How to use Einsteins Field Equations ?

[notknowing]

Suppose a spaceship is leaving earth, moves close to the speed of light, passes regions of strong gravitational fields, etc., then the Einsteins Field Equations (EFE) should be able to predict the path of the spaceship with great precision. My question is how we specify the position of the spaceship relative to Earth (in curved spacetime). The EFE are differential equations, so that lenghts are obtained as integrals of ds along some path. So, what is the relativistically correct way to describe the position and movement of the spaceship relative to Earth ?

 

[HallsofIvy]

First you solve the field equation for the metric tensor. Once you have that, solve for the geodesics. If you want "with respect to the earth", you can make sure the origin of your coordinate system is at the earth. That's going to complicate the metric tensor as opposed to using, say, the sun.

 

[notknowing]

First you solve the field equation for the metric tensor. Once you have that, solve for the geodesics. If you want "with respect to the earth", you can make sure the origin of your coordinate system is at the earth. That's going to complicate the metric tensor as opposed to using, say, the sun.

I'm not so sure how to do this in practice. Could you expand a bit more on this? How would the position then eventually be specified (in contrast to a simple (x1,y1,z1,t1) coordinate) ?

 

[cesiumfrog]

In practice, what you want to do (limited to the simplest nontrivial case) is actually a major assignment given at the end of the GR 3rd year physics course at my uni, and has a reputation as the most horribly tedious task imaginable. A computer would make it less tedious, but it's still a lot more difficult than pointing runge-kutta at Newtonian dynamics.

 

[MeJennifer]

The EFE are differential equations, so that lenghts are obtained as integrals of ds along some path. So, what is the relativistically correct way to describe the position and movement of the spaceship relative to Earth ?

Well that is the way to describe it.
Remember that spacetime is curved so the idea of some absolute position in GR is even more removed from practicality than it is in SR.

 

[pervect]

Suppose a spaceship is leaving earth, moves close to the speed of light, passes regions of strong gravitational fields, etc., then the Einsteins Field Equations (EFE) should be able to predict the path of the spaceship with great precision. My question is how we specify the position of the spaceship relative to Earth (in curved spacetime). The EFE are differential equations, so that lenghts are obtained as integrals of ds along some path. So, what is the relativistically correct way to describe the position and movement of the spaceship relative to Earth ?

You can pick *any* arbitrary set of coordinates. Just set up any arbitrary smooth labelling of events in space-time with coordinate values.

The coordinates choice defines the numerical value of the metric coefficients, which then gives rise to the exact forms of the geodesic equations.

Of course, some coordinate choices turn out computationally to be much more conveneint than others. A rule of thumb is to make as many Christoffel symbols zero as possible.

 

[notknowing]

You can pick *any* arbitrary set of coordinates. Just set up any arbitrary smooth labelling of events in space-time with coordinate values.

The coordinates choice defines the numerical value of the metric coefficients, which then gives rise to the exact forms of the geodesic equations.

Of course, some coordinate choices turn out computationally to be much more conveneint than others. A rule of thumb is to make as many Christoffel symbols zero as possible.

I'm beginning to realize that the answer to my original question can become rather complicated. Let us simplify the situation. Consider the earth-moon system. When I now ask "what is the position of the moon, relative to the Earth (at a given time)?, in what "format" should the answer be given, to be fully compatible with GR ? You can choose of course the origin of a coordinate system at the earth, but this will not be sufficient to specify the position of the moon with respect to this coordinate system as a vector (x1,y1,z1) because the curvature of spacetime will change the orientation of the axes from point to point (as given by the connection coefficients) along some line up to the moon. Would it be correct to say that the position of one object relative to another (in GR) in curved spacetime can not be given as a single vector (as in flat spacetime) but is in fact given as a kind of "road description", specifying at each new point a new direction to follow (for an infinitisemal length) ?

 

[pervect]

I'm beginning to realize that the answer to my original question can become rather complicated. Let us simplify the situation. Consider the earth-moon system. When I now ask "what is the position of the moon, relative to the Earth (at a given time)?, in what "format" should the answer be given, to be fully compatible with GR ? You can choose of course the origin of a coordinate system at the earth, but this will not be sufficient to specify the position of the moon with respect to this coordinate system as a vector (x1,y1,z1) because the curvature of spacetime will change the orientation of the axes from point to point (as given by the connection coefficients) along some line up to the moon. Would it be correct to say that the position of one object relative to another (in GR) in curved spacetime can not be given as a single vector (as in flat spacetime) but is in fact given as a kind of "road description", specifying at each new point a new direction to follow (for an infinitisemal length) ?

You don't NEED to use any particular format. You can basically assign any four numbers to every event in space-time, and use these as your coordinates. The only constraint on this process is that nearby events must have nearly the same coordinates - the labelling process must be smooth. This requreiment basically insures that the distance between any two nearby points can be represented by a quadratic form (i.e. a polynomial of degree two) in the coordinates. You can realize this by sticking only a finite number of "labels" unto events in space-time (think of sticking pins on a map), and then using some sort of interpolation process to find the coordinates of events "near" the labelled refrence points.

Once you've made your totally arbitrary assignments of labels (coordinates) to events, you will have defined a metric. Or more precisely, you will have defined the particular components of the metric in the coordinate system you have chosen.

This is described in more detail in MTW's gravitation, by the way.

The mathematical system that allows one to chose arbitrary coordinates in this way is "differential geometry", or informally "tensors". The whole point of adopting this approach is that we don't have to answer questions like you just asked - we can choose any coordinates we like, and the mathematics will give us the correct answer.

