GR space-time motion in the absence of gravity

[Alan Grayson]

In flat space, which is tantamount to assuming the absence of gravity, and non-zero curvature, a body placed at spatial coordinates x,y,z, will move because t increments. But if there is zero curvature, in which direction will it move? That is, how is the direction of motion determined? TIA, AG

[Alan Grayson]

On Saturday, August 1, 2020 at 10:35:09 PM UTC-6, Alan Grayson wrote:

In flat space, which is tantamount to assuming the absence of gravity, and non-zero curvature, a body placed at spatial coordinates x,y,z, will move because t increments. But if there is zero curvature, in which direction will it move? That is, how is the direction of motion determined? TIA, AG

CORRECTION; above, I meant to write, " ... which is tantamount to assuming the absence of gravity and ZERO curvature, ... "   AG

[Lawrence Crowell]

The periapsis or perihelion advance of Mercury is largely a result of classical perturbation theory in classical mechanics. About 10% of the perihelion advance could not be accounted for by perturbation methods in classical mechanics. 

This has to be admired in some ways. Finding the ephemeris of Mercury is tough, for the planet makes brief appearances near the sun in mornings and evenings. Finding an orbital path from its course across the sky is not easy. The second issue is that perturbation methods in classical mechanics are difficult. These were developed arduously in the 19th century and Le Verrier worked on this to find the planet Neptune from the perturbed motion of Uranus in 1848. These methods were worked on through the 19th century. The later work of von Zeipel and Poincare were used to compute the periapsis advance of Mercury, but there was this persistent 43arc-sec/year that resisted these efforts.

It was general relativity that predicted this anomaly in ways that are far simpler than the classical perturbation methods. This post-diction of GR was an initial success in the theory, followed up shortly by the Eddington expedition that found the optical effects of GR in a solar eclipse in 1919.

LC

[Alan Grayson]

On Sunday, August 2, 2020 at 5:30:36 AM UTC-6, Lawrence Crowell wrote:

The periapsis or perihelion advance of Mercury is largely a result of classical perturbation theory in classical mechanics. About 10% of the perihelion advance could not be accounted for by perturbation methods in classical mechanics. 

This has to be admired in some ways. Finding the ephemeris of Mercury is tough, for the planet makes brief appearances near the sun in mornings and evenings. Finding an orbital path from its course across the sky is not easy. The second issue is that perturbation methods in classical mechanics are difficult. These were developed arduously in the 19th century and Le Verrier worked on this to find the planet Neptune from the perturbed motion of Uranus in 1848. These methods were worked on through the 19th century. The later work of von Zeipel and Poincare were used to compute the periapsis advance of Mercury, but there was this persistent 43arc-sec/year that resisted these efforts.

It was general relativity that predicted this anomaly in ways that are far simpler than the classical perturbation methods. This post-diction of GR was an initial success in the theory, followed up shortly by the Eddington expedition that found the optical effects of GR in a solar eclipse in 1919.

LC

I appreciate your grasp of the history, but you haven't answered my question and don't seem aware of what it is (plus you posted your reply on the wrong thread). AG 

[Lawrence Crowell]

I looked at the precession question, wrote it in WORD and then posted it in the wrong thread. A big line of anti-virus defense is working off-line. I do a lot of work locally and pop on and off the internet. I try to never leave my machines on-line with an open port for anyone or any bot to enter to cause mischief.

With this the question is odd. How something moves in free and flat space and spacetime is just determined by its initial conditions.

LC

[Alan Grayson]

On Sunday, August 2, 2020 at 1:55:15 PM UTC-6, Lawrence Crowell wrote:

I looked at the precession question, wrote it in WORD and then posted it in the wrong thread. A big line of anti-virus defense is working off-line. I do a lot of work locally and pop on and off the internet. I try to never leave my machines on-line with an open port for anyone or any bot to enter to cause mischief.

With this the question is odd. How something moves in free and flat space and spacetime is just determined by its initial conditions.

