Time-rate change in relatively moving frames

[PengKuan Em]

a) Material clock

What is time? This question is tricky because in relativity time-rate
changes when frame of reference changes. Time-rate changing is puzzling
because it is in conflict with our intuition that time is the flow of
ticks of clocks of which the mechanical structure does not change. Then
how to clearly explain the contradiction between the constant flow of
ticks delivered by clocks and the relativistic time dilation?

In order to grasp the essence of time-rate, we have to understand the
fundamental property of time. In still image, there is no time. When we
see scenes of cinema time emerges. Time emerges in moving scenes because
objects in the scenes change. So, the fundamental property of time is
the ability of objects to change state in moving image as well as in
reality.

For recording the rate of change of objects, human has invented clock,
the work of which is the change of state of the clock itself. For
example, in Figure the hands of the clock change position, the pendulum
changes position. Clocks record time by counting the number of times
that one part passes a specific state, for example, the big hand at the
number 12 on the dial. If the big hand has passed n times this state, we
say that the recorded time is n hours.

From the principle of work of clock, we extract the fundamental function
of clocks: counting the number of times an oscillating object passes by
a fixed point in space. This is true for archaic sundial as well as for
modern quartz clock which makes a quartz tuning fork to vibrate around
its neutral position.

So, all material chocks can be represented by the abstract clock in
Figure, which is formed by a material point k oscillating between the
ends of the short rod a and b. The motion of k is characterized by the
length of the rod. Let us refer to this abstract clock as “k clock”. The
time recorded by the k clock is the number of times that k strikes the
point a. We define one tick of time delivered by this clock to be one
strike.

Below, we will show how the time-rate of the k clock changes while ticking at the same rate. For doing so, we pair it with a light clock.

b) Paired with a light clock

In relativity light is the reference to all motion, so we make the light
clock in Figure which is formed by a photon bouncing back and forth
between the two mirrors Ma and Mb at the end of the long rod. In order
to calibrate the rate of the k clock, we synchronize it with the light
clock by matching the length of the short rod with that of the long rod
such that if k starts from the point a simultaneously with the photon
from Ma, k gets back to “a” simultaneously with the photon back to
Ma. So, the k clock is synchronous with the light clock and they make
one pair of “k clock - light clock”.

The time recorded by the light clock is the number of strikes the photon
makes on the mirror Ma. As k clock is synchronous with the light clock,
the number of strikes k makes on the point a always equals the number of
the photon’s strikes. The lengths of the rods and the identically
repetitive motion of k stay the same for whatever motion they are
in. This way, when the pair of “k clock - light clock” of Figure is
brought into motion, they are always synchronous.

The flow of ticks is the intrinsic tick-rate of a clock. Because the
length of the short rod and the motion of k do not change, the intrinsic
tick-rate of a material clock does not change either. But the time-rate
they show can change due to motion, which we will see below.

c) Time-rate change

Let us take 2 frames of reference frame 1 and frame 2, frame 2 moves at
constant speed in frame 1. In order to show the relativistic change of
time-rate of frame 2, we will put one pair of “k clock - light clock” in
frame 1 and an identical one in frame 2, see Figure. If we stand in
frame 1 and look the pair of this frame, then we stand in frame 2 and
look the pair of this frame, we will not detect any difference, which
shows that material clock does not change when jumping frame.

Then, why is the time-rate of frame 2 different from that of of frame 1?
Let us see Figure in which a pair “k clock - light clock” moves with
frame 2 in frame 1. In frame 2 the photon goes straight upward. But due
to the motion of the light clock, the path of the same photon is slanted
in frame 1. Let us denote the length of the path (back and forth) in
frame 1 with L1 and that in frame 2 with L2. Because the path in frame 1
is slanted, L1 is longer than L2.

One strike of the photon indicates that it has done the distance L2 once
in frame 2. Meanwhile, the same photon has done the distance L1 in frame
1, see Figure. Suppose that we have counted n2 strikes, then the photon
has done n2 times the distance L1 in frame 1, which makes the length of
its total path to equal S1=n2L1, see equation.

For counting the time passed in frame 1 during the n2 strikes, we count
the ticks given by the identical pair “k clock - light clock” in frame
1, see Figure. Within the same frame, light travels simultaneously the
same distance in all direction. Then, during the n2 strikes the photon
of frame 1 will also do the distance S1. Because the length of the long
rod is also L2 in frame 1, this photon will strike n1=S1/L2 times and S1
also equals n1L2, see equation. Then, we find in equation that n1 = n2
L1/L2. As L1>L2, the number n1 is bigger than n2.

