GR explanation of Newtonian Phenomena

1. Mar 2, 2008

Xeinstein

1. How would a general relativist explain why an object falls towards the earth?

2. Is it correct to say that it is not the apple that falls towards the earth but it is the earth that accelerates towards the apple? Why is this ok to say?
Is it because, in GR, there are no preferred reference frames?

a qualitative and/or quantitative explanation would be great. thanks...

2. Mar 2, 2008

A.T.

A free falling object moves straight ahead trough spacetime which is curved. This might help to visualize it:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

The earth's surface is in fact accelerated upwards, in GR. But this doesn't mean it is moving upwards. Accelerated in GR merely means "not moving straight ahead trough spacetime". This is explained well in chapter 2.6 of this work:
http://fy.chalmers.se/~rico/Theses/tesx.pdf

3. Mar 2, 2008

Wallace

Yes, it's quite different conceptually than Newtonian mechanics. You get used to equilibrium mechanics, such as something resting on a table being at rest since the forces balance, in Newtonian mechanics, and then you try and get your head around GR in which something apparently just resting on a table is in fact being accelerated!

4. Mar 2, 2008

Xeinstein

How can something just resting on a table is in fact being accelerated upward?
The definition of acceleration is change of velocity over change of time. So if something just resting on a table is being accelerated, that means its velocity must change over time.

Last edited: Mar 2, 2008
5. Mar 2, 2008

Staff: Mentor

Interestingly, an accelerometer just resting on a table actually detects an acceleration whereas an accelerometer that is falling towards the table does not. Given that fact, are you certain that you are choosing a good coordinate system when you say that the object resting on the table is not accelerating?

6. Mar 2, 2008

Staff: Mentor

Let me expand on my previous post. Acceleration is a coordinate-system dependent quantity, i.e. it is the second derivative of the position wrt some coordinate system.

There are many different kinds of coordinate systems (aka reference frames). For example, consider a circular space station rotating about its axis to make "spin gravity". Inside the space station there is something "resting" on a table which we wish to analyze. We can do so either in the rotating reference frame where the space station is at rest, or we can analyze it in the inertial reference frame.

In the inertial reference frame the object is accelerating. We attribute this acceleration to the centrepital force exerted by the table. An accelerometer at rest in this reference frame detects no acceleration.

In the rotating reference frame the object is not accelerating. We introduce a ficticious force called the centrifugal force to balance the centripetal force exerted by the table in order to explain why the object is not accelerating. An accelerometer at rest in this reference frame detects an acceleration.

Now, apply the same analysis to an object on a table on Earth. The object is not accelerating because the force of gravity balances the normal force exerted by the table. An accelerometer at rest in this reference frame detects an acceleration.

If we were to change to a reference frame where the accelerometer did not detect any acceleration we would find that gravity disappears and the object accelerates due to the normal force exerted by the table.

So, in this sense gravity seems more like the ficticious centrifugal force than like the real normal force, and the surface of the earth seems more like the rotating reference frame than the inertial reference frame.

7. Mar 2, 2008

Xeinstein

Interestingly, an accelerometer just resting on a table actually detects an acceleration that the direction of gravity is downwards, but previous post claim something just resting on a table is in fact being accelerated upwards?

Last edited: Mar 2, 2008
8. Mar 2, 2008

George Jones

Staff Emeritus
I don't read what DaleSpam wrote as saying this.

The accelerometer detects an acceleration upward.

Take a look at the last half of post #12 in this thread

9. Mar 2, 2008

kdv

The key point is this: In Newtonian physics, the force of gravity is a physical force. IN GR, it's a fictitious force, like the Coriolis force. So if you accept this, you realize that every time you notice a force of gravity acting down (in the sense of Newton), it's because you are in fact in a frame accelerated upward. That's the key point. For the same reason that if you are in a car and you feel a (fictitious) centrifugal force pushing you against the door of the car, it's really that the car is being accelerated around a curve.

Now, the acceleration in the context of GR is a bit counterintuitive because it's through spacetime, but the basic idea remains the above.

Last edited: Mar 2, 2008
10. Mar 2, 2008

Staff: Mentor

That is incorrect, an accelerometer resting on a table detects an upwards acceleration.

Consider an accelerometer consisting of a mass between two springs, one on the left and one on the right. If the mass is accelerated to the right the spring on the right is in tension. Now consider a mass between two springs, one on the top and one on the bottom. If the mass is at rest on the surface of the earth then the top spring is in tension, so the acceleration detected is upwards.

Last edited: Mar 2, 2008
11. Mar 2, 2008

pmb_phy

Acceleration does not have an absolute meaning beyond the fact that the 4-acceleration for a particle in free-fall is always zero. To inquire beyond this one must define a frame of reference from which observations are made. For an observer who is at rest with respect to the apple then the apple is not accelerating as measured by such an observer. For an observer who is in an inertial frame of reference, which in this case is a frame of reference in free-fall, then the apple is accelerating as measured by such an observer.

Best wishes

Pete

12. Mar 3, 2008

A.T.

Understand it today in three simple steps :

1) When you draw the path of an object into a space-time diagram, you can easily say if it's accelerated (curved path) or not (straight path). This holds true in both Newtonian mechanics and GR, so it's a more general definition of acceleration.

2) In Newtonian mechanics, the object resting on the table has a straight path (-> not accelerated), while a falling object has a curved path (-> accelerated by Newtons force of gravity).

3) in GR the entire space-time diagram is curved in a way that makes the path of the falling object straight (-> not accelerated). But this also makes the path of the table object curved ( -> accelerated up by the force the table exerts on it)

Last edited: Mar 3, 2008
13. Mar 5, 2008

Xeinstein

In General relativity, we understand that gravity is described not as a force, but rather as curvature of spacetime and test particle follows geodesic path of the curved spacetime. But what is the rationale that the gravity is a "fictitious force". How do we get "fictitious force" of the rotating reference frame from curved spacetime? I think "fictitious force" seems to be a concept in Newtonian mechanics, Not in general relativity; Somehow, I Can't find it in Hartle's book, Gravity: An Introduction to Einstein's General Relativity

Last edited: Mar 5, 2008
14. Mar 5, 2008

Staff: Mentor

In GR free-falling observers are inertial. Other observers are non-inertial. Any time you have a non-inertial reference frame you must introduce ficticious forces in order to get Newton's first and second laws to work. Like gravity, such ficticious forces are not detectable by accelerometers.

The ficticious force does not stem from the curvature of spacetime but rather from the choice of a non-inertial reference frame.

15. Mar 5, 2008

George Jones

Staff Emeritus
The advanced book Relativity on Curved Manifolds by de Felice and Clarke elaborates on DaleSpam's comments in a very readable way.

Take a look at the nice prechaper Geometry and Physics: An Overview; in particular, read sections 3 and 4 from this prechapter.