The Missing Ingredient People have been pushing back on my assertion that the rest frame for a Lorentz transformation cannot be dealer’s choice. It boggled my mind for the longest time because it seemed perfectly obvious to me. Why did Hafele and Keating start with a non-rotating Earth for example? Because that context has no angular momentum. The cosmic rest frame has no net linear momentum so the same logic should apply. The universe has no net momentum so the cosmic rest frame is the best context for distinguishing between hitting the gas and hitting the brakes. But momentum doesn’t factor into the equations. Time dilation is agnostic to the direction of motion and proper time intervals are invariant so who cares about the cosmic rest frame? We can transform events into the cosmic rest frame before calculating proper time intervals: Elapsed cosmic time from launch to reunion is Tb = t / sqrt(1- Vb ^2/c^2) Elapsed traveller time from launch to waypoint is To = ( Tb /2 ) * sqrt(1
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Derive the Lorentz Transformation The fastest way is to generalize Galilean relativity like this: X = A(x - vt) T = B(t - Cvx) where the constants A, B, and C are to be determined. The reverse transformation is: x = (X/A + vT/B) / (1 - Cv^2) t = (T/B + CvX/A) / (1 - Cv^2) These equations have to work if x is swapped with X and t is swapped with T so: A = B = 1 / sqrt(1-Cv^2) The constant C puts an upper limit on the velocity v so it should be expressed in terms of a speed limit: C = 1 / c^2 The value of ‘c’ is pretty much arbitrary because it’s really just a unit conversion factor. Any non-zero real number will do the trick. You could set it equal to one with no loss of generality. You just have to measure space and time in the same units. The last step, which was Einstein’s deep insight, is to notice that the same is true of Maxwell’s equations. i.e. The phase speed of an electromagnetic wave in the absence of any external fields is also unity if you measure space and time in the same
The Brass Tacks of the Ehrenfest Paradox
Coordinate Transforms for Rotating Reference Frames Back Story Wikipedia explains the conundrum in the Ehrenfest paradox thusly. Imagine a disk of radius R rotating with constant angular velocity . The reference frame is fixed to the stationary centre of the disk. Then the magnitude of the relative velocity of any point in the circumference of the disk is . So the circumference will undergo Lorentz contraction by a factor of . However, since the radius is perpendicular to the direction of motion, it will not undergo any contraction. So {\displaystyle {\frac {\mathrm {circumference} }{\mathrm {diameter} }}={\frac {2\pi R{\sqrt {1-(\omega R)^{2}/c^{2}}}}{2R}}=\pi {\sqrt {1-(\omega R)^{2}/c^{2}}}.} This is paradoxical, since in accordance with Euclidean geometry , it should be exactly equal to π . (Image credit: By Geek3 - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=24038062 ) Conventional Wisdom I won't bore you with the details, but Wikipedia goe
The Brass Tacks of the Spaceship Paradox
Does the Thread Break? Back Story Bell's Spaceship Paradox is a lesson in formation flying at near light speed. Here's the challenge: Point two spaceships in the same direction and line them up, one behind the other Tether them with a delicate thread and pull it taut Try to get the whole assembly up to speed without breaking the thread On the one hand, the entire assembly is subject to Lorentz contraction so everything, including the thread, should survive. On the other hand, each component of the assembly is subject to Lorentz contraction so the thread should break as the spaceships shrink. It's a conundrum. Here are the worldlines for the ends of the thread from the two perspectives of interest: Rest Length Note that Lorentz contraction is an unfortunate moniker because it goes both ways. Objects perceive themselves to grow longer as they pick up speed and shrink as they slow down. When physicists talk about "rest length", they are referring to perceived lengt
The Brass Tacks of the Ladder Paradox
Lorentz Contraction and Perceptions of Simultaneity Back Story In the Ladder Paradox (aka. Pole-Barn Paradox), we are given a ladder (or pole), which is too long to fit inside our barn when both are at rest. A smart aleck physicist, who knows a thing or two about Lorentz contraction, claims he can make it fit (if only for an instant) by throwing it through the barn at a sufficiently high speed. We presume the ladder is perfectly rigid so it won't be contorted by the force of acceleration. That would be cheating. In fact, let's put rockets on both ends just to be sure. Here we go: Hmmm. What went wrong? We followed the recipe, but the ladder didn't shrink. In fact, just the opposite. In its moving reference frame, the ladder actually thinks it has gotten longer (by a factor of γ .) Our smart aleck friend is undaunted though because, he says, the rest length of the ladder must be measured from the moving frame of reference. Its prelaunch length is therefore shorter so it wi
The Brass Tacks of Black Holes
Twin Paradox with Spacetime Curvature None Shall Pass Here's what the Twin Paradox looks like in the vicinity of a Schwarzschild black hole if both twins go walkabout on radial trajectories with equal and opposite velocities. Ages are quoted in terms of bystander time, not observer time (aka. coordinate time.) Note that even the light ray, which emanates from the origin, diverges off towards the end of time without ever reaching the horizon. Escape Plan The takeaway is, anything can escape from the grips of a black hole if you are willing to wait long enough. It's just that the longer an object spends on an inbound trajectory, the longer it takes to emerge after it turns around. However, a black hole is bad news if your personal integrity relies on travellers (or QM wave functions) reuniting at the launch pad at the same time. Terminology As an aside, I was once chastised by a GR expert for describing a black hole as a region in space with an unusually high concentration of mas
The Brass Tacks of the Andromeda Paradox
The Subjective "Now" Interpretation of Special Relativity Back Story In the Andromeda paradox (aka. Rietdijk–Putnam argument), we are given to understand that there's a certain faction of hostile actors in the Andromeda galaxy who want to invade the Earth. They have presented their case to their local council and the matter has been put to a vote. An astronomer on Earth observes the verdict through a telescope and, after correcting for the propagation delay, works out the time at which our fate was sealed. A passerby sees the same image at the same time, but having a skewed perception of simultaneity, arrives at a completely different conclusion. So when exactly did the verdict make the transition from the uncertain future to the certain past? i.e. When is "now" over there? Conventional Wisdom Conventional wisdom says the question is invalid. What you see is what you get. All perceptions of "now" over there are real because, unlike regular spacetime ev