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Sep 5, 2020, 4:36:26 AM9/5/20

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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

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

Sep 8, 2020, 2:51:56 PM9/8/20

to

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

> [...] 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

Oct 22, 2020, 12:23:17 AM10/22/20

to

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
> 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?

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

Oct 26, 2020, 1:23:49 PM10/26/20

to

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:

> 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
>> 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?

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.]

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
> time (s), which we'll identify as proper time. Throw it in as another

> coordinate too.

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

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