertial frame, the mass is accelerated by an extension of the spring -- the length of the spring measures this "proper acceleration" and if the spring does not extend (i.e. the mass floats freely relative to the observer) the observer must be an inertial observer. That is one way of defining an inertial observer: small stationary particles that are close to the observer remain stationary.

In special relativity, that definition works perfectly. In general relativity, the particles have to be small enough not to be gravitationally attracted to the observer and close enough to the observer not to be affected by the tidal effects of spacetime curvature. So the definition is only approximately true and if you want a really rigorous definition you need to formulate it in terms of calculus.

It depends what you mean by "acceleration". It's true if you mean "proper acceleration", acceleration measured by an accelerometer, which is also the acceleration measured by a co-moving inertial observer. It's also true in the sense that all local inertial observers agree whether something is accelerating or not. But in general, it's not true:

1. Even in special relativity (SR), different inertial observers disagree on the value of a non-inertial object's acceleration, although they all agree it's not zero.

2. In both SR and general relativity (GR), relative to a non-inertial observer acceleration certainly is relative. And in GR, inertial objects can even accelerate relative to a non-local inertial observer -- this is the "tidal effect" of gravity.

6. Nov 3, 2008

### matheinste

Hello DrGreg.

Would it be true to say that in SR an accelerated observer knows he is accelerating and hence assumes that he is not in an inertial frame. I assume he knows this without reference to anything outside his reference frame.

Matheinste.

7. Nov 3, 2008

### pallidin

That's a good question. Here I sit, spinning around the earths axis, which is revolving around the Sun, which is revolving around the center of our galaxy.
Yet, I feel nothing.

8. Nov 3, 2008

### Naty1

It is good you realize you are a bit unclear. That's a solid start.

(Here's a first draft of some notes I'm making for my own understanding...all comments welcome.)

Frames of reference are subtle in many respects and take some time, patience and understanding. There are a number of pieces that become evident only after considerable thought. Isaac Newton understood the importance of proper frames of reference and his fundamental laws involved accelerations, changes in velocities of physical bodies rather than velocities themselves since velocities are RELATIVE. Are you moving or am I? There is no real way tell. But when I accelerate I feel a force, just like I feel gravity. So acceleration is fundamentally different from velocity and is closely related to gravity.

Frames of reference are crucial in nearly all physical measurements. Relativistic mass and kinetic energy depend on the reference frame; but potential energy does not. Gravity and velocity can each change observed space and time: Observed measures for space and time become subject to frames of reference, as does the observed speed of light. All observers will NOT agree.

Special Relativity: NO GRAVITY
Einstein spent considerable time thinking about frames of reference before starting on special relativity. In special relativity, without gravitation, an inertial frame of reference means bodies move uniformly in a straight line without acceleration whenever there are no external forces; This is because there is no gravity to curve space and curve motion and accelerate bodies. No matter how far you measure, everything is still straight line. So bodies move in straight lines and gravity does not curve (change) time either. Distance and time are affected by velocity.

General Relativity: WITH GRAVITY
The tricky part comes with General Relativity where gravity curves spacetime and all motion which causes acceleration. A photon will accelerate in gravity: speed remains "c" locally but the direction changes from straight line to a curved trajectory. When is a body undergoing no force when gravity is present? When it is free falling! So a free falling frame of reference is an (LOCAL) inertial frame when gravity is present.

Why "local"? Because all gravitational fields are curved, and non uniform, this definition only works (locally) over small distances of space and small amounts of time. That's because locally, the curve of spacetime is so small as to be neglected. (Think of measuring tangents to curves.) Einstein came to realize that an infinitely extensible inertial frame of reference might have to be abandoned in favor of the local free falling frame of reference.

All free falling observers measure the local speed of light as "c", but distant lightspeed observations are relative. Relative motion between observer and distant light source can change the apparent frequency of light, and its apparent power, causing it to appear red or blue shifted. Observing distant light in a different gravitational field will also change observations and cause them to be different than a free falling local observer at the distant light source. (I think)

To summarize: an (infinite) inertial frame without gravity corresponds to a local free falling frame with gravity.

9. Nov 3, 2008

### HallsofIvy

You mean you feel nothing that you don't feel everyday. The spin of the earth on its axis does measurably change your weight. But not by very much and you don't notice it because you always feel it. But accurate measurements at the equator, where the linear speed is greatest, and near the pole, where it would be lowest, show different weights. The effects of the earth circling the sun or the sun circling the galaxy are too small to be measured.

