Hi all, Here is the problem. I want to try to do some calculations with seeing how much time a relativistic ship has before it can dodge a dust grain. I am having some trouble applying the Lorentz Contraction Equations to get consistent results. Take two frames, the coordinates of the dust grain at rest with respect to the Earth, and a ship moving with v near c. In the below, exponents marked with a "^". In all coordinates, position is first, time is second. At x=0, t=0, x'=0, t'=0 the Earth and ship origins line up. For the Earth frame, the dust grain is at (d, 0). If you do the Lorenz Transformation Equations, you will find that in the ship's frame, the coordinates transform to: (d', -γvd/c^2) where the last term is the time offset in the ship's frame. d' = γd. This part is correct, and is referenced in Griffiths 3rd Edition. Now, the Earth says that at its x=0, t=0, the ship sends out a radar pulse. This radar pulse travels at c, being a photon, and it will arrive at the dust grain at (d, d/c). If I do a Lorentz Transform at the event on when the dust grain recieves the radar pulse, I get the following: equation 1 equation 2 d' = γ(d-vd/c) and t' = γ(d/c-vd/c^2). γ is gamma, v is velocity, d is dust grain coord for Earth, c is light speed. d' is grain coord as seen by ship, t' is time coord as seen by ship for grain to recieve radar pulse. Notice that in the ship's t' calculated here, the -γ(vd/c^2) term matches the transformed t' from transforming (d,0) to the ship's time. Now, here is the problem. Look closely at t' = γ(d/c-vd/c^2). It has two terms. The way I read this is that, in the ship's frame, at the time beginning with the offset -γ(vd/c^2) the ship fired a radar pulse at c and the radar pulse travelled a distance d'/c. The problem I have here is: doesn't the dust grain move towards the ship, so that if I didn't use a Lorentz Transform from a ground frame, I would write d'-vt'=ct' (equation 3) for the ship? If, I discard th " vt' " term in equation 3, I agree with the first term in equation 2 above. In writing d'-vt'=ct', I think I am saying that my radar pulse will collide with the dust grain moving towards me, but this doesn't agree with equation 2. Why doesn't this make sense? The first term of Equation 2 seems to say that the dust grain won't move at all in the ship's frame, but simply be contracted. Equation 2 seems to imply that in the ship's frame, the radar pulse will travel to the dust grain in time d'/c. So I don't know which to believe here. I looked up some derivations of the lorentz transform. Is the idea that, in the ship's frame, the ship stays still and the ground moves with respect to the ship already incorporated into the structure of the the Lorentz Transforms? This is all I could think of to reconciles this inconsistency. Also, when the radar pulse reaches the dust grain, I tried to figure out where the Earth and ship say the new coordinates of the ship are. For the Earth, this is easy. In the Earth's frame, the ship will be at (vt, t). The ship will move to a new coordinate vt, where t is the time it sees the ship's radar pulse reach the dust grain, i.e, t=d/c. If I transform to the ship's frame, I get x_s'=0 which makes sense; the ship percieves itself to be at its own origin even though Earth says the ship is at x_s = vt. This is mostly a check to make sure I am doing things right. But if I transform the time from Earth to ship, shouldn't I get the same transformed time of equation 2? The ship is at x_s = vt at time "t" and that is the same "t" the dust grain recieves the radar pulse in the Earth's frame. However, because the x coordinates of the ship and dust grain are different, I don't. Here's what happens: t_s'=γ(d/c-v*vt/c^2)=γ(d/c-d*(v^2)/(c^3)). The first term in this transformation is right, but the 2nd term -d*(v^2)/(c^3) does not at all match the second term in equation 2. If the ship, stays at his origin and watches the dust grain absorb the radar pulse, shouldn't he have the same transformed time for both events? Shouldn't the Earth transform to the same time in the ship's frame for both events? Or have I got caught by simultaneoity here? The ship may be at x_s = vt and time "t" and the dust grain may also recieve the radar pulse at the same time "t" in the Earth's frame, but if I go to the ship's frame, will a simultaneity issue not let the transformed times be the same? That seems very weird.