Gravitational vs geodesic proper time

[westwood]

I've been trying to learn GR and I've been back and forth through Schutz's first course book. I think I understand the basic principals, but one thing still eludes me: a traveler in free fall travels along the geodesic, the path of longest proper time. If the path between two points passes through a strong gravitational field, it should take longer (more ticks on the traveler's clock) than it would have if the field had not been in the path. But as the traveler passes through the field, gravitational time dilation makes their clock show down (fewer ticks on the traveler's clock). Do these two effects compete against each other or am I missing something? Somehow I got it in my head that gravitational time dilation would somehow _cause_ the geodesic to have the longest proper time, but that doesn't seem to work. Please help straighten me out.

 

[bcrowell]

IMO this is a good question, and shows that you're thinking about the right things and engaging actively with the material. Pat yourself on the back :-)

There's a good discussion of this kind of thing here: http://www.astro.ucla.edu/~wright/deflection-delay.html They analyze deflection of light by the sun. Newtonian physics predicts that the light will speed up (assuming light can be treated as a material particle that is initially moving at c), so it gets to its destination sooner. GR predicts that it will arrive late. There is a classic experiment by Shapiro that tested this. The slowing is directly related to the deflection according to the wave theory of light. I guess this isn't perfectly on target for your question, since you were asking about a material particle (for which proper time is meaningful), whereas this is all about light.

One thing that I think you may be messing up is that it sounds like you're interpreting least action as a comparison of field and no field, for a fixed path. Actually it's a comparison of different paths, for a fixed field.

 

[atyy]

This solution is wrong (I don't think gravitational time dilation is fundamental, and it's better to define it with some experiment in mind, and usually you need to be able to define an equivalent Newtonian potential, which is not always possible in general relativity), but I cannot resist.

Two clocks start at some height above the ground. The one that drops will follow the geodesic and fall deeper and deeper into the potential well, running slower and slower compared to the one that stays in the air.

 

[westwood]

Thanks for all your responses. I think my problem is that I was trying to compare a trajectory in the cases where it does and does not pass through a gravitational field, when in fact there must be two different trajectories.