How far away is tomorrow?


The cumulative part of spacetime is time.

It is the cumulative nature of time [for an macroscopic scale] that makes the time a minus in the spacetime interval formula?

\displaystyle{\Delta s^{2} = - (c \Delta t)^{2} + (\Delta x)^{2} + (\Delta y)^{2} + (\Delta z)^{2}}

— Me@2011.09.21


Space cannot be cumulative, for two things at two different places at the same time cannot be labelled as “the same thing”.

— Me@2013-06-12 11:41 am


There is probably no directly relationship between the minus sign and the cumulative nature of time.

Instead, the minus sign is related to fact that the larger the time distance between two events, the causally-closer they are.

— Me@2018-10-13 12:46 am


Recommended reading:


— Distance and Special Relativity: How far away is tomorrow?

— minutephysics



2018.10.13 Saturday (c) All rights reserved by ACHK

Block spacetime, 9

motohagiography 42 days ago [-]

I once saw a fridge magnet that said “time is natures way of making sure everything doesn’t happen all at once,” and it’s stuck with me.

The concept of time not being “real,” can be useful as an exercise for modelling problems where to fully explore the problem space, you need to decouple your solutions from needing them to occur in an order or sequence.

From an engineering perspective, “removing” time means you can model problems abstractly by stepping back from a problem and asking, what are all possible states of the mechanism, then which ones are we implementing, and finally, in what order. This is different from the relatively stochastic approach most people take of “given X, what is the necessary next step to get to desired endstate.”

More simply, as a tool, time helps us apprehend the states of a system by reducing the scope of our perception of them to sets of serial, ordered phenomena.

Whether it is “real,” or an artifact of our perception is sort of immaterial when you can choose to reason about things with it, or without it. A friend once joked that math is what you get when you remove time from physics.

I look forward to the author’s new book.

— Gödel and the unreality of time

— Hacker News



2018.06.26 Tuesday ACHK

Life, 3

生命 3


We exist in time because time is change.

Growing is part of the definition of life. Growing is a kind of change.


Also, without time/change, there would be no thinking and no thoughts.

— Me@2017-12-26 11:42 am

— Me@2018-05-23 10:05:03 PM


time ~ change


Time is logically necessary if change is necessary.

— Me@2018-02-04 09:07:48 PM



2018.05.23 Wednesday (c) All rights reserved by ACHK

Tree rings, 2


This file is licensed under the Creative Commons Attribution 2.0 Generic license. Author: Lawrence Murray from Perth, Australia

Time-traveling to the past is like “making an outside ring more inside”, which is logically impossible.

— Me@2011.09.18





2018.05.16 Wednesday (c) All rights reserved by ACHK

Logical arrow of time, 6.3

“Time’s arrow” is only meaningful when considering with respect to an observer.


c.f. the second law of thermodynamics

The direction of time is direction of losing microscopic information… by whom?


“Time’s arrow” is only meaningful when considering with respect to an observer.

— Me@2018-01-01 6:14 PM



2018.04.09 Monday (c) All rights reserved by ACHK

The language of Change 1.2

Energy conservation, 6.2 | Energy 5.2


time ~ change

energy ~ the ability of _keeping_ changing


constant velocity ~ the amount of an object’s change of position, measured with respect to its observer’s unit of change, is constant

s = \Delta x

v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}


kinetic energy ~ the amount of the ability of keeping changing an object’s position

\frac{1}{2} m v^2 ~ the square of (the amount of change of position, relative to the observer’s unit of change)



Energy difference is _not_ exactly a measurement of the amount of change, time interval is.

— Me@2018-02-20 09:39:30 AM



2018.02.20 Tuesday (c) All rights reserved by ACHK

The language of Change

Energy conservation, 6 | Energy 5


time ~ change

energy ~ the ability of causing change


1. a system of one single particle

2. has only kinetic energy

3. and that kinetic energy is conserved.

conservation of energy ~ an object’s potential amount of change of position, measured with respect to its observer’s unit of change, is constant

s = \Delta x

v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}

— Me@2018-02-15 02:21:20 PM



The above argument has a bug:

If the mass m is constant, the kinetic energy E_K should be proportional to velocity squared v^2, instead of velocity v.

E_K = \frac{1}{2} m v^2


However, the above argument is still technically correct:

When E_K is constant, v^2 is constant. In turn, the magnitude of v also remains unchanged.

— Me@2018-02-19 09:37:24 PM



2018.02.15 Thursday (c) All rights reserved by ACHK

Logical arrow of time, 6.2

Source of time asymmetry in macroscopic physical systems

Second law of thermodynamics



Physics is not about reality, but about what one can say about reality.

