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Light is affected by gravity, its path is bent by gravity, demonstrated by gravitational lensing. This got me thinking: speed of light is a constant, as we all know, however if its path and therefore direction is changing, is its velocity too? is light under acceleration when passing through a gravitational field?

When we illustrate the path of light in warped space-time by drawing it on a flat 2D surface, the result is a curved path. If we draw the space-time grid on that same 2D surface, it should appear warped, and the light's path should be parallel to the warped coordinates. In Minkowski 4D coordinates, the path of light is a straight line. In fact, the 4D space-time grid is defined by the straight paths of imaginary photons. So, in Minkowski space-time, light does not accelerate in either speed or direction. This doesn't mean that Euclidean space is invalid. It just isn't as useful as Minkowski space-time for calculating trajectories in gravitational fields. In Euclidean space, a photon does accelerate. It's momentum changes when passing near a star, which means the star is pulling the photon with a gravitational force. Conservation of momentum dictates the photon must exert an equal and opposite gravitational force on star. Thus, in Euclidean space, a photon has a gravitational field of its own. Unfortunately, the use of Euclidean space necessitates a whole different set of definitions for units of space, time, etc. The math is unwieldy, even more so than the math of general relativity. If I had any aptitude for math, I think I would use numerical analysis to predict paths in Euclidean space, and I'd expect to get results equivalent to those derived by Einstein's general relativity. The results should be equivalent in predicting whether two objects will collide or orbit. The two methods should agree on what a space traveler would observe on his clock and what he should see out his window.

So now im wondering in euclidean spacetime photons have mass correct?

...so can you accelerate any other massive particle to the speed of light?

very informative post thank you, euclidean spacetime is very wacky and intriguing. So now im wondering in euclidean spacetime photons have mass correct? so can you accelerate any other massive particle to the speed of light?