As has been pointed out, there is no standard way to define how the proton can be "at rest in its frame" but my favorite way is to use the radar method which works identically to the standard Inertial Reference Frames (IRFs) and yet also works seamlessly with non inertial reference frames of the type you presented.From this long debate what i understood is only one thing.
The proton hits the earth AFTER THE DEATH OF ROBERT from earths rest frame. That is sure & the proton ages only a few days
But when you are considering the rest frame of proton, you are varying time of the earth inorder to solve the paradox.
How is that possible?
If the proton is at rest in it's frame and rest of the universe is moving, then how come the time on earth suddenly jumps into 2018 from 2014?
I need to know what kind of math/relativistic principle is used here?
You mentioned in your OP that:
It's important that both planets send signals in order for the proton to determine the correct synchronization. So let's see how this works. Let's assume that Planet X and the proton at some point in time don't know the time on earth. They send a message at the speed of light to the earth requesting them to immediately send a message back with the current time on the earth. The proton (and Planet X) start their stopwatch when the message was sent to earth and stop their stopwatch when they receive the response from earth. Then they take half of the measured time and assume that earth sent the response at that time and they go back and re-assign that time to their own time at the half-way point. They also assume that earth was that same number of light-years away at the time it sent the response. Here is a spacetime diagram to show how they do this using the same scenario (0.97c and 4 ly) as I did before:Both the planets are synchronised in time by sending light pulses so that they can keep 2013 on both planets.
Note that the proton sent its request when its stopwatch read 0 and it receives the response from earth when its stopwatch read 8 years so it assumes that earth received its message and sent its response when its stopwatch was at 4 years and since the message from earth said they were at the year 2008, the proton assigns 2008 to the point where its stopwatch was at 4 years. The proton also assigns all the other years before and after accordingly.
In the meantime, the proton has sent another message one year after the first one and similarly confirms that earth's year 2009 matches its year 2009 (since it's halfway between 2005 and 2013). Likewise, the proton concludes that the earth remains 4 light-years away:
The proton continues to send a new request to the earth every year and continues to keep track of sent/received times so that it can construct a non IRF diagram:
Here is a complete list of the information the proton has gathered:
Message Response Calculated Calculated Earth's Sent Received Time Distance Time 2004 2012 2008 4 2008 2005 2013 2009 4 2009 2006 2014 2010 4 2010 2007 2014.123 2010.56 3.56 2011 2008 2014.247 2011.12 3.12 2012 2009 2014.370 2011.68 2.68 2013 2010 2014.493 2012.25 2.25 2014 2011 2014.616 2012.81 1.81 2015 2012 2014.740 2013.37 1.37 2016 2013 2014.863 2013.93 0.93 2017 2014 2014.986 2014.49 0.49 2018
And just as a sanity check, we can show how all the signals that the proton sent and the responses that the earth sent are just as valid in this non-inertial frame as they are in the inertial frame:
As you can see, there's no jumping of any times (clocks) and no times (clocks) going backwards. The math is simple, averaging two measurements and dividing the difference by two. The relativistic principle is that the proton assumes that the signals that it sends to the earth take the same amount of time as the responses the earth sends back. Very simple, isn't it?
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