What is the relationship between free falling bodies and spacetime geodesics?

[FaithSeeker]

http://www.youtube.com/watch?v=8MWNs7Wfk84&feature=PlayList&p=858478F1EC364A2C&index=2" , Edmund Bertschinger is talking about Einstein's field equations .
during the lecture , under the title of : "Gravity as sapcetime curvature (GR viewpoint) " , he wrote :
"Freely falling bodies move along spacetime geodesics ..."
this statement made me confused.
what does this actually mean ?

 

[A.T.]

"Freely falling bodies move along spacetime geodesics ..."
this statement made me confused.
what does this actually mean ?

It means that they advance straight forward trough space-time. Here a visualization:
http://www.relativitet.se/spacetime1.html

 

[FaithSeeker]

thanks
but i didn't get the point yet
what does this has to do with geodesics ?

 

[A.T.]

thanks
but i didn't get the point yet
what does this has to do with geodesics ?

Geodesics are just a generalization of straight lines. If you advance straight ahead trough space-time, your space-time path (world line) is a geodesic.

 

[FaithSeeker]

forgive my stupidity , but i don't understand
let me put the question in another form ,
why do things fall to the surface of the Earth in straight lines according to GR ?
(and straight lines are not geodesics in curved space time)

 

[xepma]

You should see it the other way around. In curved space time the notion of a straight line generalized to that of a geodesic.

In the absence of curvature a free test particle moves along a geodesic, which in this case corresponds to moving with a constant velocity along a straight line.

In a curved background the particle still moves along a geodesic. Only now this geodesic (and the motion itself) no longer correspond to "straight lines" and constant velocity (if there even is such a thing as a straight line in a curved background!).

For instance, on the Earth the motion of a free falling test particle can be along a vertical line, with an accalerating velocity. The statement then is that the movement is along a geodesic.

(Note that in the case of the earthe there are also geodesics which are not vertical. Satellites for instance are also "free falling objects". Their movement is also along a geodesic, but corresponds to an elliptic orbit.)

 

[stevebd1]

I'll give this a bash.

http://en.wikipedia.org/wiki/Schwarzschild_metric" represents curved space-time in a vacuum in the following form (assuming that $d\theta$ and $d\phi$ equal zero)-

$$ds^2 = \left(1-\frac{2M}{r} \right)dt^2 -\left(1-\frac{2M}{r}\right)^{-1}dr^2$$

where $M=Gm/c^2$ (often referred to as the gravitational radius) where G is the gravitational constant, m is the mass of the object, c is the speed of light, dt and dr are change in time and distance respectively and r is variable, reducing the closer you get to an object of mass.

proper time would be represented by-

$$dt_{shell}=\left(1-\frac{2M}{r} \right)^{1/2}dt$$

where dt_shell represents the time dilation at a specific (coordinate) radius.

and proper distance would be represented by-

$$dr_{shell}=\left(1-\frac{2M}{r} \right)^{-1/2}dr$$

where dr_shell represents the distance inflation at a specific (coordinate) radius.

Velocity is normally expressed as v=m/s (distance/time) and the velocity induced by spacetime curvature could be expressed as-

$$v_{shell}=\frac {dr_{shell}}{dt_{shell}}$$

If we substitute the above equations for dt_shell and dr_shell, we get the following equation-

$$v_{shell}=-\left(\frac{2M}{r}\right)^{1/2}$$

negative because the object is moving away from the observer towards the source

Multiply by c for m/s. Objects take the shortest path through spacetime so if it feels a time dilation, no matter how slight, to one side of it, it will tend towards the source of the time dilation (or curvature). v increases as r reduces which is in some way analogues with Newton's equation for gravity $g=Gm/r^2$ (technically, gravity is $g=dr_{shell}\cdot Gm/r^2$ but dr_shell is normally only included when calculating gravity for ultra-compact objects such as neutron stars and black holes and normally ignored for less dense objects such as planets). If you apply the above to Earth say, you'll notice that v_shell for a free-falling object at the Earth's surface is -11.2 km/s which is the negative of the escape velocity expressed as $v_{esc}=\sqrt(2Gm/r)$ which means that an object that fell from rest at infinity would hit the Earth's atmosphere at ~11.2 km/s, 'moving along the curved spacetime geodesic' caused by Earth's mass. (Note: The above velocity represents an object free-falling from infinity only and doesn't account for an object falling from rest at a specific radius) In Minkowski space (i.e. flat space) the velocity (which would be the result of work done by an external source) would remain constant and there wouldn't be any acceleration.

It's also worth noting that for an object free-falling from infinity, E/m=1, the energy required for an object to remain stationary in curved Schwarzschild space is-

$$\frac{E_{shell}}{M}=\left(1-\frac{2M}{r}\right)^{-1/2}$$

so unless there's an input of energy, the object has to move along the curved spacetime geodesic.

 

[A.T.]

forgive my stupidity , but i don't understand
let me put the question in another form ,
why do things fall to the surface of the Earth in straight lines according to GR ?
(and straight lines are not geodesics in curved space time)

I will give you an analogy:

If you stick adhesive tape nicely to a flat surface, without bends, you get a straight line (stripe). If you stick it to a curved surface (e.g. a vase) you get a geodesic. Locally the geodesic is still like a straight line with respect to the 2D-surface of the vase. The fact that the tape is curved within the embedding 3D-space, doesn't change the fact, that topologically it is always a straight line.

To understand how this relates to the apparently curved paths of free fallers, you should check out this link:
http://www.relativitet.se/spacetime1.html

 

[FaithSeeker]

ok ,i'll tell what i have understood here
space time is a 4-manifold
the space time around the Earth is locally flat , so the geodesics of freely falling bodies in this region are straight lines , so they fall in straight lines .
but globally the sapce time around the Earth is not flat , so the geodesics of freely falling bodies (e.g. the moon) are not straight lines , they are curves , so the moon orbits around the Earth in this curve .
did i get it right ?

 

[A.T.]

ok ,i'll tell what i have understood here
space time is a 4-manifold
the space time around the Earth is locally flat , so the geodesics of freely falling bodies in this region are straight lines , so they fall in straight lines .

I think you mix up things a bit:

trajectory : the path of the object in space
worldline : the path of the object in space-time
geodesic : a locally straight path in a manifold
line : special case of a geodesic in a flat manifold

For a free falling object the worldline is always a geodesic. The trajectory is just the projection of this worldline onto the spatial dimensions, and can be curved or not.

This picture shows how a vertically falling object moves straight in curved space-time (worldline is a geodesic):
http://www.physics.ucla.edu/demoweb...alence_and_general_relativity/curved_time.gif