Notice that n1 and n2 concern only the length of the photon’s paths, not
time. For knowing the time-rate in frame 1 and 2, we define the quantity
of time passed as the number of ticks delivered by light clocks which
equals the number of strikes by their respective photons. As n2 ticks is
delivered by the one of frame 2, the quantity of time passed in frame 2
equals n2 ticks. Simultaneously, the photon of the light clock of frame
1 has struck n1 times, so the quantity of time passed in frame 1 equals
n1 ticks, see equation and.

So, when the light clock of frame 2 delivers n2 ticks, simultaneously
the light clock of frame 1 delivers n1 ticks. If 2 clocks deliver
different number of ticks simultaneously, we say that the one that
delivers fewer ticks is slower. Using this image, we say that time is
slower in frame 2 than in frame 1 because n2 is smaller than n1. But
“time slowing” is only an image to describe this phenomenon, it is not
an appropriate term and it confuses people for understanding relativity.

Notice this difference: the n2 ticks are delivered by the light clock of
frame 2 but we count them in frame 1, the n1 ticks are delivered by the
light clock of frame 1 and also counted in frame 1.

d) Moving material clock

What about the moving k clocks? As it is synchronized with the paired
light clock, the number of ticks it delivers equals that of the paired
light clock and the k clock of frame 2 delivers fewer ticks than that of
frame 1 too, although the 2 “k clocks” are identical, which means that
material clock shows slower time-rate when moving while keeping the same
mechanical structure.

If we really want to find what object causes time to slow, we would say
the culprit is our standpoint. The path of the photon is straight upward
when we see it in frame 2. The path of the same photon is slanted when
we see it in frame 1. So, it is our standpoint that makes the path to
appear slanted and longer, which makes it to contain more ticks. In
consequence, the intrinsic mechanical structure of clocks and time
itself do not change, only their appearance changes depending on our
standpoint.

Figure and equation are in the article below.
https://www.academia.edu/44018092/Time_rate_change_in_relatively_moving_frames
https://pengkuanonphysics.blogspot.com/2020/09/time-rate-change-in-relatively-moving.html

[Tom Roberts]

On 9/5/20 3:36 AM, PengKuan Em wrote:
> [...] in relativity time-rate changes when frame of reference
> changes. [...]

This is incorrect. Using standard English words as they apply to
physics, the "time-rate" is the same in every locally inertial frame --
it never "changes".

[It is best to avoid such wishy-washy phrases as "time
rate". Talk instead about definite, unambiguous, and
directly measurable quantities such as clock tick rates.]

Einstein's first postulate, solidly confirmed experimentally, implies
that clocks always tick at their usual (standard) rate, regardless of
where they are located or how they might be moving (because the laws of
physics that govern their ticking are the same). Since "Time is what
clocks measure [Einstein and others]", this also applies to "time rate".

The rest of your article is useless because it fails to recognize this
very basic and fundamental aspect of relativity.

Tom Roberts

[rockbr...@gmail.com]

On Saturday, September 5, 2020 at 3:36:26 AM UTC-5, PengKuan Em wrote:
> a) Material clock
>
> What is time? This question is tricky because in relativity time-rate
> changes when frame of reference changes. Time-rate changing is puzzling
> because it is in conflict with our intuition that time is the flow of
> ticks of clocks of which the mechanical structure does not change. Then
> how to clearly explain the contradiction between the constant flow of
> ticks delivered by clocks and the relativistic time dilation?

We can fix that. Make it both. A coordinate time (t) and historical
time (s), which we'll identify as proper time. Throw it in as another
coordinate too. The Minkowski line element for proper time

ds^2 = dt^2 - (1/c)^2 (dx^2 + dy^2 + dz^2)

then becomes

ds^2 - dt^2 + (1/c)^2 (dx^2 + dy^2 + dz^2) = 0.

Now ... let's regularize it by transforming the coordinate to the
*difference* of proper time and coordinate time, defining

u := c^2 (s - t).

Then, the line element can be rewritten as the quadratic invariant:

dx^2 + dy^2 + dz^2 + 2 dt du + (1/c)^2 du^2 = 0

and, in addition, we have a linear invariant

ds := dt + (1/c)^2 du.

which may be interpreted as a "soldiering form" which ties the
historical "flowing" time onto the space-time geometry.

In the resulting geometry, it's *all* *four* *dimensions* which
flow in time, not just space! It's both "block time" and "flowing
time" at the same time. The entire block, itself, is flowing in
time!

