# Seeming contradiction between Lorentz equations and length contraction

I'm currently teaching IB physics and I'm preparing to cover relativity in my class as it's recently been added to the syllabus.

I'm experimenting with different ways to explain relativity and while doing so I was toying with a scenario used on Khan academy to explain some ideas. But I've run into a seeming contradiction I can't reconcile.

Here's the scenario.

An astronaut is adrift in space and a UFO passes him by moving in the positive x direction. It is being trailed by a spaceship and both vessels are traveling at half the speed of light. In the ships's reference frame the vessels are separated by 1 light second

In the ship's frame I set t = 0 to be the moment it passes the astronaut. In this frame we'll say that at t = -1 s it shoots a photon at the UFO. I believe I am correct in identifying the ship's frame of reference as the measure of the proper distance between it and the UFO and the proper measure of time it takes the photon to reach the UFO. Here's what I don't understand.

I think that in both frames of reference (the ships and the astronauts) the ship fires its photon at the moment when the ship and UFO are symmetrically positioned around the astronaut. So this means that I should be able to take the contracted length in the astronauts frame and cut it in halve to get the position of the ship in the astronauts frame. Resulting in 1 light second divided by the Lorentz factor as the distance from the astronaut to the ship in the astronaut's frame.

But when I use the Lorentz Transformation equation I get half a light second times the Lorentz factor as the result rather than divded by.

What am I missing here? Can someone explain how I should use the Lorentz transform to get the same result as my logic with length contraction? Or is my length contraction logic flawed in some way?

I think that in both frames of reference (the ships and the astronauts) the ship fires its photon at the moment when the ship and UFO are symmetrically positioned around the astronaut.

This is false.

You have been working in the ship-UFO frame, and when the photon is fired, the astronaut was at $$(t,x)=(-1,\frac12)$$. However, the astronaut would not agree that the photon was fired when the astronaut is at the exact middle.

If you drew this out on both Minkowski diagrams, it will become obvious.

In general, however, the takeaway is that a length measurement is subtraction of two places at the same time; The astronaut will totally disagree with same time. There is also no proper time for light; proper time is subtraction of two clock readings at the same place. Obviously these two notions will be vehemently disagreed upon by different observers. The Lorentz $$\gamma$$ factor will appear in the wrong place if you make a mistake here. These are very standard mistakes.

• Very clear answer and I kind of suspected that may be my problem but couldn’t wrap my head around it. Thank you very much. Nov 30, 2023 at 19:25
• I do have one more question though and sorry if this is me being dumb. But I’m not sure what you mean by “both” Minkowski diagrams. Generally there’s just one. Do you mean a diagram taking the astronaut to be the frame which is draw with the typical coordinate system and another in which the ship is? Nov 30, 2023 at 19:33
• Yes. In general, if you want to really hammer down the confusions in Minkowski diagrams, it is often helpful to draw the same situation for both frames, whatever you consider as Lab frame, and the other frame that is moving. Otherwise, it is not always easy to understand what the slanted frame is seeing, say. Dec 1, 2023 at 4:09