Lorentz Jr

That sounds too technical to me. I've always assumed he meant "I have subjective experiences, therefore I exist." Maybe he said "think" to distinguish human beings from animals, but I would generalize what he said. The "I" he had in mind (no pun intended) would be the conscious entity making the claim, doing the thinking, inferring its/his own existence, and having whatever other perceptual experiences it/he has.

Yep. You can't be faulted for feeling that way. In physics, there's a longstanding debate about whether there's such a thing as "absolute space" or whether space is only relative. The relativists have dominated for the last century or so, but not all theorists are convinced. Beyond physics, there's the dominance of social issues in most people's lives. It's said that we only use 10-20 percent of our brains, but I think what's really going on is that we're using the other 80-90 percent unconsciously, for dealing with intuitive issues in the social world. So physicists tell you we live in a physical universe (or at least they used to!), but most people's daily lives are dominated by subjective issues of politics (both public politics and "micropolitics" with our friends, family, neighbors, and coworkers), religion, and other issues that have no obvious connection to physics. It feels like we live in two completely different worlds, and it's hard to say what is the best way of dealing with them.

Instantaneous acceleration is a solution, not a problem. I take the problem as wanting to accelerate the train quickly and the idea of instantaneous acceleration as the OP's attempt to solve it. Or, if you take the problem more specifically to be wanting to accelerate the cars instantaneously, then yes, I'm saying that's obviously impossible, and I'm showing why it's impossible and providing a solution to a more realistic problem. As soon as a car accelerates, it would crash into the one ahead of it if that one isn't also accelerating. I'm not sure what "apply Born rigidity" means. It's defined mathematically, but it can't be maintained by objects that are made of ordinary materials and being forcefully pulled from ahead or pushed from behind. I only mentioned it because eliminating stresses between the cars is mathematically equivalent, at least if the cars aren't stressed too much internally by their propulsion systems. That wouldn't work. At t=0, the cars are all stationary, so the formula I posted is the proper acceleration of each car at that moment, and the formula shows that they have to be different from each other. And then, as the train approaches its target speed, the front cars have to keep accelerating while the back cars are already up to speed. That's an interesting suggestion though. I suppose it might turn out to be approximately true for some parts of the train's trajectory when the accelerations are all converted to each car's reference frame. I couldn't say for sure though. I seriously doubt there's an exact analytic solution to the problem. I'm still not sure what you mean. A "solution for Born rigidity" is only defined mathematically. It's physically impossible for ordinary materials that are subjected to high accelerations by external sources. The point is that a solution that works for individual cars doesn't work for the train as a whole. Compression in the front part of the car can compensate for stretching in the back part, and the car is short enough that those effects occur almost instantaneously. So the proper length of the car can stay relatively constant without too much trouble, but there's no way that can happen to a train with only engines at the ends.

This should be [math]\displaystyle{-\frac{d\nu}{dn} = k_1 \nu^2}[/math]. And a reminder that this isn't the whole solution. It's only the beginning, when the train is just getting started.

