To calculate the time it takes for a photon to travel from the Sun to the Earth as seen from a rocket moving at 0.9999c (99.99% of the speed of light), we can use the concept of time dilation from special relativity. However, it's important to note that as an observer approaches the speed of light, the time dilation effects become extremely significant, and relativistic calculations can be quite complex.
In this scenario, we will assume that the distance between the Sun and the Earth is approximately 149.6 million kilometers.
The time experienced by the observer on the rocket, known as the "proper time" (Δt_0), will be dilated or "slowed down" compared to the time measured by an observer at rest relative to the Sun (Δt). The time dilation factor (γ) can be calculated using the Lorentz factor:
γ = 1 / √(1 - v^2/c^2)
where v is the velocity of the rocket and c is the speed of light.
Given that v = 0.9999c, the time dilation factor can be calculated as:
γ = 1 / √(1 - (0.9999c)^2/c^2) ≈ 70.711
Now, to determine the time it takes for the photon to travel from the Sun to the Earth as observed from the rocket, we divide the distance (d) by the speed of light (c) and then multiply by the time dilation factor (γ):
Δt = (d / c) * γ
Substituting the values, we have:
Δt = (149.6 million km / c) * γ
Assuming the speed of light is approximately 299,792 kilometers per second, we can calculate:
Δt ≈ (149.6 million km / 299,792 km/s) * 70.711 ≈ 235.7 seconds
Therefore, from the perspective of the rocket moving at 0.9999c, it would take approximately 235.7 seconds (or about 3.93 minutes) for a photon to travel from the Sun to the Earth.
Regarding your second question, time dilation does not depend on the direction of motion. It is a symmetric effect that occurs regardless of the direction of relative motion between the observers. The time dilation factor is solely determined by the relative velocity between the two frames of reference.