18th Mar, 2022

Universitat Politècnica de Catalunya

Discussion

Started 6th Feb, 2022

Consider two particles A and B in translation with uniformly accelerated vertical motion in a frame S (X,Y,T) such that the segment AB with length L remains always parallel to the horizontal axis X (X_{A} = 0, X_{B} = L). If we assume that the acceleration vector (0, E) is constant and we take the height of both particles to be defined by the expressions Y_{A} = Y_{B} = 0.5 ET^{2}, we have that the vertical distance between A and B in S is always (see fig. in PR - 2.pdf):

1) Y_{B} - Y_{A} = 0

If S moves with constant velocity (v, 0) with respect to another reference s(x,y,t) whose origin coincides with the origin of S at t = T = 0, inserting the Lorentz transformation for A (Y = y, T = g(t - vx_{A}/c^{2}), xA = vt) into Y_{A}= 0.5 ET^{2} and the Lorentz transformation for B (Y = y, T = g(t - vx_{B}/c^{2}), x_{B} = vt + L/g) into Y_{B}= 0.5 ET^{2} we get that the vertical distance between A and B in s(x,y,t) is:

2) y_{B} - y_{A} = 0.5 E (L^{2}v^{2}/c^{4}- 2Lvt/c^{2}g)

which shows us that, at each instant of time "t" the distance y_{B} - y_{A} is different despite being always constant in S (eq.1). As we know that the classical definition of translational motion of two particles is only possible if the distance between them remains constant, we conclude that in s the two particles cannot be in translational motion despite being in translational motion in S.

More information in:

- 424.64 KBPR-2.pdf

Stam Nicolis, any comment will be appreciated: using hyperbolic motion in frame S(X,Y,T) we may write:

1) J^{2} = Y^{2} - c^{2} T^{2}

where “J” is the high in the corresponding inertial commoving frames. Using Born´s definition of rigidity, “J” must be invariant under Lorentz transformations. It is straightforward to see that this is the case just for boosts along the Y direction. For a boost along X, inserting Y = y, T = g (t - vx/c2) we get:

2) J^{2} = y^{2} - c^{2} (t - vx/c^{2})^{2}

which is clearly non invariant (only x^{2} - c^{2}t^{2} is invariant). Thus, we get a violation of Born´s rigidity.

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Exercises in first year mechanics seldom describe research problems and this one’s no exception. One, usually, learns in first year classes about Newton’s second law and how to use vectors.

Later on one learns that global Lorentz transformations relate inertial not non-inertial frames. That's why claiming that by performing a global Lorentz transformation it's possible to pass from an inertial to a non-inertial frame just doesn't make sense. And a local Lorentz transformation isn't given by the expressions provided.

A Lorentz boost involves the time coordinate and one spatial coordinate-and it acts on all points of a frame. The claim that the transformations displayed are Lorentz transformations is, simply, wrong. So it's no surprise that only confusion can result.

They're not even Galilean boosts.

[The motion of particles in a non-inertial frame is given by the geodesic equation. So one learns to write down that equation and solve it for two different initial conditions, that correspond to the two particles.

The starting point, therefore, would be to write down the metric that describes the non-inertial frame and solve the corresponding geodesic equation.

This is, still, within the framework of any course in physics and no longer a research problem, at least after 1915 (OK, maybe after 1921, when Pauli wrote his review article). ]

Of course the problem may be solved within Newtonian mechanics quite trivially, by realizing that the only way the segment can remain (a) horizontal and (b) of fixed length is if the initial conditions are y_{A}(0)=y_{B}(0), v_{A,x}(0)=v_{B,x}(0) and v_{A,y}(0)=v_{B,y}(0).

Therefore it doesn't make sense demanding that the segment can satisfy conditions (a) and (b) if the initial conditions are different. It suffices to write Newton's equations along x and y for the two particles to show this.

Then Newton's equations ensure that y_{A}(t)=y_{B}(t) and that |x_{A}(t)-x_{B}(t)|=|x_{A}(0)-x_{B}(0)| which every first year student learns in the first few weeks.

Newton's equations, also, imply that a Galilean transformation along x is consistent with these relations, since the force along x is zero...

And the statement that the vertical distance could change is, just, wrong. There's just no way it could change; it can be shown to be a constant of motion, by writing the equation satisfied by the relative distance, y_{A}(t)-y_{B}(t), namely d^{2}(y_{A}(t)-y_{B}(t))/dt^{2}=0, since the acceleration is constant along y and the same for both particles-and both particles are assumed to have the same initial velocity along y.