To find what these things mean, consider first that these both have
non-relativistic limits:

dx^2 + dy^2 + dz^2 + 2 dt du = 0,
ds = dt.

In non-relativistic theory, by this account, it is justified to
treat coordinate time as the "flowing" time. On account of this,
the two may be safely confused in non-relativistic theory.

But by stepping forward into relativity and then moving back into
non-relativistic form, an extra item has cropped up that wasn't
there before and now, suddenly, you also actually have a space-time
metric for non-relativistic theory ... and one that morphs continuously
into the metric for relativity by adjusting a parameter alpha from
0 to (1/c)^2, with the generic case being

dx^2 + dy^2 + dz^2 + 2 dt du + alpha du^2 and ds = dt + alpha du.

The quadratic invariant identifies a metric that *always* has a 4+1
signature, for *all* values of alpha.

Second, consider what the most general linear transforms are that
leave these invariants intact; denoting infinitesimal transforms
by D(...):

dx D(dx) + dy D(dy) + dz D(dz) + (dt + (1/c)^2 du) D(du) + du D(dt) = 0,
0 = D(ds) = D(dt) + (1/c)^2 D(du).

Substituting the second equation D(dt) = -(1/c)^2 D(du) into the
first yields:

dx D(dx) + dy D(dy) + dz D(dz) = -dt D(du).

The general solution is:
D(dx, dy, dz) = omega x (dx, dy, dz) - upsilon dt
D(du) = upsilon . (dx, dy, dz)
D(dt) = -alpha upsilon . (dx, dy, dz)
where alpha is the above-mentioned parameter and the two new vectors are:

omega = infinitesimal rotations,
upsilon = infinitesimal boosts,
() . () denotes 3-vector dot-product,
() x () denotes 3-vector cross-product.

This is the 5-dimensional representation of the Lorentz group (when
alpha = (1/c)^2) and of the Galilei group (when alpha = 0) ... which
*not* the Galilei group, but the Bargmann group!

All cases are restrictions of the symmetry group SO(4,1), for the
4+1 metric, that correspond to the little group for the linear
invariant ds.

Now, to determine what this means, consider how the mass (m),
momentum (p) and kinetic energy (H) transform in non-relativistic
theory:

D(m) = 0, D(p) = omega x p - upsilon m, D(H) = -upsilon.p.

Under the classical version of the correspondence rule
p-hat = -i h-bar del, H-hat = i h-bar @/@t
(@ denotes the curly-d partial derivative symbol)
one has the correspondence
p <-> -del, H <-> @/@t
and this leads to the consideration of the corresponding one-form:
p.dr - H dt.

What is the transform of this?
D(p.dr - H dt)
= (omega x p - upsilon m).dr + p.(omega x dr - upsilon dt)
- (-upsilon.p) dt - H (0)
= -m (upsilon.dr)
= -m D(du)
= -D(m du),
This puts the spot-light on the one-form
p.dr - H dt + m du
showing that it is actually an invariant.

If we adopt these same quantities for Relativity and *continue* to
assume that this is the case for the relativistic form, by turning
on the parameter alpha = 0 to alpha = (1/c)^2, then the assumption
that this be invariant leads to:

0 = D(p.dr - H dt + m du)
= Dp.dr + p.(omega x dr - upsilon dt)
- DH dt - H (-alpha upsilon.dr)
+ Dm du + m (upsilon.dr)
= (Dp - omega x p + upsilon (m + alpha H)).dr
- (DH + upsilon.p) dt + (Dm) du
then we obtain the following transforms
Dp = omega x p - upsilon M
DH = -upsilon.p
Dm = 0
which singles out the "moving" mass M = m + alpha H as the mass
that goes with the momentum in the formula (momentum = mass times
velocity). Its transform, derived from those above, is
DM = -alpha upsilon.p
and, as a consequence, we find the following as the two invariants
under these transforms:

Linear invariant: mu := M - alpha H = m,
Quadratic invariant: lambda := p^2 + 2MH - alpha H^2 = 0.

The coordinate u is conjugate to m and in a quantized theory, m
would be represented as i h-bar @/@u. The coordinates (s,u), when
used in place of (t,u) produce the one-form

m du - H dt = m du - H (ds - alpha du) = (m + alpha H) du - H ds = M du - H ds.

So the corresponding operator forms would be, respectively,
M <-> i h-bar (@/@u)_s, H <-> -i h-bar (@/@s)_u.

The energy term H - which is the relativistic form of the *kinetic*
energy (not the total energy) is conjugate to the proper time s,
provided that it and u be taken together as the coordinates.

