A Brain Overload: Comparing Proper Time, Ephemeris Time & Coordinate Time

[Matter_Matters]

Currently reading the following document which is a bit of a brain overload at the minute!

Im considering Equation (4.61). It is the general relativistic correction due to the Schwarzschild field for a near Earth satellite when the parameters \beta, \;\gamma \equiv 1. However, as you will see, for the equations of motion they use geocentric coordinate time as the independent variable instead of the proper time as is usually the case. This is also the same for the IERS convention given http://iers-conventions.obspm.fr/2010/2010_official/tn36.pdf by equation 10.12. Has anyone seen this before? I am struggling to get my head around this.

 

[Paul Colby]

However, as you will see, for the equations of motion they use geocentric coordinate time as the independent variable instead of the proper time as is usually the case.

Geocentric coordinates are quite common in satellite ephemeris and tracking calculations. They are looking for small modifications to the usual situation.

 

[pervect]

Currently reading the following document which is a bit of a brain overload at the minute!

Im considering Equation (4.61). It is the general relativistic correction due to the Schwarzschild field for a near Earth satellite when the parameters \beta, \;\gamma \equiv 1. However, as you will see, for the equations of motion they use geocentric coordinate time as the independent variable instead of the proper time as is usually the case. This is also the same for the IERS convention given http://iers-conventions.obspm.fr/2010/2010_official/tn36.pdf by equation 10.12. Has anyone seen this before? I am struggling to get my head around this.

I'm not sure I understand exactly what you're asking. Let's consider a typical problem where one might want to use an ephermis. Suppose one has a telescope, and a clock, here on Earth, one looks at the clock and wants to know in what direction to point the telescope to see some satellite or celestial body at that particular instant.

The end result of thinking about this problem about how to point the telescope is that one wants to know the position of the satellite at some instant as judged by a clock on the Earth (which is easily accessible, in our example case perhaps it is a wall clock by the telescope).

To go further, judging by your question, you may be confused about why the sort of clock that one has on the wall measures a coordinate time, and not proper time. (Most likely a wall clock would read UTC, Universal Coordianted Time, which you can look up, and note that it is a coordinate time standard). I hope the following answer will explain the necessity. One of the functions of a coordinate system is to be able to synchronize multiple clocks. Proper time can never do that, proper time encapsulates the notions of what a clock reads, but it does not incorporate any notion of how to synchronize different clocks. The notion of synchronizing clocks requires more than a knowledge of proper time, it requires the use of a simultaneity convention. Recall from special relativity (hopefully it's familar) that different frames of reference have different simultaneity conventions. In the presence of gravity, we've moved beyond special relativity, but we still have the issue of the need to synchronize clocks. When we have a coordinate system, this issue is solved - we can regard events that have the same time coordinate as being simultaneous in that particular coordinate system. The remaining issues involve exactly what coordinate system we want to use. In some problems, we might be using several different coordinate systems. This is a lot of detail, but it's just something you have to slog through, after you have some idea of what it is you're trying to accomplish.

I hope this helps, at least to give grounds to clarify what your question is.

 

[Matter_Matters]

I'm not sure I understand exactly what you're asking. Let's consider a typical problem where one might want to use an ephermis. Suppose one has a telescope, and a clock, here on Earth, one looks at the clock and wants to know in what direction to point the telescope to see some satellite or celestial body at that particular instant.

The end result of thinking about this problem about how to point the telescope is that one wants to know the position of the satellite at some instant as judged by a clock on the Earth (which is easily accessible, in our example case perhaps it is a wall clock by the telescope).

To go further, judging by your question, you may be confused about why the sort of clock that one has on the wall measures a coordinate time, and not proper time. (Most likely a wall clock would read UTC, Universal Coordianted Time, which you can look up, and note that it is a coordinate time standard). I hope the following answer will explain the necessity. One of the functions of a coordinate system is to be able to synchronize multiple clocks. Proper time can never do that, proper time encapsulates the notions of what a clock reads, but it does not incorporate any notion of how to synchronize different clocks. The notion of synchronizing clocks requires more than a knowledge of proper time, it requires the use of a simultaneity convention. Recall from special relativity (hopefully it's familar) that different frames of reference have different simultaneity conventions. In the presence of gravity, we've moved beyond special relativity, but we still have the issue of the need to synchronize clocks. When we have a coordinate system, this issue is solved - we can regard events that have the same time coordinate as being simultaneous in that particular coordinate system. The remaining issues involve exactly what coordinate system we want to use. In some problems, we might be using several different coordinate systems. This is a lot of detail, but it's just something you have to slog through, after you have some idea of what it is you're trying to accomplish.

I hope this helps, at least to give grounds to clarify what your question is.