This goes back to some of your earlier questions, I think, the question about multiplying the components of a metric by a constant. There is no physical significance to the particular coordinates used to represent a physical system, just as there is no geometrical signifance to drawing lines or labels on a map. The map is not the territory. Thus the particular coordinates of the metric don't have any physical significance, they depend on the human choice of what lables one chose to give particular events. Thus multiplying all the components of the metric is a statement about coordinates (labels), not a statement about physics.

[add]
OK, now that I've mentioned that you don't have to use any particular coordinate system, I should add something about convenient or standard choices for coordinates.

One standard choice for the solar system (which could also be applied to your Earth-moon system) are post-Newtonian PPN coordinates. Unfortunately I'm not quite sure exactly how they work empirically.

The basic idea is to make the metric fit into a simple and standard form, when expanded as a series in epsion (where epsilon^2 is the Newtonian potential U, which is a dimensionless number in geometric units).

One has terms of order epsilon^4 in g_00, epsilon^3 in g_0j, and epsilon^2 in g_jk.

This is only an approximation - AFAIK there isn't any simple analytical metric for the two body problem, though.

 

[notknowing]

You don't NEED to use any particular format. You can basically assign any four numbers to every event in space-time, and use these as your coordinates. The only constraint on this process is that nearby events must have nearly the same coordinates - the labelling process must be smooth. This requreiment basically insures that the distance between any two nearby points can be represented by a quadratic form (i.e. a polynomial of degree two) in the coordinates. You can realize this by sticking only a finite number of "labels" unto events in space-time (think of sticking pins on a map), and then using some sort of interpolation process to find the coordinates of events "near" the labelled refrence points.

Once you've made your totally arbitrary assignments of labels (coordinates) to events, you will have defined a metric. Or more precisely, you will have defined the particular components of the metric in the coordinate system you have chosen.

This is described in more detail in MTW's gravitation, by the way.

Thanks Pervect for this clear answer. I'll read again the relevant part in MTW.

 

[HallsofIvy]

Eddington, in his "Mathematical Theory of Relativity" devotes one chapter to a detailed calculation of the metric tensor in the simple case of a single massive body, starting by altering the spherical metric by unknown functions, then using the gravitational tensor equations to find those functions. He then calculates orbits showing that a test mass close to the origin (think "Mercury") has an orbit that is, approximately, an ellipse with apogee moving.

 

[notknowing]

Thanks Pervect for this clear answer. I'll read again the relevant part in MTW.

I finally had the time to read the relevant part in MTW (pages 5-10, par. 1.2, Fig.1.2, Fig.1.3).
From page 10 : "A more detailed diagram would show a maze of world lines and of light rays and the intersections between them. From such a picture, one can in imagination step to the idealized limit: an infinitely dense collection of light rays and of world lines ..."
This is all nice and well but to be able to give the position of an event, one really needs worldlines of other particles. So, following the same line of reasoning, one would run into trouble when one would encounter a region where there are no worldlines (of photons, particles, etc.) at all. Such a situation is of course very unlikely, though one could imagine it. I think this is were quantum physics could come to the rescue: since the vacuum is full of virtual particles they provide the necessary world lines even if there is no real photon around. At the same time, they probably provide a limit to the smoothness of space. At the time general relativity was "constructed", such considerations were probably not be made. So, how then did people envisage the world to be filled with an almost infinitely dense maze of worldlines ?

 

[Los Bobos]

At the time general relativity was "constructed", such considerations were probably not be made. So, how then did people envisage the world to be filled with an almost infinitely dense maze of worldlines ?

The effects of gravity and acceleration are locally indistinguishable (postulate) -> spacetime is locally flat (free falling observers, geodesic coordinates and so on).

And to the original question, if i remember correctly, this is quite "easy" to calculate in spherically symmetric case, you just choose a suitable plane so that you are just left with the azimuthal angle and radial part of the Schwartzschild metric, then but this in the geodesic equation, calculate some pages and you have the geodesic.

 

[pervect]

I finally had the time to read the relevant part in MTW (pages 5-10, par. 1.2, Fig.1.2, Fig.1.3).
From page 10 : "A more detailed diagram would show a maze of world lines and of light rays and the intersections between them. From such a picture, one can in imagination step to the idealized limit: an infinitely dense collection of light rays and of world lines ..."
This is all nice and well but to be able to give the position of an event, one really needs worldlines of other particles. So, following the same line of reasoning, one would run into trouble when one would encounter a region where there are no worldlines (of photons, particles, etc.) at all. Such a situation is of course very unlikely, though one could imagine it. I think this is were quantum physics could come to the rescue: since the vacuum is full of virtual particles they provide the necessary world lines even if there is no real photon around. At the same time, they probably provide a limit to the smoothness of space. At the time general relativity was "constructed", such considerations were probably not be made. So, how then did people envisage the world to be filled with an almost infinitely dense maze of worldlines ?

Suppose you have a piece of graph paper. It could be considered to be a "maze of lines". If you want to specify a coordinate very precisely, it would be an "infinitely dense maze of lines".

MTW is just expanding the graph paper idea to more than 2 dimensions - and to curved surfaces. Thus we can imagine the "maze of lines" as being drawn on the surface of a sphere (lattitude and longitude) as well as on a planar piece of paper.
line.

 

[notknowing]

Suppose you have a piece of graph paper. It could be considered to be a "maze of lines". If you want to specify a coordinate very precisely, it would be an "infinitely dense maze of lines".

MTW is just expanding the graph paper idea to more than 2 dimensions - and to curved surfaces. Thus we can imagine the "maze of lines" as being drawn on the surface of a sphere (lattitude and longitude) as well as on a planar piece of paper.
line.

Yes, but the point which is made in MTW is that you can not "stick" mark points on space like you can on a piece of paper. Instead one uses the worldlines of particles, photons, etc which are already there IN ORDER to be able to locate an event. This is the thing I wanted to expand on (virtual particles, quantum physics, ..).