# Can an accelerating frame of reference be inertial?

In physics problems, the earth is usually considered to be an inertial frame. The earth has a gravitational field and the second postulate of the general theory of relativity says:

In the vicinity of any point, a gravitational field is equivalent to an accelerated frame of reference in gravity-free space (the principle of equivalence).

Does this mean that accelerating frames of reference can be inertial?

Accelerating frames are never truly inertial; however, in many situations the acceleration is sufficiently small that we can assume the accelerating frame to be inertial. It largely depends on the scale relevant to the problem.

For example, for purposes of projectile motion, we can consider the Earth to be an inertial reference frame and still model the projectile's path accurately. However, in orbital mechanics, we definitely cannot consider the Earth to be an inertial frame, since it constantly accelerates in its orbit around the Sun.

An inertial frame is equivalent to a frame's velocity at any given time. An accelerating frame still has intertial frames for the same reason that we can calculate instanteous slopes of a function. An accelerating frame is changing inertial frames constantly but that doesnt mean it isn't an inertial frame at a given point it time.

No. By definition an accelerating frame of reference cannot be an inertial frame of reference.

The Earth is only approximately an inertial frame of reference over sufficiently small distances and times.

• Do you think the two paragraphs are contradictory? – Rumplestillskin Jan 20 '17 at 5:39
• @Rumplestillskin : No. 22/7 is not an irrational number, but it is an approximation to one. – sammy gerbil Jan 20 '17 at 5:44
• What if I say that the Earth in truly intertial, moreover, it is at rest. and it is the universe which is rotating around it so everything else in a non-inertial frame. I suspect the answers (ie, spacetime length between two events) would still be same after the necessary changes in the equations. Am I right? – Kalpak Gupta Jan 20 '17 at 6:40
• @KalpakGupta definitely not. Acceleration, unlike velocity, has an absolute meaning in the sense of its being the reading on an accelerometer (insert standardized design for such an instrument here: Any modern technology for this week do). Take an accelerometer into space and either put it into orbit freefalling around Earth or floating in deep space. It will read nought; put it on the Earth's surface it will read 1g. Matter settled. General relativity also reflects this absolute sense (naturally). – WetSavannaAnimal Jan 20 '17 at 8:26
• Oops. That was stupid of me. An inertial frame won't have rotating cyclones, after all! – Kalpak Gupta Jan 20 '17 at 8:28

I think I understand what you mean. In an inertial frame, an object on which no net forces act, moves with a constant speed along a straight line. On earth this doesn't even approximately hold.

If you still want to consider this as approximately inertial, the conclusion must be that the earth itself is present (as mass, matter or a potential) inside your frame. In other words, the frame is not equivalent to one that is accelerating, but there is a source of acceleration present within the approximately inertial frame.