Excellent thanks for clearing that up for me. However, what I am struggling with is to grasp why the geocentric coordinate time is the affine parameter. So, let me try say it like this. Given a spacetime interval (In the GCRS) the line element of Minkowskian space is given by
ds^2 = c^2 d \tau^2 = \underbrace{c^2dt^2}_{\text{Is this a geocentric coordinate time?}} -d\mathbf{r}^2,
where d\mathbf{r}^2 = dx^2 +dy^2 +dz^2. Now the Lagrangian given in the above document is for a slightly different metric but let's not worry to much as I believe the result. I just don't get understand the parameterisation. So are the coordinates in the above line element are given by
x^\mu = (ct(t_{GC}),x(t_{GC}),y(t_{GC}),z(t_{GC}))
where t_{GC} is the geocentric coordinate time. The above line element would have a Lagrangian given by
c^2 - \dot{r}^2,
where the dot notation is used to note a derivative with respect to geocentric coordinate time? Does that make sense?

 

[Matter_Matters]

Geocentric coordinates are quite common in satellite ephemeris and tracking calculations. They are looking for small modifications to the usual situation.

Thanks Paul. However, the part I don't understand is the parametrisation of the spacetime coordinates. Normally, in GR they are parametrised by the proper time for the geodesic equation at least.

 

[Paul Colby]

However, what I am struggling with is to grasp why the geocentric coordinate time is the affine parameter.

There is quite a bit of freedom in how one chooses an affine parameter[1]. The proper time seems a compelling choice but it really isn't. It doesn't work for null geodesics for example. The mapping to proper time which comes straight from the metric,

$ds = \sqrt{c^2 - dr/dt \cdot dr/dt}dt.$​

With the freedom to choose the parameter it's not surprising they choose geocentric coordinate time.

[1] I only pretend to understand what "affine" means and suspect it's some mathematical term to make one sound knowledgeable.

 

[Matter_Matters]

There is quite a bit of freedom in how one chooses an affine parameter[1]. ...

...[1] I only pretend to understand what "affine" means and suspect it's some mathematical term to make one sound knowledgeable.

This is brilliant. I enjoyed this a lot.

Okay thanks mate. Now, you've opened up two different cans of worms...

You can derive equations of motion for a null geodesic with the proper time as the "affine" parameter no problem I thought? Isn't it a standard calculation in undergrad GR?

Secondly

With the freedom to choose the parameter it's not surprising they choose geocentric coordinate time.

Can you elaborate please? Why is it not surprising?

 

[Paul Colby]

You can derive equations of motion for a null geodesic with the proper time as the "affine" parameter no problem

Well how does that work exactly? The proper time is the time as seen by an observer traveling along the geodesic. That's straight out for a null geodesic, right? However, one may choose a parameter, $x^\mu(\lambda) = (\lambda,0,0,\lambda)$ to parameterize a null geodesic of a light beam along the +z axis. $\lambda$ isn't a proper time at all.

Can you elaborate please? Why is it not surprising?

As I said the people that wrote that document track satellites and such. Relativistic effects are small and so they choose coordinates and conventions that they are familiar with. That would be geocentric coordinates as one relatively standard choice.

 

[PAllen]

There is quite a bit of freedom in how one chooses an affine parameter[1]. The proper time seems a compelling choice but it really isn't. It doesn't work for null geodesics for example. The mapping to proper time which comes straight from the metric,

$ds = \sqrt{c^2 - dr/dt \cdot dr/dt}dt.$​

With the freedom to choose the parameter it's not surprising they choose geocentric coordinate time.

[1] I only pretend to understand what "affine" means and suspect it's some mathematical term to make one sound knowledgeable.

The set of all affine parameters are connected by linear transformations, so 'almost all' parametrezations are excluded. Using an arbitrary parameter is possible but then the variation of the action produces a more complex form of the geodesic equation.

 

[Matter_Matters]

The set of all affine parameters are connected by linear transformations, so 'almost all' parametrezations are excluded. Using an arbitrary parameter is possible but then the variation of the action produces a more complex form of the geodesic equation.

Can you illustrate that with an example? For example using \lambda over \tau as a parameter for the world line of a particle or satellite as it were.

 

[PAllen]

Can you illustrate that with an example? For example using \lambda over \tau as a parameter for the world line of a particle or satellite as it were.

See page 71 of:

http://xxx.lanl.gov/abs/gr-qc/9712019

Ask further questions as needed.

 

[Matter_Matters]

Well how does that work exactly? The proper time is the time as seen by an observer traveling along the geodesic...

I was a bit hasty with my response there. You cannot parametrise a null geodesic with the proper time as it will assign the same value to all points along the geodesic. Just like you said :)