10. Nov 3, 2008

### Naty1

I just posted the following in another thread....another example of reference frame on observation:

11. Nov 3, 2008

### DrGreg

Yes.

Although, to be precise, I would prefer to say "non-inertial" or "undergoing proper acceleration" instead of "accelerated" (as acceleration is relative).

If you are not moving inertially (i.e. are properly-accelerating) your own accelerometer will indicate this. (And, if the acceleration is large enough, your own body will feel the proper acceleration, such as you feel when a car rapidly accelerates, brakes or corners. Or, in GR, your own weight.) If you drop an apple, it won't float in front of you, it will fall away.

Also, to be precise, I don't like the phrases (a) "in an inertial frame" or (b) "outside his reference frame". It's better to say (a) "at rest in..." or "stationary relative to..." or (b) "moving relative to...". A reference frame is just a coordinate system, so everything is "in" every frame.

12. Nov 3, 2008

### DrGreg

Actually...

The circling of the sun and of the galaxy are both inertial motions, so there is nothing to feel. Or, rather, there wouldn't be if the Earth were a single point. But the earth isn't a point so there are some small "tidal effects". The biggest source of tidal effects on the Earth's surface is the Moon. The surface of the sea moves almost inertially (considering the tides only and ignoring local waves -- it would be exactly inertial if it were frictionless) so, on dry land, relative to the sea you are periodically rising and falling (i.e. moving non-inertially) which has a tiny effect on your perceived weight (along with the "centrifugal" effects of the Earth's spin).

13. Nov 3, 2008

### calculus_jy

atyy what do you mean by 'results match the "simple laws.' is it not true that if a body is not travelling in straight line, then the observer conclude that a force is acting on it...
and doctorGreg if you measure acceleration you already assume that the acceleration is in respect to a preset inertial frame??

14. Nov 3, 2008

### atyy

I mean the laws look like what freshman textbooks say they are. Yes, if you can verify Newton's first law, then you are in an inertial frame. But to verify the law you need to know that your straight line is straight, and that clock you use to isn't slowing down. If you use an accelerometer, it has to be calibrated against a known acceleration. To check that your apparatuses are working, you need the laws of physics.

15. Nov 3, 2008

### calculus_jy

but is it not true that if newton's first law is not satisified, we conclude that a force acts upon it ??? and so how to really 'verify' first law??

16. Nov 3, 2008

### atyy

I think you misunderstand my point. I mentioned the first law because you mentioned it. For me, the simple laws include Maxwell's equations and the Lorentz force law. Those laws will not hold unless the first law holds, so if you can prove the first law holds, then Maxwell's equations and the Lorentz force law will hold.

17. Nov 4, 2008

### atyy

The question you ask actually has meaning even without special and general relativity. The standard form of Newton's laws require an inertial frame to work. The first law is only a special case of the second law. We require all three laws to have their standard form in order for the frame to be a Newtonian inertial frame.

One thing you may like to read about is the reduced mass in the two body Newtonian gravity problem. There the problem is formulated for two bodies in an inertial frame. However, it is convenient to work in an non-inertial frame attached to one mass. As a result, Newton's 2nd law does not have its standard form, because the "reduced mass" is used instead of the mass.

Another situation where it is convenient to use a non-inertial frame is for calculating the angle a pendulum bob makes with the vertical in an accelerating train. There we introduce a force which does not obey Newton's 3rd law.

In both cases once you've solved the problem in the non-inertial frame, you can do a coordinate transformation to an inertial frame in which all of Newton's laws hold, confirming that our assumption that such a frame exists is correct.

General relativity, which is our best classical theory, says that there are no Newtonian inertial frames and no special relativistic inertial frames, and in fact no inertial frames of any finite size at all. So basically whether you can consider a frame to be inertial depends on how accurate your experiment is going to be.

Last edited: Nov 4, 2008
18. Nov 4, 2008

### DrGreg

The "crude accelerometer" I described in post #5 will tell you if you are an inertial observer or not. You don't need to calibrate it to decide whether the spring is in tension or not. Used in this way, that device is essentially a test for Newton's first law. (In practice you would need to add a bit of damping (friction) to prevent the mass oscillating.) Once you have found an inertial observer you could then calibrate the accelerometer to convert spring-extension into metres-per-second (by calculating acceleration relative to the inertial observer).

19. Nov 4, 2008

### Naty1

atyy posted:
Is this a way of saying only LOCAL free falling frames are (almost) inertial? Are you equating :...no inertial frames of any finite size..." to local?? That makes sense to me...