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I’ve also pointed this out to him already on his “Norm of 4-Acceleration” thread, and gave the explicit expressions for the necessary transforms of x and ct, but he ignored it completely. In fact, so far he has ignored pretty much everything that has been pointed out to him - it doesn’t seem like there is any genuine openness to feedback. It feels more like a personal blog to me. Just for the casual reader’s reference - in the special case of motion starting from rest at the origin, with uniform acceleration, the integral becomes straightforward, and the expressions evaluate to: \[v( \tau ) =c \tanh\left(\frac{a\tau }{c}\right)\] \[v(t) =\frac{at}{\sqrt{1+\left(\frac{at}{c}\right)^{2}}}\] Which is, of course, hyperbolic motion, as we would expect in a Minkowski spacetime.

I was reading through Mordred's long standing thread on space and I came across some posts from Mike in the early part. Members may wish to know that Mike passed away earlier this month after a long standing degenerative illness (not covid). Mike was an interesting character, an artist with a degree in Physics and the founder of a successful manufacturing business before retirement to Cornwall. His artistic (dreamy) side gave him an unusual and sometimes frustrating perspective on Physics, especially later in life when we knew him. But he was a genuine character and sometimes offered suprising insights as well.

Well and there are reports that Trump might have been groomed as a Russian asset since the 80s. https://www.businessinsider.com/russia-cultivating-trump-asset-40-years-says-ex-kgb-spy-2021-1

I read they are worried about kickback for "disloyalty". It seems they are in a 'mob rules' situation. I've a feeling they are going to paint themselves into a corner.

To add to the panoply of excellent comments by Kino and Markus, you cannot expect an arbitrary linear combination of 4-vectors to be a physically significant 4-vector. Both vectors must be timelike \( \left(u_{\left(i\right)}^{0}\right)^{2}-\boldsymbol{u}_{\left(i\right)}\cdot\boldsymbol{u}_{\left(i\right)}\geq0 \) and orthochronous \( u_{\left(1\right)}^{0},\:u_{\left(2\right)}^{0}>0 \). Also, the resulting 4-vector must be normalised to \( c^2 \). In that sense, when you're working with 4-velocities, you're not working on a plain linear space --Minkowski space--, but in some kind of "unitary quotient of it." This distinction is referred-to in physics by means of the buzzwords "on-shell" and "off-shell." Adding vectors off-shell can lead you to vectors on-shell, and vice versa. This point has arisen before --Ghideon has been particularly persistent. Off the top of my head, you can derive a common (CoM) 4-velocity for 2 material particles moving every which way by calculating the common 4-momentum, and then dividing by \( m_1+m_2 \) --which are relativistic invariants. Another thing you could do is calculate the centre-of-energies motion and then impose that it be normalised as to become a physical 4-vector. One last thing you could do is use Einstein's addition of velocities --which doesn't involve the masses--, to obtain a physical 4-vector, by multiplying by the appropriate observer-dependent factor as to obtain a 4-vector. I don't know. I'm just trying to help you so that your effort is not in vain. So far, it is in vain, simply because you're not distinguishing with any care what's on-shell and what's off-shell.

Yes, and it also doesn’t imply that Riemann is always identically zero for all manifolds. Which it evidently isn’t.

Well, in a pure mathematics context, the metric has a pretty abstract definition in terms of something called the “first fundamental form” - which essentially boils down to defining the notion of an “inner product” on your manifold. Physically, this means that a metric enables you to quantify lengths, areas, volumes, angles etc - so it makes it possible to relate physical laws to real-world measurement outcomes. More specifically in GR, it is used to quantify the separation between events in spacetime. Given a connection, this determines all the various curvature tensors etc. I must reiterate that it is not my intention to deny that some coordinate system needs to be chosen in order to actually perform the calculation - however, the point is that the physics aren’t in the coordinate system itself, they are in structure of the metric, i.e. in how the components of the metric tensor (in a given coordinate basis) are related to one another. This is an invariant property, unlike the value of the components themselves. I meant it in this sense. Indeed, that’s pretty much what I am attempting to point out, with the addition that in the context of GR, the metric is the more fundamental object. Indeed.

Correcting myself here: \(m\) can be anything it likes because it does exactly nothing. However if you insist that \(k^2\bar c^2=c^2\), then noting that \(\bar c^2\) is negative since \(a\) is spacelike, then \(k\) is imaginary.

Can have a long life yes, if properly manufactured and installed. They said this about early transistors but then came the 'tin whisker' effect due to migration of metal from the leads into the semiconductor material and disabling the transistor. https://nepp.nasa.gov/whisker/anecdote/af114-transistor/index.html I said maunfactured and installed because this effect can also effect inappropriately chosen circuit board materials, solder, leads and other components as John has noted. We do not yet know if another unforseen effect will rear up and bite us in the ass in years to come. I have never seen a long life compact flourescent that achieved more than half the promoted service life bulb for instance.

This is where my head was UNTIL I read that previous post of yours. “Nah, they were just there like at a rally... sure, they doubt the election results, but whatever. They’re just regular old Tom, Dick, and Sally’s out holding signs and taking photos.” But you made a point that gave me pause... that made me view things differently. Majority of them DID want to overturn a free and fair election, to throw away votes they didn’t like, and circumvent the process. That in itself is a type of sedition. It’s a desire to overthrow our democratic principles and our republic. Sure, everyone exists along a spectrum and some were more aggressive/extreme than others who were meek and mild, but NONE of them were there supporting our legal processes or allowing the justice system to sort through the claims of fraud. The sincerity of their beliefs isn’t relevant. They felt they knew better than those in charge and were acting like vigilantes who desired to replace the actual election results with their own personally preferred winner. In short, they were trying to overthrow our government. That’s a well made point I can’t simply ignore or dismiss. Dude, I’m an experienced wood worker now. I wear safety glasses. No splinters in these eyes. Matthew 13:13 Therefore I speak to them in parables: because they seeing see not; and hearing they hear not, neither do they understand

It sounds to me that it might be about folks who profess a superficial love for science or anything sounding "sciency" (or like science ficton-y) but do not have actual interests in that area as such. Essentially folks that assume that liking science memes being the same as liking science. I also disagree that one needs to have a higher degree to get into science. There are many hobbies that provide in-depth knowledge about certain parts of that natural words, including e.g. bird watching or wildlife photography. It is more about to what depth you involve yourself into it. Fundamentally anyone running an aquarium or doing birdwatching is learning and applying more science that self-professed geeks who want to use Crispr to make superhumans.

Yes, they are; I didn't mention the height of the step. 😉