— Bohr

— paraphrased



Physics should deduce what an observer would observe,

not what it really is, for that would be impossible.

— Me@2018-02-02 12:15:38 AM



1. Physics is about what an observer can observe about reality.

2. Whatever an observer can observe is a consistent history.

observer ~ a consistent story

observing ~ gathering a consistent story from the quantum reality

3. Physics [relativity and quantum mechanics] is also about the consistency of results of any two observers _when_, but not before, they compare those results, observational or experimental.

4. That consistency is guaranteed because the comparison of results itself can be regarded as a physical event, which can be observed by a third observer, aka a meta observer.

Since whenever an observer can observe is consistent, the meta-observer would see that the two observers have consistent observational results.

5. Either original observers is one of the possible meta-observers, since it certainly would be witnessing the comparison process of the observation data.

— Me@2018-02-02 10:25:05 PM




2018.02.03 Saturday (c) All rights reserved by ACHK


無額外論 5
“God” is your self at the farthest future.

That’s why we are all becoming gods.

— Me@2018-01-12 7:08 PM
Cooper : Did it work?

TARS : I think it might have.

Cooper : How do you know?

TARS : Because, the bulk beings are closing the tesseract.

Cooper : Don’t you get it yet, TARS? They’re not *beings*… they’re us! What I’ve been doing for Murph, they’re doing for me, for all of us.

TARS : Cooper, people couldn’t build this.

Cooper : No. No, not yet. But one day. Not you and me, but a people, a civilization that’s evolved beyond the four dimensions we know.

[the tesseract closes around him in a brilliant flash of light]

Cooper : What happens now?

[he sees the Endurance on its flight through the wormhole, touches Brand’s hand through the space-time distortion]

— Interstellar (film)
2018.01.14 Sunday (c) All rights reserved by ACHK

Euler Formula

Exponential, 2


general exponential increase ~ the effects are cumulative

natural exponential increase ~ every step has immediate and cumulative effects

— Me@2014-10-29 04:44:51 PM

exponent growth

e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n

~ compound interest effects with infinitesimal time intervals

multiply -1

~ rotate to the opposite direction

(rotate the position vector of a number on the real number line to the opposite direction)

~ rotate 180 degrees

multiply i

~ rotate to the perpendicular direction

~ rotate 90 degrees

For example, the complex number (3, 0) times i equals (0, 3):

3 \times i = 3 i
(3, 0) (0, 1) = (0, 3)

multiplying i

~ change the direction to the one perpendicular to the current moving direction

(current moving direction ~ the direction of a number’s position vector)

exponential growth with an imaginary amount

e^{i \theta} = \lim_{n \to \infty} \left( 1 + \frac{i \theta}{n} \right)^n

~ change the direction to the one perpendicular to the current moving direction continuously

~ rotate \theta radians

— Me@2016-06-05 04:04:13 PM
2016.06.08 Wednesday (c) All rights reserved by ACHK

注定外傳 2.6

Can it be Otherwise? 2.6 | The Beginning of Time, 7.3











— Me@2016-05-18 11:40:31 AM

2016.05.18 Wednesday (c) All rights reserved by ACHK

注定外傳 2.5

Can it be Otherwise? 2.5 | The Beginning of Time, 7.2



4. 即使可以追溯到「時間的起點」(第一因),所謂的「可以」,只是宏觀而言,決不會細節到可以推斷到,你有沒有自由,明天七時起牀。


















— Me@2016-03-15 08:43:58 AM

2016.03.31 Thursday (c) All rights reserved by ACHK

注定外傳 2.4

Can it be Otherwise? 2.4 | The Beginning of Time, 7








1. 「宇宙大爆炸」是一件事件,有一個過程,並不是時間上的「一點」,所以不算是「起點」。「宇宙大爆炸這件事的開始那刻」才算是起點。



2. 物理學家根據愛因斯坦的「廣義相對論」推斷,「宇宙開端」那一刻,開始發生的第一件事,是「宇宙大爆炸」。所以,如果「廣義相對論」不正確,「宇宙大爆炸」就未必為真。

3. 即使「廣義相對論」是可信的,普朗克時期(Planck epoch),即是開端後的頭\(10^{−43}\)秒之內,以現時的物理知識,是處理不到的。所以,物理學家推斷不到,那段時間內,發生了什麼事。




— Me@2016-02-15 07:04:56 PM

2016.02.15 Monday (c) All rights reserved by ACHK