LC

If one starts with SR and zero curvature of spacetime, and places a test particle in that spacetime spatially at rest, how will spacetime tell matter how to move if spacetime isn't curved? AG 

[Alan Grayson]

On Sunday, August 2, 2020 at 5:00:22 PM UTC-6, Alan Grayson wrote:

On Sunday, August 2, 2020 at 1:55:15 PM UTC-6, Lawrence Crowell wrote:

I looked at the precession question, wrote it in WORD and then posted it in the wrong thread. A big line of anti-virus defense is working off-line. I do a lot of work locally and pop on and off the internet. I try to never leave my machines on-line with an open port for anyone or any bot to enter to cause mischief.

With this the question is odd. How something moves in free and flat space and spacetime is just determined by its initial conditions.

LC

If one starts with SR and zero curvature of spacetime, and places a test particle in that spacetime spatially at rest, how will spacetime tell matter how to move if spacetime isn't curved? AG 

I think in this situation the direction of motion is ambiguous. AG 

[Alan Grayson]

On Monday, August 3, 2020 at 8:55:17 AM UTC-6, Alan Grayson wrote:

On Sunday, August 2, 2020 at 5:00:22 PM UTC-6, Alan Grayson wrote:

On Sunday, August 2, 2020 at 1:55:15 PM UTC-6, Lawrence Crowell wrote:

I looked at the precession question, wrote it in WORD and then posted it in the wrong thread. A big line of anti-virus defense is working off-line. I do a lot of work locally and pop on and off the internet. I try to never leave my machines on-line with an open port for anyone or any bot to enter to cause mischief.

With this the question is odd. How something moves in free and flat space and spacetime is just determined by its initial conditions.

LC

If one starts with SR and zero curvature of spacetime, and places a test particle in that spacetime spatially at rest, how will spacetime tell matter how to move if spacetime isn't curved? AG 

I think in this situation the direction of motion is ambiguous. AG 

No. It doesn't spatially move, but it moves in space-time since the observer's clock continues to advance. AG 

[Alan Grayson]

On Monday, August 3, 2020 at 12:15:23 PM UTC-6, Alan Grayson wrote:

On Monday, August 3, 2020 at 8:55:17 AM UTC-6, Alan Grayson wrote:

On Sunday, August 2, 2020 at 5:00:22 PM UTC-6, Alan Grayson wrote:

On Sunday, August 2, 2020 at 1:55:15 PM UTC-6, Lawrence Crowell wrote:

I looked at the precession question, wrote it in WORD and then posted it in the wrong thread. A big line of anti-virus defense is working off-line. I do a lot of work locally and pop on and off the internet. I try to never leave my machines on-line with an open port for anyone or any bot to enter to cause mischief.

With this the question is odd. How something moves in free and flat space and spacetime is just determined by its initial conditions.

LC

If one starts with SR and zero curvature of spacetime, and places a test particle in that spacetime spatially at rest, how will spacetime tell matter how to move if spacetime isn't curved? AG 

I think in this situation the direction of motion is ambiguous. AG 

No. It doesn't spatially move, but it moves in space-time since the observer's clock continues to advance. AG 

What bothers me about this is that the spatial coordinates generally depend on each other, and time. In this situation will the geodesic equations yield a solution where the spatial coordinates remain fixed? AG 

[Lawrence Crowell]

The motion is determined by initial conditions. It is not something determined by some other physics. 

LC

[Brent Meeker]

You can choose coordinates so that a particular geodesic is a coordinate
axis.

Brent

[Alan Grayson]

Maybe this will clear things up. EE has 10 independent equations, so one needs 10 initial conditions to define the path of a test particle in spacetime. What are they, and what would distinguish a geodesic from a non-geodesic solution? TIA, AG

[Lawrence Crowell]

The Einstein field equations are 10 in number and they describe the dynamics of a spatial surface, and its conjugate momentum metric, for a total of 6 variables. The coordinate fixing condition, analogous to a gauge condition, are 4 additional equations that fix the initial spatial surface. 

This is different from geodesics, which correspond to the geodesic motion or separation of two masses. 

LC