Mordred

\[e^-+e^-\overbrace{\rightarrow }^{photon} e^-+e^-\] There is the Moeller scattering of two electrons fired at generating a scattering event of two electons. You will subsequently generate an off shell mediator photon. So you will be generating light, however assuming that you can generate sufficient mass density to have the mass below the Schwartzchild radius the photons would not escape. See Swansont's reply above However electrons will repel each other so simply firing them into a specific region will not be sufficient. I don't think a magnetic trap will suffice either as the EM field interactions will cause further scatterings and particle generation/ annihilation events. However lets assume you can generate a BH by some method of containment. The Hawking radiation will be fastest the smaller the BH. So the initial BH will want to decay extremely quickly however Hawking radiation only applies when the Blackbody temperature of the BH is higher than the Blackbody of the surrounding region... Not sure how that would work as your firing beams into the confined space.

As time is a property that describes rate of change you must have the requirement of some state, object, field, space (as volume) that changes. as long as you can measure something then time is applicable. Obviously measurement isn't a requirement for time to be involved either as things change even without being measured. However you must have something that changes even if that something is another property.

One thing I've always found annoying about how entanglement is described in numerous papers involving the various paradoxes and interpretations is that very few of the papers ever mention that the entangled particle pair had a causal past connection at the moment of entanglement. That causal connection also prepares the allowable states in accordance to the conservation laws. (charge,energy-momentum, flavor, color, isospin, lepton number ) etc. So lets examine that using the parametric down conversion of a monochrome light beam through the beam splitter common to the EPR experiment. As photons are used were dealing with polarity states (left and right circular polarity states). Ok so we don't know the which particle is which so naturally we have a superposition probability state that will be common to both wavefunctions as its fired to the detectors. (treat as one state shared by both) or treat as two identical states. (seems to me shouldn't matter which descriptive is used in this case. The probability is further increased by the detector alignment at detector A and B in so far as the angle is concerned. That gets factored into the correlation function. Now lets stop for a second and think about this. The state sent to each detector is a probability state. It is not the physical state. We do not know the physical state so we can only describe the probability state. Once you observe/measure (in QM measure is identical to observe), you have determined the physical state so naturally the probability state is no longer required. Applying the previous conservation laws the person at detector A will automatically know wheat state should be detected at detector B as being the opposite polarity. There is no cause or communication needed beyond the original preparedness of the entangled particle pair. The very act of generating an entangled particle pair in the first place requires a causal connection. (the two particles must interact in order to become entangled). the particles themselves do not communicate between each other nor change states as a result of the superposition wavefunction collapse. The communication that occurs is when you go to communicate the results of one detector to the other detector or for timing purposes. if for example observer at detector A does their measurement. detector B doesn't know the results until detector A informs them. Until detector B gets the results that detector will still treat the state as a superposition state. A probability state isn't a physical measured state. Any measured stated is a determined state. A key note you can have a superposition of waveforms that are physical (ie a collective of different frequencies in the same space) thats a bit different than the entangled superposition state which is a probability function.

I don't see how you would get a string loop to represent charge.... In string theory itself charge is represented by where the endpoints lie on the graph/ brane. However string theory doesn't apply a single string either. Quite frankly having a single string represent all of the periodic table would be an impossibility.

We have tested the weak equivalence principle on the moon and found it holds as \[m_i=m_g\] held on the moon test it follows that the time dilation aspects of GR will also hold

on that I fully agree with you, I enjoy a good scientific discussion its natural and part of the scientific process to examine different view points for validity etc. A good discussion should include alternate views for examination

I got rid of that neg rep you got above you didn't deserve it there even though rep is a trivial thing

ok no offense taken and I had no intention of giving offense so My apologies on that by the way I dont feel you deserve that neg rep point so I'm going to remove it.

my apologies but did you not correct how I verbally described the above ?

Sure we can focus if you like on language. Either way the flow of charge is a useful descriptive and it is part of the high school curriculum

charge isn't carried by only electrons with flow of charge you are describing the effective average of each individual atom in a conductor. The photons are the mediator of charge in interparticle interactions. In this case specifically atom to atom. This is effectively the same as your EM field descriptive. With flow of charge you are not specifying the flow of a specific particle but the flow of the effective charge at a given locale

I'm prepared to hear your argument why do you feel the flow of charge doesn't have a rate of c ?

yes you are correct I specified for of charge as that's the lanquage used in a lot of high school lessons. A lot of the high school physics textbooks also apply the flow of charge terminology and subsequently some of their exam questions will trip a student up if they don't realize the question applies flow of charge vs flow of electrons. ( there is one specific lightning question on flow direction for a specific charge that students will get wrong if they don't pay attention to the flow of charge specifically). lol at least in Canada flow of charge in high school is still taught today, as I often help volunteer as an assistant instructor to high school and undergraduate students

