Oh, how this has vexed me. For the last year, this has vexed me. That just goes to show how dim and stubborn I can be. Discovery takes time. Acceptance takes time. So, I finally accepted what was there all along.
Gravitational Time Dilation. The final, hardest step along this journey. It’s right there in the diagram. It always was, had I but the wit and willingness to see it.
The field is not the coordinate line at the top. That’s what a distant observer would perceive as space. No, the red curve is the the surface of the field. The surface of the field is “curved spacetime”.
But where in the diagram is gravitational time dilation? Why does light travel more slowly near a massive object like the sun? It doesn’t. Time isn’t part of the diagram. Time travels along at the universal tick, uncaring of your private perception of time.
Light doesn’t travel more slowly. It always travels at the same rate of one unit distance per one unit time - along the curve. Not along the coordinate grid. And the curve is always longer than the grid. The difference is usually minuscule. But when your near a massive object, the curvature becomes enough to notice. The red line is longer than its projection onto the coordinate grid above. Just barely. By how much?
That’s really not much. At 10 units from the mass, it’s 1.0000414. At 50 units from the mass, it’s 1.0000001. Further than that, my calculator just gives up and says it’s one.
It’s enough to make the time it takes a radar signal to pass the sun twice on its way to and from Mercury to take about 200 microseconds longer.1 It’s enough to make light bend twice as much towards the sun as Newtonian gravity would allow.2 It’s enough to accelerate the orbit of Mercury.3 It’s enough to make the GPS satellites have to update their clocks every pass.4
Wait - If there is more “distance” in the space near a massive object, then how does Mercury’s orbit accelerate near the sun? Shouldn’t it slow down like light?
Remember how every particle has a fixed and finite radius of one unit? That’s not measured against the coordinate grid. It’s measured against the field - the curvature caused by everything else.
That shrinks the base of the particle (length contraction). That, in turn, increases the angle formed by the energy differential inside the particle. That increases the velocity (and decreases the perceived time) of the particle. Without increasing the particle’s energy. That’s the effect of spacetime “flowing” towards the massive object.
We are now operating entirely outside the imagination of Newton. And we’re still looking at the same picture.
This is why the math of General Relativity is so famously difficult. Everything depends upon everything else, and it’s all varying all the time.
https://en.wikipedia.org/wiki/Shapiro_time_delay
https://www.einstein-online.info/en/spotlight/light_deflection/
https://en.wikipedia.org/wiki/Tests_of_general_relativity#Classical_tests
https://en.wikipedia.org/wiki/Gravitational_time_dilation#Experimental_confirmation
