I thought up a variation of the Twin Paradox, which I call “A One-way Trip to Proxima Centauri.” Is it a Real or just a Seeming Paradox?
Summary: As Gracie gets very close to Proxima Centauri, a radio message from Earth tells her one thing, but her super-telescope tells her another.
Let’s start with some numbers, which I got from Professor Greene’s lecture: Proxima Centauri is 4.25 light years from Earth. A rocket ship heading there at 80 percent of the Speed of Light would take 5.31 years to get there (according to a clock on Earth). But the rocket ship’s clock, upon arrival at Proxima Centauri will claim that only 3.19 years have gone by.
Picture Planet Earth near the left side of your page or screen, and picture Proxima Centauri on the right side. Gracie starts her journey in a rocket ship far to the left of Earth (to use scientific terminology) and she has accelerated to a final and constant 80 percent of the speed of light as she passes Earth, heading toward Proxima Centauri on the right.
I had Gracie get a running start for a reason — to avoid the complication of acceleration on the “important” part of her trip. To be clear: she does not accelerate on the part of her trip from Earth to Proxima Centauri, and we are not concerned with any possible return trip.
As she passes near Earth, she sends this message to George: “I am setting my Rocket-Clock to Zero.” He sends her a simultaneous message: “I am setting my Earth-Clock to Zero.”
Throughout her trip, George looks through his super-telescope and he notices that Gracie’s Rocket-Clock is moving slowly. Why? Because she is traveling at a very high velocity, and Special Relativity says this causes time dilation.
However, when Gracie looks back at Earth with her super-telescope, she notices that the Earth-Clock is moving slowly. Why? Because from her inertial Frame of Reference, she is standing still and the Earth is traveling at a very high velocity away from her.
After approximately one year and 22 days on Earth (or 1.06 years), George sends this radio message to Gracie: “My Earth-Clock now reads one year and 22 days.” That message takes 4.25 years to get to Proxima Centauri, and arrives there around the same time Gracie arrives there.
She does some math – adds 1.06 years (on the Earth-Clock) to 4.25 years (the radio signal’s travel time at the speed of light) and gets 5.31 years. She figures that is how much time has passed on Earth since she and George synchronized their clocks.
But now she is confused. She looks in her super-telescope and it tells her that Earth-Clock is NOT reading 1.06 years – it shows considerably less time (possibly .6 years ?). I assume that light reaches her telescope at the same speed as the radio waves reach her radio receiver.
Her super-telescope has told her continuously that the Earth-Clock was moving slowly compared to her Rocket-Clock. But the radio message from George tells her that the Earth-Clock did NOT slow down. To me, that is a real paradox, not a seeming paradox.
When she receives George’s radio message, she looks at her Rocket-Clock and sees that, because of Special Relativity’s time dilation, only 3.19 years have passed on her clock. HER clock slowed down, not the one on Earth. She re-checks her math: 1.06 + 4.25 = 5.31 years.
Gracie sends this message back to Earth: “My Rocket-Clock reads 3.19 years.” George gets that message 4.25 years later. At that time, his Earth-Clock reads 9.56 years (5.31 + 4.25); but when he adds her “3.19” to 4.25, he gets only 7.44 years.
It seems to me that only one Frame of Reference is correct: George’s. That’s because Gracie is the one who is really moving.
So, what happened to the notion that all inertial Frames of Reference are equal? I think Einstein said something like: “There is no experiment that can be performed to determine whether one is at rest or moving with constant velocity.” Isn’t the radio message just such an experiment? Or am I, a non-scientist, missing something?
Related to that, Professor Greene stated: “The only time that you can claim to be at rest and the rest of the world is moving by you is if you are going at constant velocity, constant speed in a fixed direction.” Well, Gracie is going at a constant velocity.
I have a question for Gracie: “When you rendezvoused with Proxima Centauri, and you looked at your Rocket-Clock, it told you that only 3.19 years have passed in your space ship. But Proxima Centauri is 4.25 light years from Earth. How do you explain that? If you were standing still, and Proxima Centauri was moving towards you, was it moving at a speed faster than the speed of light? That’s a no-no.
And, hanging over this whole trip is another nagging question: how could she not know that she is the one who is moving? I mean, really. She’s the one in a rocket ship!