I agree the earth spins slowly enough that it wouldn't make any significant differences. For the first part if your applying the line elements already derived such as he Schwartzchild metric etc the coordinates vs proper time are already incorporated. You still have to be aware that those same line elements will return the proper time. Same goes for expressions for proper time already accounting for coordinate time The commonly used proper time to coordinate time expression is as follows \[\Delta \tau =\int \sqrt{1- \frac{1}{c^2} \left ( \left (\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2+ \left (\frac{dz}{dt}\right)^2\right )}dt\] which works fine until you have other factors such as frame dragging, rotation or acceleration. For rotation as above \[d\tau=\sqrt{1-\frac{\omega^2 r^2}{c^2}}dt\]

as the others mentioned there is a distinction between flow of electrons and flow of charge. As you noticed flow of electrons is extremely slow however flow of charge is near c.

The history of Eather theory is quite interesting. I have a copy of a 1918 physics textbook that described it. It didn't include anything involving relativity and the entire particle model only comprised of protons and electrons. Neutrons were discovered roughly 1935 if I recall.

Your work sheet looks good I don't see any problems or mistakes in that analysis well done.

It simply means there is always something new to learn and if your dedicated in your studies you never stop trying to learn new aspects of a given theory. from a preliminary quick research it seems to me applying the Carter Constant for the elliptical orbit may give me a methodology. I wonder if Markus or Jadus agrees with that approach https://en.wikipedia.org/wiki/Carter_constant there we go I think I have an applicable method https://en.wikipedia.org/wiki/Boyer–Lindquist_coordinates#Spin_connection edit speaking of challenges I wonder at the steps to get get proper time for an observer at infinity to a clock orbiting Earth with the subsequent transverse blue/redshifts.

lol its how I test myself for improvement. If I can help others also learn then bonus. One thing about any physics theory you never know enough

I I like a good challenge lol and as the Earth also orbits the observer at sea level has numerous interesting effects though most are negligible in the case of the earth I am curious how to handle situations not so trivial

hopefully the proper acceleration example I provided aids you both in this GL I'm looking into finding a method to handle the elliptical orbit to have an observer at sea level while an observer is in orbit. I have found a couple of Kerr metric examples in elliptical orbit still examining them to see if they will be useful. currently examining this one https://arxiv.org/pdf/0903.3684.pdf

Continuing from here for the orbiting Observer to Observer at CoM for simplicity. Once again will be using the Lewis Ryder reference in the previous post above. were going to set the orbital rotation along the z axis in cylindrical coordinates \[ds^2=-c^2dt^2+dr^2+rd\phi^2+dz^2\] \[\acute{t}=t, \acute{r}=r, \acute{\phi}=\phi-\omega t, \acute{z}=z\] \[\acute{ds}^2=-ct^2d\acute{t}^2+\acute{r}^2(d\acute{\phi}+\omega d\acute{t})^2+d\acute{z}^2\] \[=-(c-\omega^2\acute{r}^2)d\acute{t}^2+2\omega\acute{r}^2d\acute{\phi}^2 d\acute{t}^2+d\acute{r}^2+\acute{r}^2d\acute{\phi}^2+d\acute{z}2\] dropping the primes we have the invariant spacetime in a rotating frame \[ds^2=-(c^2\omega^2 r^2)dt^2+2\omega r^2d\phi^2 dt+dr^2+r^2d\phi^2+dz^2\] this becomes \[ds^2=g_{\mu\nu}dx^\mu d^\nu=g_{00}(dx^0)^2+g_{0i}dx^0dx^j+g_{ik}dx^i dx^k\] ijk sums over 1-3 where \[x^\mu=(x^0,x^1,x^2,x^3)=(ct,r,\phi,z)g_{\mu\nu}\] \[g_{\mu\nu}=\begin{pmatrix}-(1-\frac{\omega^2 r^2}{c^2}&0&\frac{\omega r^2}{c}&0\\0&1&0&0\\\frac{\omega r^2}{c}&0&r^2&0\\0&0&0&1\end{pmatrix}\] time interval between events \[ds^2=-c^2d\tau^2\] as world time not proper time relation between world time and proper time is given by \[d\tau^2\sqrt{-g_{00}}dt\] giving in the above case \[d\tau=\sqrt{1-\frac{\omega^2 r^2}{c^2}}dt\] lengthy process but hope that helps better understand the 3 clock scenario described by the OP... all the material as mentioned is from Lewis Ryder "Introductory to Relativity" Sagnac effect as shown above further detail here https://en.wikipedia.org/wiki/Sagnac_effect I'm also considering applying the Kerr metric to the rotating case to place observer at sea level instead of CoM have to think about that though

