Yes, while that was key, it is worth emphasising. It is of course only possible for the proper times to be different in the two diagrams because they start at different points in the lives of the twins (or they are different pairs of twins).
You rightly draw attention to something that I should...
Not quite. A new theory must give predictions (what you meant by "results"?) that are compatible with the existing empirical data to within experimental uncertainty. These predictions way well be slightly different to those of an existing theory that is also compatible (to within experimental...
I don't have any problem agreeing with your unambiguous statements.
"their starting ages are frame dependent"
"there is always of which was born first according to any given pair of synchronized clocks at the events"
" different pairs of synchronized clocks at the births disagree on their...
Yes.
But if they are essentially stationary relative to each other there is an objective answer to the question of which was born first according to either of the clocks they carry with them (because these clocks run at the same speed and can be synchronised at any time in a unique, consistent...
There is an objective answer to the question of whether A or B is younger (age in proper time) when they meet. Thus, with all due respect, this answer simply cannot be frame dependent.
With the planets having relative velocity zero and with the watches being synchronised (by say one twin sending...
Yes, I made my post more precise as you were posting! The reason for referencing the rider's anatomy was that back somewhere in this discussion we were discussing a claim that a rider had to rest their head on something.
The tilt of the rider's head in your penultimate photo makes some of this...
You agree the 2g ( (1+k)*g in my post), as measured by an accelerometer on the rider say, is near enough along your blue line (presuming that goes through the centre of gravity).
Do you accept the point that if you pick your frame moving with the bike, so that the plane through the points of contact and the centre of gravity is considered as "vertical", then in this frame, the artificial centrifugal force (which results from the fact that the frame is rotating) and the...
Yes, but you can combine gravity and centrifugal force (the fictitious force in the non-intertial frame of reference is the relevant one), as I did earlier. A consequence is that the angle of the road changes.
For a bike, this force is along the line between the tyre's contact with the road and...
Short answer: not correctly! Long answer leaves me skeptical for a different reason.
For a car, in the frame where the road is horizontal, you can think of the component of the centrifugal force that is perpendicular to the road as being an increment k * g to gravity so g is replaced by g * (1 +...
This cannot be strictly true. The sideways component of the force is of the order of 1.5 g, because it is limited by the friction of the tyres on the road. See https://www.f1technical.net/forum/viewtopic.php?t=20627 for data for F1. The support would make it a lot more comfortable and less...
If the net mass is positive, all the masses (both negative and positive) would gravitate towards the centre of mass, at which point you can imagine the (positive) total net mass existing. While this is an imprecise description, I think it might clarify the issue.
Well, both particles accelerate in the direction of the positive mass particle but, if the masses are equal, their relative velocity remains constant. So, given your description, they get further apart. The distance between them is thus O(time) and the interaction forces and the accelerations...
No, I was referring to the inverse square law of gravitation which determines the acceleration of the each of the two particles just like for positive masses, except both accelerations are in the same direction. This law gets very accurate as two objects move apart in general relativity (it's...
Note that my program had used the same procedure and come to the same conclusion except it is inaccurate to say there is a "runaway" solution when there is any initial relative velocity. By contrast, the velocity (in any inertial frame) of each of the particles is asymptotic to a constant value...
There is no doubt about the answer to your question which may have been motivated by a little confusion between the relative velocity of the particles and the velocity of one of the particles relative to a Galilean/inertial frame.
Whatever the initial relative velocity of two identical particles...
You can do the calculation for a small δ. If it is very similar for sufficiently small δ, all is fine. In fact what you find is the finite force determined by the product of the two transformed masses acts on the reduced mass, which grows to infinity as δ shrinks. The conclusion is that the...
Glad we can finally agree on that!
We are not obliged to use the same frame all the time: different ones are useful for different things. I don't see any flaw in my reasoning that is due to switching frames. The centre of the two particles accelerates with respect to an observer in a...
Good job! I have now corrected the typo. This effectively made the gravitational constant time-dependent without changing the sign of any interactions. I have verified that the main conclusion about the way the relative velocity fails to change over time stands.
I agree with you about the potential energy (I was editing my last post when you were replying). My deleted reasoning had one sign flip too few! Or maybe it was too many.
However, negative masses tend to accelerate to where they have more potential energy according to Newton's laws, so it would...
When the two particles are moving at velocity v, the kinetic energy of the first is 0.5mv2 and the kinetic energy of the second is -0.5mv2.
I imagine you see the answer to your question now. ;)
[None of this implies I am convinced that negative masses exist, but this is what an extrapolation of...
Firstly, with two body system with total mass zero, the centre of the system does exhibit the weird acceleration given Newtonian dynamics, because the two forces are in opposite directions, but F=ma makes both accelerations in the same direction. The centre of the particles has the average of...
I thought I should make it a little more user friendly first. Python with standard numpy and matplotlib libraries only. It was written in a jupyter notebook, so should work nicely if pasted into one. The graphs show the behaviour quite nicely when there is an initial relative velocity. The...
A short light discussion of some more anti-intuitive (but consistent) behaviour of negative masses. The observation that if you were able to push a negative mass like a normal mass, this would cause acceleration of the mass towards you is bizarre. However, the notion of pushing involves contact...
google is your friend, kent. ;) Sabine Hossenfelder is a German researcher in quantum gravity.
Note that Sabine made what I identified as an slip in her blog post. It was about whether negative masses repel each other, where she claimed that Franes had got it wrong (but in fact he had got it...
It doesn't really matter why I would guess the particles continue to drift apart over time if their initial relative velocities are not equal: it's a fact, as my elementary finite difference simulation, based on the definitions of the forces and Newton's laws (with negative masses) confirms...
Why would you think that? I have just written a little simulation where the velocities do not perfectly match to start off with, and the particles steadily drift apart over time as I would have guessed. Note that the two particles are not bound: intuitively, as they get further apart, the...
In physics we presently deal with gravity as a classical phenomenon acting on particles that are quanta of field theories and the same would be true here. We don't have a quantum gravity theory, so it would be unreasonable to object to Franes not having one for gravity extended to negative...