# B How earth is a non-inertial frame

1. Sep 15, 2016

### parshyaa

How (more accurately) earth is a non inertial frame?
• A frame of reference(FOR) will be a non-inertial when a=0 ⇔ F =0
• Suppose a book on a table is our object and earth as a Frame of reference, generally we take accelaration of book w.r.t earth as 0 but more accurately it is not 0(because earth accelarates), therefore force acting on a book will also be not equal to 0. Then how it shows that earth is a non-inertial frame , it only tells that a ≠0 and F ≠ 0, it does not tell that if a ≠0 , F=0 or vice versa

2. Sep 15, 2016

### parshyaa

Sorry , a frame of reference is non-inertial if it does not follow a=0 ⇔ F=0.

3. Sep 15, 2016

### DrStupid

A frame of reference is not inertial if it does not follow all three laws of motion. Checking the first law only is not sufficient.

4. Sep 15, 2016

### pixel

The earth is not an inertial frame of reference because of its rotation which, for example, gives rise to Coriolis forces. In many experiments in small regions of space, inertial is usually a good approximation.

5. Sep 15, 2016

### jbriggs444

In the context of general relativity where gravity is not a force and is, instead, modeled as a geometric effect, a book on a table is subject to a net force but does not accelerate. Ergo, the earth does not define an inertial frame.

6. Sep 15, 2016

### Staff: Mentor

Who says that the 4 acceleration of the book is zero?

7. Sep 15, 2016

### jbriggs444

Dunno. But its 3-acceleration relative to the table sure is.

8. Sep 15, 2016

### Staff: Mentor

9. Sep 15, 2016

### jbriggs444

Yes and no. It is at rest in an accelerating frame (the one where the surface of the earth is at rest) and it is accelerating outward in the tangent inertial frame (the one in which a freely falling elevator would be at rest).

10. Sep 15, 2016

### Staff: Mentor

I think you're right, but I'd like to get a second opinion from @Dale. As a relative novice to GR, I await the reply of the guru.

11. Sep 15, 2016

### Staff: Mentor

Ooh, I've never been a guru before!

The four-acceleration is a tensor, so it is covariant and therefore is non-zero in all frames. The three-acceleration is a coordinate dependent quantity, and in the rest frame of the ground it is 0. That discrepancy does identify the rest frame of the ground as being non inertial.

Last edited: Sep 15, 2016
12. Sep 15, 2016

### Staff: Mentor

Yes you have. You just didn't know it.

13. Sep 15, 2016

### parshyaa

Do you mean that there is a net force on a book due to earth(how) , and acceleration of a book with respect to earth is 0(because book is at rest with the earth surface). Therefore F≠0 but a=0 (therefore its a non-inertial frame)

14. Sep 15, 2016

### Staff: Mentor

Note, this answer is in the context of general relativity. Newtonian physics gives a different answer.

In GR the 4-acceleration (or proper acceleration) is the acceleration measured by an accelerometer. A book sitting on a table therefore has a 4-acceleration of 9.8 m/s^2 upwards.

The 3-acceleration (or coordinate acceleration) is the second derivative of position. So in the ground's coordinates the book's 3-acceleration is 0. So the ground's coordinates are non-inertial (in GR)

15. Sep 15, 2016

### parshyaa

What does newtonian physics say , I think you are trying to say that with context to newtonian mechanics earth is a inertial frame but in context to general relativity it is a non-inertial frame

16. Sep 16, 2016

### jbriggs444

Yes. In Newtonian mechanics, gravity is seen as a force. The book is subject to a downward force from gravity and an upward contact force from the table upon which it rests. The net force is zero and the acceleration is zero. Newton's laws are satisfied and the frame qualifies as inertial.

At least if we ignore rotation.

The force of gravity is different from most other forces. It is proportional to the mass of the object upon which it acts. Forces with this property are called "inertial" forces. Such forces (e.g. centrifugal, Coriolis forces) can typically be eliminated by choosing a different coordinate system. This is difficult to do with gravity, but Einstein managed to do so with General Relativity.

17. Sep 16, 2016

### parshyaa

Thanks Dale and jbriggs you both are guru ^_^ ^_^ ^_^

18. Sep 16, 2016

### olivermsun

But rotation does seem like the key thing that makes the earth frame non-inertial in Newtonian mechanics.

19. Sep 16, 2016

### Staff: Mentor

Yes.

In both Newtonian mechanics and GR non inertial frames have fictitious forces. These fictitious forces have the properties that they are proportional to mass, they are undetectable by accelerometers, they disappear if you change reference frames, and they don't have an equal and opposite interaction.

The difference between GR and Newtonian mechanics comes in the treatment of gravity. So gravity has three of the four properties of fictitious forces, and the fourth property goes away when considering gravity as spacetime curvature instead of a force. So GR classifies gravity as a fictitious force, and this is what leads to the different classification of inertial vs non inertial frames.