That generalizes the non-relativistic prescription of taking H to
be the generator of "flowing time".

This clean, continuous deformation from non-relativistic to
relativistic form is obscured because in Relativity, one normally
takes the *total* energy E, instead of the kinetic energy H as the
relevant component of momentum.

Here, that arises from the fact that the 4D sub-representation (M,p)
of the 5D representation (M,p,H) closes under the transforms, so
(M,p) forms a Minkowski 4-vector. M is, of course, converted to E
by way of the equation

E = M c^2.

So the corresponding transforms would read:

D(p) = -(1/c)^2 upsilon E, D(E) = -upsilon.p

and we may find the rest-mass as the square of the "mass shell"
invariant, which can be constructed from the quadratic invariant
lambda and linear invariant mu as:

mu^2 - alpha lambda
= (M - alpha H)^2 - alpha (p^2 - 2MH + alpha H)
= M^2 - alpha p^2
= (E/c^2)^2 - (p/c)^2 = m^2.

Strictly speaking, this construction goes BEYOND relativity, since
it has 5 components. The difference can be brought out clearly by
considering what the rest-frame form of the 5-vector (H,p,M) is in
non-relativistic theory

(H,p,M) -> (U,0,m), in the rest frame; U = internal energy.

If we adopt the same assumption here, then the respective invariants
would, generalize to:

lambda = p^2 - 2MH + alpha H^2 -> 0^2 - 2mU + alpha U^2,
mu = M - alpha H -> m - alpha U.

The constructs of Relativity are obtained by constraining U = 0.
The inclusion of a non-zero U corresponds to the inclusion of a 5th
coordinate (be it s or u) and of the splitting of the energy E into
two components: moving mass M and kinetic energy H.

This generalization allows one to consider more general systems
that may not have a rest-frame, and it continues to make sense in
those context, while the notion of "rest mass" (m) no longer carries
any meaning, except for systems whose mass shell invariant is
non-negative M^2 >= alpha p^2 (i.e. tardions, luxons and the
"vacuons", a.k.a. homogeneous states where M = 0, p = 0).

For curved space-times ...

If you repeat all of the same processes above with the Schwarzschild
solution:

proper time metric:
ds^2 = dt^2 (1 + 2 alpha U)
- alpha (dr^2/(1 + 2 alpha U) + r^2 ((d theta)^2 + (sin theta d phi)^2))
and

soldiering form: ds = dt + alpha du

where U = -2GM/r is the gravitational potential of a body of mass M...

and substitute and regularize, you obtain:

dr^2/(1 + 2 alpha U) + r^2 ((d theta)^2 + (sin theta d phi)^2)
+ 2 dt du + alpha du^2 - 2U dt^2 = 0

In the non-relativistic form of this - for alpha = 0 - the spatial
coordinates reduce to Euclidean form and can be replaced by Cartesian
coordinates to yield the metric:

dx^2 + dy^2 + dz^2 + 2 dt du - 2U dt^2 = 0.

The geodesics for this metric are *precisely* the motions of a body
moving under the influence of an energy potential U per unit mass;
i.e. moving under the influence of a potential energy in a way that
respects the equivalence principle. For U = -GM/r, that's the field
given by Newton's law of gravity.

I'm not the only one doing things this way. As I discovered a short
while ago there are these...

5D Generalized Inflationary Cosmology
L. Burakovsky∗ and L.P. Horwitz
https://arxiv.org/abs/hep-th/9508120

Their tau is my s. They take it out to a more general context -
curved space-times. I've toyed with this before, and they got the
same expression (equation 2.6) as I encountered, for the 5D form
of the radiation-dominant case of the FRW metric.

This *might* all tie into the MacDowell-Mansouri formulation - they
succeed in wrapping up the connection and frame field into a single
gauge field for gravity which works whenever the cosmological
coefficient Lambda is non-zero.

MacDowell–Mansouri Gravity and Cartan Geometry
Derek K. Wise
https://arxiv.org/abs/gr-qc/0611154

One of the reasons I say it's probably related, is because Mansouri
is already known as one of the people involved with dealing with
"signature changing" geometries. The FRW Big Bang metrics - especially
the radiation dominant one - has a null surface at time t = 0 and
its metric passes continually from one corresponding to alpha > 0
for t > 0, to alpha = 0 at t = 0, to Euclidean 4D form alpha < 0
at t < 0. Mansouri showed that the sectors of a signature-changing
metric with a null initial surface can only be consistently stitched
together under a "junction condition" that *forces* the cosmology
to be inflationary.

