Time Dilation different depending on cause?

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My question concerns gravitational time dilation and time time dilation due to motion. If an outside observer views a person falling into a black hole to stand still when the person falling into the event horizon due to time standing still at the horizon, then why don't particle physicists view particles they accelerate to slow down...as they speed them up. I know that the half-life of particles is increased when they accelerate them, but why don't they view the motion of the particles to slow down as well. The same would qualify for light wouldn't it? Light experiences 100 percent time dilation, meaning that no time passes for light, but if it experiences time dilation, then why do we see it moving? Shouldn't it appear stationary like the man falling into the event horizon? The same question applies for accelerated particles.
 
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Consider the photon with which you measure either the black hole infalling particle or the particle accelerated in the laboratory. In the former case, we observe a particle that must pass through the black hole gravitational field, time dilating near the point of "freezing". The latter situation involves photons passing through the reference frame of the laboratory, itself at rest relative to us. The acceleration of these particles may be the same at v~0 or v~c; acceleration is absolute, whereas in general velocity is not.
 


The concept of time dilation is a crucial aspect of Einstein's theory of relativity, and it can be a bit confusing to understand at first. The key thing to remember is that time dilation occurs due to differences in the relative motion or gravitational fields between two observers.

In the case of gravitational time dilation, it is important to note that it is not the act of falling into a black hole that causes time to stand still at the event horizon. Rather, it is the extreme gravitational pull of the black hole that causes time to slow down. From the perspective of an outside observer, it may appear that the person falling into the black hole is standing still, but this is only because their perception of time is different due to their distance from the black hole.

In contrast, when it comes to particle accelerators, the particles are being accelerated to high speeds, but they are not experiencing a significant change in gravitational pull. Thus, the time dilation effect is much smaller compared to the extreme gravitational pull of a black hole. This is why particle physicists do not observe a significant slowdown in the motion of these particles.

Similarly, the speed of light is not affected by time dilation in the same way as matter. The speed of light is a constant and is not affected by changes in relative motion or gravitational fields. This is why we still see light moving at its normal speed, even though it experiences time dilation.

In summary, the key difference between the time dilation observed in the case of a person falling into a black hole and the time dilation observed in particle accelerators is the difference in the strength of the gravitational fields involved. The stronger the gravitational field, the greater the time dilation effect. This is why we see a more noticeable time dilation effect in the case of a black hole compared to a particle accelerator.
 
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In Philippe G. Ciarlet's book 'An introduction to differential geometry', He gives the integrability conditions of the differential equations like this: $$ \partial_{i} F_{lj}=L^p_{ij} F_{lp},\,\,\,F_{ij}(x_0)=F^0_{ij}. $$ The integrability conditions for the existence of a global solution ##F_{lj}## is: $$ R^i_{jkl}\equiv\partial_k L^i_{jl}-\partial_l L^i_{jk}+L^h_{jl} L^i_{hk}-L^h_{jk} L^i_{hl}=0 $$ Then from the equation: $$\nabla_b e_a= \Gamma^c_{ab} e_c$$ Using cartesian basis ## e_I...

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