# How would you define an inertial frame of reference?

1. May 1, 2014

### Luffy

I've researched about it and watched a few videos, but I can't seem to get my head around it. Would saying that "it's a marker that is fixed relative to your position, in which Newton's first law holds" be an accurate way to define it?

2. May 1, 2014

### HallsofIvy

Staff Emeritus
Not necessarily. If you are accelerating, then "fixed relative to your position" will not give an inertial frame of reference.

3. May 2, 2014

### tom.stoer

An inertial frame is a reference frame in which free particles (w/o any external force) move along geodesics (which is a generalization for arbitrary curved manifolds)

4. May 2, 2014

### vanhees71

To make it very clear: What tom.stoer means are geodesics in fourdimensional general relativistic space-time.

This is, however, not the way an inertial frame is understood in general relativity. If there is a non-negligible gravitational field present, space-time manifold is curved and there is no global inertial frame for entire space time but according to the (weak) equivalence principle you can always find a local one, i.e., you restrict yourself to the neighborhood around your position in space-time which is small compared to the "curvature radii" of spac-time at this point. Then you can always find an equivalence class of reference frames where the local laws take the form of special-relativistic physics. These reference frames are given as the freely falling ones. In such a frame Newton's 1st law holds (locally): A body that is not subject to any forces is moving along a straight line with uniform velocity (or stays at rest).

5. May 2, 2014

### tom.stoer

Yes, this is exactly what I mean: geodesic motion defines local inertial frames

6. May 2, 2014

### WannabeNewton

Just for minor clarification, it's not enough to just have geodesic motion. That only constrains the frame to be non-accelerating. There is still residual freedom in how the spatial axes of the local Lorentz frame are oriented and more importantly how their orientation changes relative to local gyroscopes. One needs geodesic motion as well as Fermi-transported (which reduces to parallel transport for geodesics) spatial axes so as to keep the frame non-rotating and hence inertial.

7. May 2, 2014

### sathwik

The co-ordinate system which you had considerd should have its acceleration i zero

8. May 2, 2014

### sathwik

EXAMPLE:Earth

9. May 2, 2014

### sathwik

The co-ordinate system which you had considerd should have its acceleration zero

10. May 2, 2014

### D H

Staff Emeritus
Acceleration with respect to what? And what exactly do you mean by "acceleration"?

tome.stoer's answer, coupled with the modification suggested by WannabeNewton, suggest one way to define an inertial frame: Co-locate an accelerometer and rate gyro. From the perspective of general relativity, if the accelerometer reports zero acceleration and rate gyro reports zero rotation, a frame based on the accelerometer+gyro to within measurement error is a local inertial frame.

That's one of the nice things about GR: It provides an experimental method for defining an inertial frame, and the experiment is based solely on local measurements. One of the not so nice things: That inertial frame is local. The concept of an inertial frame has lost a lot of the meaning it had in Newtonian mechanics and special relativity.

Newtonian mechanics provides a means for determining whether a frame is inertial: Does any particle with no forces acting acting on it move along a straight line at constant speed? There's a slight downside here: Gravity acts on everything. There is no such thing as a particle with no forces acting on it.

11. May 2, 2014

### duordi134

If newtons laws of inertia hold within time length width and height limits within experimental accuracy then it is an inertial system.

As I remember it there is no true inertial frame of reference we can only approximate it.

Duordi

12. May 2, 2014

### D H

Staff Emeritus
There's a big difference between a Newtonian inertial frame and an inertial frame in general relativity. A frame based on a non-rotating object in free fall is a local inertial frame in general relativity but is not inertial in Newtonian mechanics.

13. May 2, 2014

### atyy

For particles acted on by forces, a more general definition of an inertial frame is one in which the laws of physics have their "standard form" (ie. no Christoffel symbols).

14. May 2, 2014

### bcrowell

Staff Emeritus
The Newtonian and relativistic answers to this question are not the same. This was asked in the relativity forum, but the OP's question actually sounds like it's being asked in a Newtonian context. In the Newtonian context, an inertial frame is one in which Newton's first law holds. For anyone who wants to understand this concept better in the Newtonian context, I'd suggest watching the classic PSSC Frames of Reference films:

For the relativistic case, I have a discussion in ch. 5 of my SR book: http://www.lightandmatter.com/sr/ .

This is both incorrect and at the wrong level for the OP, who just seems confused about the Newtonian notion of an inertial frame. It's incorrect because neither the definition of a geodesic nor the question of whether a test particle moves along a geodesic is dependent on the coordinates used.

Last edited by a moderator: Sep 25, 2014
15. May 3, 2014

### tom.stoer

Sorry for the confusion I started.

16. May 4, 2014

### JustinLevy

No. That is a necessary, but not sufficient condition. Otherwise any linear transformation from an inertial coordinate system would give another inertial coordinate system.

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It is incredibly hard to define an inertial frame precisely. Everyone 'knows' what we mean, so most books just describe some properties of it (and sometimes the symmetries relating different inertial frames) and then move on. Heck, even Einstein in his 1905 paper defined inertial frames as those in which the laws of mechanics hold, then went on to show that the laws of mechanics in inertial frames needed to be changed slightly. There was even a debate for awhile if parity transformations should be considered as relating inertial frames. Because there was no precise definition, this was more a philosophical debate ... until it turned out the weak force violated parity symmetry. So our concept of inertial frames changed slightly.

I think the Landau and Lifshitz definition wikipedia uses is fairly decent (given the alternatives). That definition is basically saying an inertial frame of reference is a choice of coordinates such that our description of space and time is maximally uniform. Check out the length between two coordinate points (x,y,z,t) and (x+1,y,z,t) ... it is the same regardless of x (or of y, z, or t as well). Similarly if we had looked at points that differed in y, or z. Same with t, but now measuring time. Do some experiment which fires off a bullet with some consistent energy, and it will have the same velocity regardless of the direction we point it (with the uniformity in spatial coordinates already given, this shows we have the 'most uniform' clock synchronization), alternatively you could check clock synchronization by slow transport of clocks and see you get the same result in all directions. In inertial frames our description of space and time are maximally uniform.

This holds fine for GR as well, but an inertial frame can now only be defined locally and becomes more approximate the larger the extent of the "local" frame is.

Last edited: May 4, 2014
17. May 4, 2014

### ghwellsjr

How about: an inertial frame of reference is what the Lorentz Transformation operates on?

18. May 4, 2014

### sathwik

Here is the another defination for inertial frame : an imaginary system which is either rest or in uniform motion and where newton's law are valid

19. May 4, 2014

### sathwik

Hey can any one help me how to post the question

20. May 4, 2014