I How do non-inertial frames affect special relativity?

[PAllen]

Thanks. Indeed
I remember reading these two conditions in Weinberg's Relativity book.

But wouldn't that lead to a wrong conclusion? I mean, suppose I found out that the curvature tensor is not zero. Then the frame is non inertial. But regarding GR, since the curvature tensor is non zero, I would conclude that gravity is present. But that's not necessarily the case.

I think an answer to this would be: "but according to GR gravity is the curvature in space-time. So if I found out that the curvature tensor is not zero, I automatically discovered that that frame is non inertial and that there's gravity present".

But I would argue that it's possible to have a non inertial frame without gravity. For instance, a accelerating spaceship traveling through out empty space.

I was describing criteria for global inertial coordinates. In GR, as many here have noted, there is no such thing as global inertial frame. Pease don’t construe more than what I wrote - how to look at a metric and expression and determine if can reasonably be considered that the coordinates and metric are a global inertial coordinate system. An accelerated frame is by definition not inertial in either SR and GR.

 

[Mentz114]

I see

My point is that only by following the discussed procedure we are not able to discern between a non-inertial frame without gravity and one with gravity. How can we do that?

a non-inertial frame without gravity = proper acceleration
a non-inertial frame with gravity = proper acceleration + gravity (Riemann ≠ 0)

But gravity cannot cause proper acceleration so you need rocket motors in both cases ( I would guess).

 

[PeterDonis]

only by following the discussed procedure we are not able to discern between a non-inertial frame without gravity and one with gravity

"Gravity" means that condition #1 is violated (the curvature tensor is nonzero).

 

[PAllen]

I see

My point is that only by following the discussed procedure we are not able to discern between a non-inertial frame without gravity and one with gravity. How can we do that?

If what you are looking for is a general classification of coordinates for both SR and GR based on metric expression in those coordinates, there is no clear answer. It is all a matter of definitions on which authors differ and physics doesn’t care. I’ll throw out a few ideas that I am sure someone could find arguments with for various cases:

1) If all the criteria I gave earlier are met, call it a global inertial coordinate system.
2) If the first two conditions are met, you might call it an SR noninertial coordinate system, but if you want to say it is adapted to a particular noninertial observer, you probably want more conditions, e.g. that surfaces of constant timelike coordinate are orthogonal to the spatial origin world line everywhere.
3) If there is curvature, and the 1 by 3 criterion is met, and constant time surfaces are orthogonal to the origin world line, then it is locally inertial if the origin world line has zero proper acceleration, else it is locally noninertial.

In both SR and GR, there are common coordinates which would not fit any of these criteria. Most coordinates used in GR would not fit into any of these categories. They are simply not based on any particular observer, inertial or not.

 

[Dr Whom]

Inertial frame. A frame of reference in which bodies are not accelerated. The Penguin Dictionary of Science. In free fall under the action of gravity objects are generally accelerated.

 

[Nugatory]

Inertial frame. A frame of reference in which bodies are not accelerated. The Penguin Dictionary of Science.

Something is wrong with that definition - at the very least it must include a qualifier about the object not being subject to external forces.

In free fall under the action of gravity objects are generally accelerated.

Objects in free fall under the action of gravity are not accelerating in a (local) inertial frame. An object dropped on the surface of the Earth is accelerating downwards in the frame in which the surface of the Earth is at rest. That's how we know that that frame is not an inertial frame.

 

[Ibix]

Inertial frame. A frame of reference in which bodies are not accelerated. The Penguin Dictionary of Science. In free fall under the action of gravity objects are generally accelerated.

In addition to @Nugatory's comments, this fails to make a distinction between proper acceleration and coordinate acceleration. Free-falling bodies do not experience proper acceleration, and whether or not there is coordinate acceleration is up to your choice of coordinates. So your statement is oversimplified at best.

 

[A.T.]

Inertial frame. A frame of reference in which bodies are not accelerated. The Penguin Dictionary of Science.

Says pretty much nothing. Bodies can be accelerating in inertial and non-inertial frames.

In free fall under the action of gravity objects are generally accelerated.

Which frame is inertial in a gravitational field depends on whether you use Newton or GR.

 

[PeroK]

Inertial frame. A frame of reference in which bodies are not accelerated. The Penguin Dictionary of Science.

The Penguin definition is incorrect. I suggest it was written by a lexicographer and not a physicist.

Wikipedia gives a better version:

"An inertial frame of reference, in classical physics, is a frame of reference in which bodies, whose net force acting upon them is zero, are not accelerated; that is they are at rest or they move at a constant velocity in a straight line."

In free fall under the action of gravity objects are generally accelerated.

If we take gravity to be a force, then this is correct. But, taking gravity as a fictitious force, objects in freefall are accelerating in a non-inertial reference frame: rest frame of the Earth's surface, say.. And objects, as they fall, are in a (local) inertial reference frame.

 

[Dragon27]

The Penguin definition is incorrect. I suggest it was written by a lexicographer and not a physicist.

It would look much better if one just inserts the (very important) word "free" before "bodies". Like here.

 

[kent davidge]

Which frame is inertial in a gravitational field depends on whether you use Newton or GR.

Can you elaborate on this? Is it because in GR we can make whatever frame locally inertial while in Newton we can't?

If we take gravity to be a force, then this is correct. But, taking gravity as a fictitious force, objects in freefall are accelerating in a non-inertial reference frame: rest frame of the Earth's surface, say.. And objects, as they fall, are in a (local) inertial reference frame.

Interesting point. Thanks for bringing this up.

 

[Ibix]

Can you elaborate on this? Is it because in GR we can make whatever frame locally inertial while in Newton we can't?

In GR gravity isn't a force, so a particle in a gravitational field isn't experiencing a force and is inertial. In Newtonian physics gravity is a force, so a particle in a gravitational field is experiencing a force so is not inertial.