To calculate the time dilation experienced by a massive particle traveling at speeds close to the speed of light (C), we can use the Lorentz factor. The Lorentz factor, denoted by the symbol γ (gamma), is given by the formula:
γ = 1 / √(1 - (v^2 / c^2))
where: γ = Lorentz factor v = velocity of the particle c = speed of light in a vacuum (approximately 299,792,458 meters per second)
In this case, we are interested in the time dilation experienced by the particle relative to an observer at rest. Let's assume the particle is traveling at a velocity very close to the speed of light (C), such that v ≈ C.
To find the time dilation, we need to compare the time experienced by the particle (τ) with the time experienced by the observer (t). The relationship between τ and t is given by:
τ = γ * t
Now, you mentioned that a photon only gains a foot (approximately 0.3048 meters) while crossing our galaxy. We can use this distance as the length traveled by the particle (L) and calculate the corresponding time experienced by the observer (t) as follows:
t = L / c
Substituting the value of t into the equation for τ, we can calculate the time dilation:
τ = γ * (L / c)
To find the Lorentz factor (γ), we need to plug in the velocity (v) into the γ formula. However, since v is very close to C, we can approximate it as C for simplicity. Thus:
γ = 1 / √(1 - (C^2 / c^2))
Now, let's calculate the time dilation using the given values:
γ = 1 / √(1 - (C^2 / c^2)) ≈ 1 / √(1 - (299,792,458^2 / 299,792,458^2)) ≈ 1 / √(1 - 1) ≈ 1 / √(0) ≈ 1 / 0 (undefined)
This calculation results in an undefined value because the Lorentz factor becomes infinite when the velocity approaches the speed of light.
However, it's worth noting that objects with mass cannot reach or exceed the speed of light, according to our current understanding of physics. The equations we used here are based on special relativity, which applies to massless particles like photons. Therefore, the concept of a massive particle traveling at the speed of light or experiencing time dilation in the same way as photons is not applicable in our physical reality.