Ha! I was typing "\displaymath" instead of "\displaystyle". 🙄

Well, I don't have anything better to do right now, so let's work out these results. We're trying to quickly accelerate the train to relativistic speeds without the cars crashing into each other or the couplings between them breaking apart. [######]-[######]-[######]-[######] --> The condition is that there's little or no stress on the couplings between the cars, so we're effectively simulating Born rigidity. If the propulsion for each car is near the middle of the car, maybe compression in the front half will roughly cancel out stretching in the back half. So, defining [math]L_0[/math] as the uncontracted length of each car, the contracted length should be pretty close to the special-relativistic value [math]L = L_0/\gamma[/math]. To keep the equations as uncluttered as possible, we'll pretend that [math]L_0[/math] and c are equal to 1. In other words, we'll do the problem using [math]L_0[/math] as our unit of distance and [math]L_0/c[/math] as our unit of time. These are the "natural" units for this problem. The units in the final answer will be wrong, but that's easy enough to fix, because there's only one way to fix them using [math]L_0[/math] and c. So, with these conventions, the contracted length of each car is [math]L = 1/\gamma = \sqrt{1-v^2} [/math] And the rate at which each car's length contracts is the time derivative of this formula, which is equal to the difference between the speed of the coupling in front of the car and the speed of the one in back of it: [math]\displaystyle{\frac{dL}{dt}= \frac{1}{2} \left( \frac{-2va}{\sqrt{1-v^2}}\right) = −\gamma va = v_{n+1}−v_n }[/math] The tapering off of the acceleration along the length of the train is the difference between the speed of one car and the speed of the one in front of it. This will be pretty close to the difference between the speeds of the couplings, as long as the acceleration isn't too extreme, so we can calculate that: [math]\displaystyle{\frac{\partial v}{\partial n} = \frac{(v_{n+1}−v_n)}{1}=−\gamma va}[/math] We can also ignore the [math]\gamma[/math] term, because it only becomes important at higher speeds (the approximation will be linear in v). [math]\gamma = 1 + O(v^2) [/math] [math]\displaystyle{\frac{\partial v}{\partial n} = -v \frac{\partial v}{\partial t}}[/math] Then we'll try to separate n and v by taking a wild guess that maybe we can express the speed [math]v(n,t)[/math] as the product of two simpler functions, [math]\nu(n)[/math] and [math]\tau(t)[/math]: [math]v(n,t) \equiv \nu(n)\tau(t) [/math] And then we'll plug & chug through the math. One side of the separated equation is independent of t, and the other side is independent of n, so they must both be constant. [math]\nu'\tau = -(\nu\tau)(\tau'\nu) = -\nu^2 \tau'\tau[/math] [math]\displaystyle{- \frac{\nu'}{\nu^2} = \tau' = k_1} [/math] [math]\tau = k_1 t + k_2 [/math] [math]\tau(0) = 0[/math], so [math]k_2 = 0[/math]. [math]\displaystyle{-\frac{d\nu}{dn} = k_1 w^2}[/math] [math]\displaystyle{- \frac{d\nu}{\nu^2} = k_1 dn}[/math] Now integrate both sides of the equation, [math] \displaystyle{\frac{1}{\nu} = k_1(n + k_3)} [/math] and plug [math]\nu[/math] and [math]\tau[/math] back into the definition of v: [math] \displaystyle{v = \nu\tau = \frac{k_1 t}{k_1 (n+k_3)} = \frac{t}{n+k_3}} [/math] Now define [math]a_0[/math] as the (constant) acceleration of the last car. Then [math]\displaystyle{ v_0 \equiv a_0 t = \frac{t}{0+k_3}} [/math], so [math] k_3 = 1/a_0 [/math]. And now we have the final solution for the speed, [math]\displaystyle{v(n,t)=\frac{a_0 t}{1 + a_0 n}} [/math] except we need to fix the units. The problem is the [math] a_0 [/math], which is an acceleration, in the denominator, so the way to fix it is to divide it by [math] c^2/L_0 [/math], which is also an acceleration: [math]\displaystyle{v=\frac{a_0 t}{1+\frac{a_0 L_0 n}{c^2}}} [/math] And differentiate with respect to t to get the acceleration: [math] \displaystyle{a=\frac{a_0}{1+\frac{a_0 L_0 n}{c^2}}} [/math]

You say the dumbest things sometimes, Joe. If two light beams leave Earth and the sun at the same time, observers on the sun and Earth will see those beams at the same time. That's what you said. But it's obvious, and it has nothing to do with the transit time, except that it's the same (except for a tiny gravitational correction) in both directions.

No offense, buddy, but you're talking about yourself. 😉

It is. Incompatibility between p and q doesn't imply anything about r or s. If silence is golden, then 1 + 1 = 2. If silence is not golden, then 2 + 2 = 4. In formal logic, disjunctions are always inclusive. Exclusive disjunctions may be written as "xor". The only ambiguity is in informal conversation.

8.3 😋

So your theory is that mass has been disappearing from galaxies all this time? I don't think there's any variation in sizes or rotational speeds of galaxies, so that would be a counterargument. If mass had been "evaporating" from galaxies, younger ones (farther away) would have been larger or rotating faster than older ones (nearer) are now, and that hasn't been observed. People occasionally hypothesize that maybe current theory inverts what's really happening, so things like the speed of light and the gravitational "constant" have been changing instead of space expanding. But that's not mainstream theory, and no one has presented any compelling arguments for preferring those ideas, so you should post questions like this in the Speculations area.

