Faster than light - under what circumstances?

[pixel]

Let’s say I am on the earth’s surface and observing a galaxy at a distance of d=10 billion light years away. In the course of one day, I will have observed the galaxy to move 2*pi*d in my frame of reference. Of course, this is >> c.

Now it will be argued that the Earth is rotating so I am not in an inertial frame of reference. I could say that I can approximate an inertial frame to a high degree by viewing the galaxy from a hypothetical planet that takes 10 billion years to complete one rotation. The galaxy would still have an apparent speed of 2*pi*c in that frame of reference.

I’m trying to get a clear understanding of just when the limitation of speeds < c holds. I’ve heard things like “only locally,” “only in an inertial frame of reference,” etc. It appears that the Earth is usually a good enough inertial frame for most observations and v < c holds, but not in the example I gave.

 

[bcrowell]

The thread has the "A" marker, but the question seems to be asked at more like the "B" or "I" level. "A" is supposed to be at the level of a graduate course.

Do you want an answer using only SR, or GR as well?

The fundamental answer certainly can't be "only in an inertial frame of reference," since frames of reference are completely optional in SR and basically don't exist in GR.

If you really want an answer at the "A" level, then one way of putting it is that the world-line of a massive particle is always timelike, i.e., it always has a timelike tangent vector. This is clearly a local thing. If you don't know what a tangent vector is or what it means for it to be timelike, then you probably don't want an "A" answer.

 

[Herman Trivilino]

I’m trying to get a clear understanding of just when the limitation of speeds < c holds.

Communications. You cannot send a signal from one place to another at a speed faster than c.

 

[Mark44]

The thread has the "A" marker, but the question seems to be asked at more like the "B" or "I" level.

Now changed to "I".

 

[Nugatory]

When we're talking about speed in the context of not being possible to exceed the speed of light, we're talking about $\frac{\mathrm{d}x}{\mathrm{d}t}$ where $x$ and $t$ are the position and time coordinates of the moving object in an inertial frame; that's the speed relative to an object at rest in that inertial frame and it cannot exceed $c$. So the answer is "in an inertial frame...

... But there's a twist. Only in flat spacetime can you extend an inertial frame across all of spacetime. If there are any significant gravitational effects present, then frames are only locally inertial, so when we specify "inertial" we're getting "local" for free. You shouldn't conclude from this that locality is a requirement - it's more that there is no unique way of defining the relative speed between two objects at different locations in a curved spacetime, and thus no way of applying the speed limit.

 

[Dale]

I could say that I can approximate an inertial frame to a high degree by viewing the galaxy from a hypothetical planet that takes 10 billion years to complete one rotation.

The accuracy of the approximation depends on $r$ as well as $\omega$. So at large distances even very small angular velocities become poor approximations.

 

[pixel]

The fundamental answer certainly can't be "only in an inertial frame of reference," since frames of reference are completely optional in SR...

In what sense "optional?"

 

[bcrowell]

In what sense "optional?"

They play no foundational role. You can develop the entire theory without reference to them.

 

[pixel]

They play no foundational role. You can develop the entire theory without reference to them.

A reference please, since to me SR is all about relating the measurements in one frame of reference to those in another.

 

[Dale]

I’m trying to get a clear understanding of just when the limitation of speeds < c holds. I’ve heard things like “only locally,” “only in an inertial frame of reference,” etc.

The limitation holds in inertial frames. In curved spacetime you cannot generally make an inertial frame that covers all of spacetime. Instead you can construct local inertial frames, in which the restriction would apply locally.

 

[Nugatory]

A reference please, since to me SR is all about relating the measurements in one frame of reference to those in another.

There's a very thorough description in chapter 2 and following of Misner, Thorne, and Wheeler's "Gravitation". There are two reasons why this geometrical approach is relatively unknown: First, it is very different than the approach by which Einstein and his contemporaries arrived at SR (an example of hindsight - "Now that we know where we're going, here's how we could have gotten there"); and second, it is usually only taught to people who are planning to move on to GR.

 

[mathman]

Let’s say I am on the earth’s surface and observing a galaxy at a distance of d=10 billion light years away. In the course of one day, I will have observed the galaxy to move 2*pi*d in my frame of reference. Of course, this is >> c.

Now it will be argued that the Earth is rotating so I am not in an inertial frame of reference. I could say that I can approximate an inertial frame to a high degree by viewing the galaxy from a hypothetical planet that takes 10 billion years to complete one rotation. The galaxy would still have an apparent speed of 2*pi*c in that frame of reference.

I’m trying to get a clear understanding of just when the limitation of speeds < c holds. I’ve heard things like “only locally,” “only in an inertial frame of reference,” etc. It appears that the Earth is usually a good enough inertial frame for most observations and v < c holds, but not in the example I gave.

Relativity has nothing to do with it. You are spinning around so things very far away appear to move at enormous speed, but they are not actually moving. Only your line of sight is changing.

 

[PeterDonis]

I’m trying to get a clear understanding of just when the limitation of speeds < c holds.

The fully general way to state the limitation in GR is that the worldlines of all objects must lie within the light cones at every event they pass through. In other words, at every event in spacetime, there is a geometric structure called a "light cone", which marks out the limits of where the worldlines of objects can go. If we pick a local inertial frame centered on a chosen event, then the speed of any object whose worldline is within the light cones at that event will be less than $c$; that is how the geometric constraint results in the "speed limit" constraint as it appears locally. But the geometric constraint can be applied in curved spacetime, where there are no global inertial frames, and where there is no invariant way to compare the speeds of spatially separated objects. The light cones are always there regardless.

 

[pixel]

Relativity has nothing to do with it. You are spinning around so things very far away appear to move at enormous speed, but they are not actually moving. Only your line of sight is changing.

It's not safe to say what is "actually moving." My frame of reference should be as valid as any other. But I think others have already answered my question.

 

[Dale]

My frame of reference should be as valid as any other.

It certainly is as valid as every other frame. But it is not equivalent to every other frame.