In fact it's, also, trivial to see from Newton's equations, that, if the two particles are subject to the same force along x, the segment will remain horizontal and of fixed length; it will just be accelerating along x. Motion along x is independent of motion along y...

The equation for y_{A}(t)-y_{B}(t) isn't invariant under transformations to a non-inertial frame; in Newtonian mechanics these would be transformations of time that aren't linear functions of time. But that's no surprise. (And Lorentz transformations are transformations that are linear in the coordinates.)

Stam, thanks a lot for your response. Please, note that:

1) It says that S moves with constant velocity v with respect to s. Both frames S and s are inertial. There is no non-inertial frame in the problem.

2) Particles A and B cannot have the same height in both inertial frames due to relativity of simultaneity.

Don’t, you agree with 1)2) ????

It seems that the only solution to this problem consists on using a re-synchronization of clocks of inertial frames. The only reference I know on this kind of synchronization is Kassner, 2013. Any additional information will be appreciated.

I agree with you that in any book like Rindler or Carroll one can easily find information on how to calculate the Chrisstofel symbols and the equations for a geadosic in accelerated frames but I think there is not so much information on re - synchronization of clocks in flat and curved spacetimes. For example, can we consider the Painlevé - Gullstrand coordinates for blackholes to be a re-synchronization??

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In the document "A brief ..." I used a resynchronization together with a global Lorentz transformation but it might be interesting to try to define a local Lorentz transformation to show the meaning of local Lorentz invariance applied to coordinates. Local Lorentz transformations are usually appied to tensors, not to coordinates. This is the reason why I use two clocks. When both clocks are very close, we get Lorentz invariance.

We may consider the change of coordinates:

1) T = g (t_{P} - vx_{P}/c^{2})

2) X = g (x - vt_{P})

where t is the time of a clock placed at x and t_{P }is the time of a clock placed at x_{P}. If we place both cloks at x = x_{P} we get for an event (x_{P} , t_{P}) happening at x = x_{P}:

3) T = g (t_{P} - vx_{P}/c^{2})

4) X = g (x_{P} - vt_{P})

which shows us that we have local Lorentz invariance for (x_{P},t_{P}). On the other hand, if we consider an event (x,t) happening at an arbitrary coordinate x and we place the clock P at the origin x_{P} = 0, using 1)2) yields:

5) T = g t_{P}

6) X = g (x - vt_{P})

which:

- is a a global change that clearly violates Lorentz global invariance for space-time coordinates.

- it is compatible with the Sagnac effect and gps simultaneity.

- it is equivalent to re-synchronizing the global Lorentz transformation.

11th Feb, 2022

Let us first define what dark matter and dark energy are. From generally accepted ideas, it can be assumed that the concepts of dark and white are directly related to physical laws. It can be assumed that dark energy and dark matter are associated with the gravitational energy of absorption, and white energy and white matter --- with the gravitational energy of radiation. It is obvious, therefore, that each force (energy) has oppositely directed actions (cold-heat, attraction-repulsion...) We live in a very small part of the galactic space, where there is a gravitational attraction energy. But we cannot deny that another kind of gravity (repulsion) may exist. White energy and white matter can be related to the white energy of repulsion. It can be shown that white energy is emitted by neutron stars and pulsars. This result is obtained by solving the field equations (Einstein equations) for the de Sitter metric. It is shown that the Newtonian laws of motion are realized in the solar system. But we cannot assume that our lives conform to Newtonian laws. The galaxy includes not only stars such as the Sun, but also neutron stars, pulsars, and so on. This object should not move according to Newtonian theory of gravitational attraction alone.

Larissa, I might be wrong but I believe that you wanted to post in another quest on dark matter and dark energy.

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Stam Nicolis, any comment will be appreciated: using hyperbolic motion in frame S(X,Y,T) we may write:

1) J^{2} = Y^{2} - c^{2} T^{2}

where “J” is the high in the corresponding inertial commoving frames. Using Born´s definition of rigidity, “J” must be invariant under Lorentz transformations. It is straightforward to see that this is the case just for boosts along the Y direction. For a boost along X, inserting Y = y, T = g (t - vx/c2) we get:

2) J^{2} = y^{2} - c^{2} (t - vx/c^{2})^{2}

which is clearly non invariant (only x^{2} - c^{2}t^{2} is invariant). Thus, we get a violation of Born´s rigidity.

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