There are also Big Bounce models, where the universe was large before 14 billion years ago, undergoing some kind of Big Crunch, and some as-yet-unknown (elastic) source of internal support prevented it from collapsing any further. I believe it provides an explanation for the uniformity of the universe's current state without requiring an inflationary phase, because the bounce prevented the quantum chaos from getting too extreme in the first place. There's also Roger Penrose's Conformal cyclic cosmology, where each "aeon" of the universe somehow emerges from what was perceived as an infinite (or very long) amount of time in the previous aeon. The main fact is that red-shifting of light from distant galaxies clearly suggests expansion of ... something ... regardless of whether it's "space itself" or whether there's some other explanation. The idea that other galaxies are red-shifted because they're at lower gravitational potentials than the Milky Way implies that our perspective is special. That's considered an unscientific hypothesis, because we haven't seen any evidence that the Milky is fundamentally different from other galaxies.

Here's a question for you. It's totally unscientific, no basis in anything at all, but I find it interesting: If you compare c (the speed of light) to speeds of ordinary human activities here on Earth, it seems very fast. But if you divide cosmological distance scales by typical human time scales, c is agonizingly slow. So relativity offers no hope at all for human beings to explore any significant portion of the visible universe in any amount of time that we would currently consider "reasonable" for us. Doesn't that seem silly? Doesn't it seem rather pointless to have a universe that we can see but we can never explore? I like to think that there may be a way to travel, or at least communicate, at superluminal speeds. It seems absurd if you believe in relativity, but Hendrik Lorentz's own interpretation of his transformation equations hasn't been scientifically disproved. It's not as rigid as relativity is about forbidding FTL phenomena, and I would imagine that things like objective interpretations of quantum entanglement and wavefunction collapse might contain clues for someday developing superluminal technologies.

I like the way it is. A little squinting is okay. 😊

Here's another one: Albert Einstein - Leiden Lecture I wonder if he gave this talk in German. If he did, maybe I read a different translation some time ago that had "atoms" instead of "parts". I guess "atoms" would be confusing if people thought he meant atoms of matter.

[math]\displaystyle{v=\frac{a_0 t}{1+\frac{a_0 Ln}{c^2}}}[/math] Hmm..... I thought I got an "unknown environment" error when I did that before. Thanks, that's perfect.

[math]v = \frac{a_0 t}{1 + \frac{a_0 L n}{c^2}}[/math] Letters too small, eqn too tall. I have at least 15 lines of math, plus explanatory text. Don't want to make too long of a post or make people squint too much.

Trump knows his audience. Supporters have been slowly getting picked off by RDS, maybe a few just getting tired of losing, some still with him in spirit but waiting for more win before getting involved again, or broke and out of work, or sick and broke from Covid-related medical bills. So the only ones left to send him their life's savings are those who have been picturing him like this in their minds all along. The "crème de la crème", so to speak, in whatever sense is relevant to members of his movement, which is beyond my powers of imagination.

But those are completely different meanings. Electrical grounding means providing a conducting pathway for charges to flow through. Grounding airplanes means a human being orders the pilots to land and/or not take off. You need to figure out what your questions are and explain what you mean by the terms in them before criticizing how people answer them. And this is a good place to start. How do you define "grounded" and "the grounding process"? Citing meanings in electrostatics and aviation is worthless in discussions about gravity and bulk matter.

It's up to the repliers to present the facts to you. If you're unable to understand them or unwilling to accept them, that's your business.

That's a good question, I don't know. You'd think the US would know about Iranian drones and develop something to defend against them. Small drones in general. It shouldn't have to be that exact model. Ukraine really needs something though. They needed it at least a month ago.

I hear it's a 1700-unit storm.

"If they are adding charge , they aren't measuring the elementary charge" is a lecture, not a question.

To me, lecturing people about things you don't understand is cheating.

Maybe this thread should be moved to the sandbox.