# Inertial frame of reference

1. Dec 26, 2012

### Drake711

hello, in relativity something can go faster than the speed of light as long as it is not in the observer's frame of reference which would say that a FOR has a size but i read that a FOR was infinite in all directions. Someone explain this contradiction please!

Last edited by a moderator: Dec 26, 2012
2. Dec 26, 2012

### Staff: Mentor

The part that I've put in bold is so imprecise, that you can't conclude much of anything from it; there's not enough there to say that a frame of reference has a size or not. Faster than light relative to what? What does it mean to "not be in" a frame of reference? Even the phrase "observer's frame of reference" is a bit sloppy, although I and most people will assume that "a frame in which the observer is at rest" was intended.

1) In special relativity and in flat space-time, if a frame is inertial it is inertial everywhere, throughout the entire flat space-time.
2) In general relativity and curved spacetime, a frame will be inertial only within an infinitesimal volume around a given point, and the farther away from that point you move the more non-inertial the frame will become. That is, the frame may cover spacetime out to infinity in all directions, but it won't necesarily be inertial except as an approximation in the neighborhood of a single point.
3) You shouldn't think of things being IN a frame of reference; a frame of reference is just a convention for assigning numerical coordinates to points in spacetime so that we can do calculations. You cannot make any meaningful statement about the relative velocity of two objects if you're not using the same frame of reference to calculate both velocities.

3. Dec 26, 2012

### Drake711

so what your saying for general relativity is that a single point's frame of reference degrades over a certain amount of space? If this is what you're saying then what is the equaton for the amount it degrades over a certain volume and also why is it different for special relativity since the theories coincide?

4. Dec 26, 2012

### Staff: Mentor

In special relativity a frame of reference is infinite in space and time, so there is no such thing as being "out" of a frame of reference. Light travels at c in any inertial frame, and any massive objects travels with v<c in any inertial frame. No massive object can go faster than the speed of light in any inertial frame. So I would say that the contradiction lies in the fact that the first statement is wrong (assuming we are talking about special relativity and inertial reference frames).

Last edited by a moderator: Dec 26, 2012
5. Dec 26, 2012

### Drake711

@DaleSpam The edge of the universe expands faster than the speed of light from our frame of reference because of dark energy. What I'm saying is how does this happen if a FOR is infinite; is it because a reference frame degrades over a certain amount of space, if so at what rate does it degrade.

6. Dec 26, 2012

### Staff: Mentor

SR applies to flat spacetime. GR applies to flat and curved spacetimes. Thus, the two theories coincide and say the exact same things about flat spacetime where they both apply, but GR also has something to say about curved spacetime where SR doesn't apply. Thus, you can think of SR as a special case of GR; in SR, there's no curvature and in GR if you set the curvature to zero it will reduce to SR.

I can always find a frame that is inertial in an infinitesimal volume around any given point in spacetime, but unless the spacetime is flat the frame may not be inertial across larger volumes. How much larger? It depends on the amount of curvature, or in classical terms, the strength of the gravitational field.

For example, if I am free-falling towards the surface of the earth, I would say that a frame in which I am at rest is inertial. But is it really? The gravitational force at my toes is very slightly stronger than that at my head because my toes are six feet closer to the center of the earth, and the force of gravity weakens with distance from the center. So a test particle in free fall near my toes will fall faster and move away from a test particle in free fall near my head - and that shouldn't happen in an inertial frame. This effect is altogether unnoticeable when we're talking about distances of tens of feet in the earth's gravitational field; but with stronger fields or greater distances it becomes more significant, and for large enough distances I'll have to recognize that the free-fall frame is only inertial in the region near me. Of course, none of this would happen if there were no gravity and spacetime curvature, so in the flat spacetime of special relativity the frame in which I am at rest is inertial no matter what the distance scale is.

But, to return to you original question: There is nothing magic about a frame being inertial; it's just easier to solve some problems by working in an inertial frame. So you should not conclude from the discussion above that a frame has a certain size, or that a frame cannot extend out to infinity.

7. Dec 26, 2012

### Staff: Mentor

(The bolded part is pretty much inaccurate/misleading, but for present purposes that's a digression - there's a pretty decent FAQ in the cosmology section).

You missed the part of DaleSpam's post where he specified "in special relativity". There's no edge of the universe and no expansion in SR, because it's an infinite flat spacetime solution.

A curved spacetime might not be infinite at all, and if it's not infinite then it's pretty obvious that even a frame that covers the entire spacetime will itself not be infinite. But if there's nothing left to cover, who cares?

8. Dec 26, 2012

### Naty1

Drake posts:
In GR [and so cosmology] if you and I are in different gravitational fields our relative time passes at different rates. Our clocks tick at different rates. So if we observe a distant phenomena, we'll measure different, say, velocities!!! That does not happen in special relativity[SR]. We also measure different distances because of different spacetime curvatures for each observer.

Observations and calculations of distant velocities may exceed 'c' in curved spacetime. But LOCALLY, in flat spacetime, the speed of light is always 'c'. So in SR the speed of light is always 'c'; but in GR, and cosmology in general, distant observations of lightspeed may vary. There is no 'edge to the universe' but what you likely mean is that very distant galaxies are seen to be moving away from us faster than the speed of light'. Unlike flat space [called Euclidean] where everyone has the same shortest distance [a straight line between objects] in GR each observer has a different curvature between him and any object. So distance and time is rather vague; mathematically we say the Lorentz transform [for space and times] don't work in GR.

Saying ‘space’ expands is just a quick way of stating Hubble's Law. Relativity places the limit ‘c’ on local frames of reference, but there is no limit on global frames in an expanding universe: so distant galaxies may recede at superluminal velocities using our standard models and methodologies.

In other models, such speeds may NOT result. This takes some getting used to, just like in SR it is NOT obvious that space and time vary and that LIGHT speed is the constant. Our standard model is called the FLRW model. [See Wikipedia ] and it has a LOT of agreed upon conventions so scientists can speak from a common reference.

We use the cosmological time parameter of comoving coordinates, moving with the cosmological microwave background radiation, because it's convenient mathematically. That sets one universal frame of reference convenient to all because it is everywhere. There are other time measures that could also be used.

[See the Wikipedia ‘Metric distance’ and especially the illustrations. They REALLY helped me understand the 'expansion'.

For example, recession velocity is a coordinate-dependent number...what that means is if you use a different time coordinate [or a different model metric] you get a different answer. Some are greater than c, some are not.

9. Dec 27, 2012

### Staff: Mentor

Cosmological solutions are part of general relativity, not special relativity, so let's discuss that a bit.

First, according to modern cosmology there is no edge of the universe, so that statement as written is simply false.

Second, in curved spacetime there is no unique coordinate independent meaning for the relative velocity of two distant objects. You can, of course, make a coordinate system and calculate coordinate velocities, but you can also choose different coordinate systems and get completely different relative velocities.

So, in modern cosmology there is a commonly used coordinate system, and according to that coordinate system there are distant objects which are moving away from us with v>c. However, that is only an artifact of the usual coordinate system and has no physical significance.

Third, there is a coordinate independent meaning for the relative velocity of two nearby objects. This is always less than c for any pair of nearby massive objects or equal to c for any beam of light passing any massive object.

Fourth, there is a coordinate independent way to classify an individual object as being slower than light (timelike), as fast as light (null or lightlike), or faster than light (spacelike). All massive objects are timelike.