Signature Change, Inflation, and the Cosmological Constant
Reza Mansouri and Kourosh Nozari
https://arxiv.org/pdf/gr-qc/9806109.pdf

Also in the same category and potentially related:

Particles as Wilson lines of gravitational field
https://www.researchgate.net/publication/1971397_Particles_as_Wilson_lines_of_gravitational_field

which works within the MacDowell-Mansouri gravity gauge-theory
formulation.

I've used this as a means to continuously deform electromagnetic
and gauge theory from relativistic to non-relativistic form. It
just so happens that mostly the same result of that process is
described here:

Galilean Geometry in Condensed Matter Systems
Michael Geracie
https://arxiv.org/abs/1611.01198
(Note, particularly, the discussion of Bargmann Geometry - the
alpha = 0 case of the geometry I described. He also associates the
extra coordinate with mass, using M as the coordinate index.)

Curved non-relativistic spacetimes, Newtonian gravitation and massive
matter
Michael Geracie, Kartik Prabhu, Matthew M. Roberts
https://arxiv.org/abs/1503.02682

also works with Bargmann geometry and the extra coordinate.

Newton-Cartan Gravity Revisited
Roel Andringa
https://www.rug.nl/research/portal/files/34926446/Complete_thesis.pdf

Section 4.4 deals with the Bargmann algebra. There are 11 generators.
It may be derived as the algebra associated with the transforms of
the coordinates that I described above:

D(dx,dy,dz) = omega x (dx,dy,dz) - upsilon dt,
D(dt) = -alpha upsilon.(dx,dy,dz),
D(du) = upsilon.(dx,dy,dz).

by integrating, which produces constants of integration:

D(x,y,z) = omega x (x,y,z) - upsilon t + epsilon,
Dt = -alpha upsilon.(x,y,z) + tau
Du = upsilon.(x,y,z) + psi

that correspond to:
epsilon: infinitesimal spatial translations,
tau: infinitesimal time translations,
psi: infinitesimal translations along the u direction.

That adds in 5 extra generators: 3 for the components of the vector,
epsilon, and 1 each for tau and psi.

That's in section 4.5. Their Z is my "mu".

The attempt to use this as a device to continuously morph between
General Relativity (suitably extended to 5D) and Newton-Cartan
Gravity (of a Bargmann geometry) -- along with the issues that crop
up when doing so -- is described here:

Bargmann Structures and Newton-Cartan Theorem
Duval, Burdet, Kuenzel, Perrin
Physical Review D 31(8), 1985 April 15
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.31.1841

[Tom Roberts]

On 10/21/20 11:23 PM, rockbr...@gmail.com wrote:
> On Saturday, September 5, 2020 at 3:36:26 AM UTC-5, PengKuan Em
> wrote:

>> a) Material clock What is time? This question is tricky because in
>> relativity time-rate changes when frame of reference changes.
>> Time-rate changing is puzzling because it is in conflict with our
>> intuition that time is the flow of ticks of clocks of which the
>> mechanical structure does not change. Then how to clearly explain
>> the contradiction between the constant flow of ticks delivered by
>> clocks and the relativistic time dilation?

The answer to this is simple and straightforward: "time dilation" does
not affect any clock or any "time rate"; it only affects how a moving
clock is OBSERVED/MEASURED from an inertial frame. The "conflict" and
"contradiction" here are PengKuan Em's alone, in misunderstanding what
"time dilation" actually is.

[As I said earlier: It is best to avoid such wishy-washy

phrases as "time rate". Talk instead about definite,
unambiguous, and directly measurable quantities such as
clock tick rates.]

> We can fix that. Make it both. A coordinate time (t) and historical
> time (s), which we'll identify as proper time. Throw it in as another
> coordinate too.

That is nonsense. Attempting to use FIVE coordinates on a 4-D spacetime
is useless. And why didn't you add a SIXTH "coordinate" in an attempt to
deal with "length contraction"? -- after all that is essentially the
same as "time dilation" (both are simple geometrical projections,
differing only in orientation).

Moreover, you use an unacknowledged PUN on "coordinate": your s, which
you identify as proper time, is not a coordinate, and can never be one
-- coordinates are a 1-to-1 map from a region of an N-dimensional
manifold to a region of R^N. Proper time is path dependent and cannot
possibly participate in such a map.

[FYI: I put "time dilation" and "length contraction" in
"scare quotes", because they are rather poor names for
the actual phenomena, fostering the mistake PengKuan Em
made above. No time actually dilates, and no length ever
contracts, only measurements and relationships do so.]

> [...ignored: long, involved elaboration of that basic mistake]

Tom Roberts