Re: I need a formula to get, time elapsed on earth while at speed of light

Depends how the spaceship works.

If you can accelerate at an infinite rate without killing the passengers, getting up to lightspeed instantly and then instantly turning around when you get to where you're going...

In this case, the time experienced by the people in the ship (T') is equal to the time experienced by people on earth (T) divided by a factor gamma:

gamma = 1/[(1-v^2/c^2)^0.5]

Here v is the constant velocity that the ship travels at (relative to Earth), and c is the speed of light. For 99% of the speed of light (v/c= 0.99), gamma is 7.08, so when the ship returns the people on Earth have aged seven times more than the people in the ship.

That's the easy equation. If you want to do this right, you need to factor in a finite acceleration that the people in the ship can endure. In hard science fiction, this is sometimes done by having the ship accelerate at a constant 1g, the same acceleration we feel on the surface of the Earth. When the ship is halfway there, it has to stop accelerating and start to decelerate (at 1g) for the rest of the way. (Never mind exactly where you get the energy and fuel to constantly accelerate for years at a time!)

Here are the equations for this. Again T is the time as seen by Earth (in units of years for the full roundtrip journey), T' is the time as seen by the spaceship (also in roundtrip years), a is the acceleration (in units of g, so a=1 is an acceleration/deceleration of 1g), v is the peak velocity (halfway through the journey), and tanh is the hyperbolic tangent (hit "hyp" then "tan" on a calculator).

v/c = 0.25aT[(1 + (0.25aT)^2)^-0.5] = tanh (0.25aT')

The total distance one can travel (one-way, in lightyears) is:

2{([1 + (0.25aT)^2]^0.5) - 1}/a

In this last equation, T is still the total roundtrip time (as seen from Earth, in years), even though the answer only gives you the one-way distance. If you want to get distance as a function of T', you need to convert from T to T' using the tanh equation.

Here's a page you might find useful. Good luck with the story!