With the rotations and boosts above of the Lorentz tranformations we can now examine the different clocks under proper acceleration which I will detail below for each case. (I will need to set one observer at CoM instead of sea level for simplicity) four velocity \[u^\mu=\frac{dx^\mu}{d\tau}=(c\frac{dx}{d\tau},\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau})\] \[A=\frac{du}{d\tau}\] without going through all the relations a constant acceleration along the x direction with constant acceleration g gives the following \[c\frac{dt}{d\tau}=u^0,\frac{dx^1}{d\tau}=u^1,\frac{du^0}{d\tau}=a^0,\frac{du^1}{d\tau}=a^1\] \[a^\mu a_\nu=-(a^0)^2+(a^1)^2=g^2\] gives two solutions \[a^0=\frac{g}{c}u^1,,a^1=\frac{g}{c}u^0\] from which \[\frac{da^0}{d\tau}=\frac{g}{c}\frac{du^1}{d\tau}=\frac{g}{c}a^1=\frac{g^2}{c^2}u^0\] gives solution between observer A and B (A set at CoM, B falling observer) \[u^1=Ae^{g\tau/c}+Be^{g\tau/c}\] hence \[\frac{du^1}{d\tau}=\frac{g}{c}(Ae^{g\tau/c}-Be^{g\tau/c}\] hence \[x=\frac{c^2}{g}cosh(\frac{g\tau}{c}),, ct=\frac{ct}{g}sinh(\frac{g\tau}{c})\] space and time coordinates fall onto the hyperbola \[x^2-c^2t^2=\frac{c^4}{g^2}\] \[\frac{dx}{d\tau}=c \sinh(\frac{a\tau}{c})\] \[\tau=\frac{c}{a}\sinh^{-1}\] there is your proper time under constant acceleration for the falling clock Lewis Ryder pages Introductory to General Relativity pages 23 to 28 even thoug its from the twin paradox solution the Hyperbolic rotation is identical in the linear acceleration case https://en.wikipedia.org/wiki/Lorentz_transformation https://en.wikipedia.org/wiki/Hyperbolic_functions

Lorentz transformations list spherical coordinates (rotation along the z axis through an angle ) \[\theta\] \[(x^0,x^1,x^2,x^3)=(ct,r,\theta\\phi)\] \[(x_0,x_1,x_2,x_3)=(-ct,r,r^2,\theta,[r^2\sin^2\theta]\phi)\] \[\acute{x}=x\cos\theta+y\sin\theta,,,\acute{y}=-x\sin\theta+y \cos\theta\] \[\Lambda^\mu_\nu=\begin{pmatrix}1&0&0&0\\0&\cos\theta&\sin\theta&0\\0&\sin\theta&\cos\theta&0\\0&0&0&1\end{pmatrix}\] generator along z axis \[k_z=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}\] generator of boost along x axis:: \[k_x=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}=-i\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0 \end{pmatrix}\] boost along y axis\ \[k_y=-i\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0 \end{pmatrix}\] generator of boost along z direction \[k_z=-i\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0 \end{pmatrix}\] the above is the generator of boosts below is the generator of rotations. \[J_z=\frac{1\partial\Lambda}{i\partial\theta}|_{\theta=0}\] \[J_x=-i\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&-1&0 \end{pmatrix}\] \[J_y=-i\begin{pmatrix}0&0&0&0\\0&0&0&-1\\0&0&1&0\\0&0&0&0 \end{pmatrix}\] \[J_z=-i\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0 \end{pmatrix}\] there is the boosts and rotations we will need and they obey commutations \[[A,B]=AB-BA\]

hrrm seems to me we will need the rotating frame described by the Sagnac effect. This will take me a bit of time to get down so please be patient I'm going to apply the form given by Lewis Ryder from his textbook as its layout is one of the easiest to follow also going to have to break it down further for readers not familiar with the Lorentz transforms in the x, y and z direction. (its worth the additional effort as it is informative to all readers.) the dust solution I simply had handy as it describes an unaccelerated frame stress energy tensor treatment as mentioned it wont work in this case by itself but the details of how to fill the two tensors are included in differential form Lorentz transformations list spherical coordinates (rotation along the z axis through an angle ) \[\theta\] \[\acute{x}=x\cos\theta+y\sin\theta,,,\acute{y}=-x\sin\theta+y \cos\theta\]