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

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

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