Does space-time have an energy itself?

[Quarlep]

BTW I couldn't reply to your posts earlier because I went outdoors to watch the eclipse around 5 AM pacific time on Saturday morning and missed a lot of sleep. So yesterday I got sleepy early in the evening and couldn't keep my eyes open. Otherwise I would have replied earlier.

Yeah I saw your last seen.Never mind Its not a problem for me

 

[marcus]

Yeah I know simple calculus.I know find a function derivative (every type of them arccosx,sinx,lnx,e^x ...

Great! I want to mention a few things to you about the hyper-trig functions sinh(x) and cosh(x)

they have the same symmetry and antisymmetry as sin and cos.

and the same kind of completeness under differentiation:
sinh' = cosh

sinh( -x) = -sinh(x)

things like that.

 

[Quarlep]

Great! I want to mention a few things to you about the hyper-trig functions sinh(x) and cosh(x)

Thanks And I understand your 33 post https://www.physicsforums.com/threads/is-space-time-has-a-energy-itself.806636/page-2#post-5064711 thanks for that too.

 

[marcus]

cos(x) is called an "even" function because cos(-x) = cos(x). I am not sure "even" is a good word. I think of it as symmetric. But my teachers called it "even" and they called sin(x) "odd". Because it has this antisymmetry sin(-x)=-sin(x). If you run it backwards, it flips over.

Imagine Nature is a person and has likes and dislikes. Nature seems to like symmetry and anti-symmetry, and completeness under differentiation.
The function e^x is by itself complete under differentiation because its derivative is the function e^x itself.

But e^x is not symmetric

However you can symmetrize it and MAKE it an even function by averaging with its backwards version and that is cosh(x)
cosh(x) = (e^x + e^-x)/2

now when you differentiate that you get an ODD function sinh(x) = (e^x - e^-x)/2

and the pair of them taken together are complete under differentiation because when you differentiate sinh(x) you get cosh(x) back.

and you can get e^x back again just by adding sinh and cosh together. So this is a very simple basic pair of functions .

If Nature were a person it would not be surprising for it to like these functions. They also have nice power series. As an even function, cosh(x) power series consists of the EVEN power terms of the series for e^x. And the series for sinh(x) consists of the ODD power terms.
So together they make up the power series for e^x

It's all very basic. e^x is the solution to the world's simplest differential equation y' = y
and it represents the simplest kind of growth---growth at a constant rate--exponential growth.
and sinh and cosh are simply the even and odd parts of e^x.

 

[Quarlep]

cos(x) is called an "even" function because cos(-x) = cos(x). I am not sure "even" is a good word. I think of it as symmetric. But my teachers called it "even" and they called sin(x) "odd". Because it has this antisymmetry sin(-x)=-sin(x). If you run it backwards, it flips over.

Imagine Nature is a person and has likes and dislikes. Nature seems to like symmetry and anti-symmetry, and completeness under differentiation.
The function e^x is by itself complete under differentiation because its derivative is the function e^x itself.

But e^x is not symmetric

However you can symmetrize it and MAKE it an even function by averaging with its backwards version and that is cosh(x)
cosh(x) = (e^x + e^-x)/2

now when you differentiate that you get an ODD function sinh(x) = (e^x - e^-x)/2

and the pair of them taken together are complete under differentiation because when you differentiate sinh(x) you get cosh(x) back.

and you can get e^x back again just by adding sinh and cosh together. So this is a very simple basic pair of functions .

If Nature were a person it would not be surprising for it to like these functions. They also have nice power series. As an even function, cosh(x) power series consists of the EVEN power terms of the series for e^x. And the series for sinh(x) consists of the ODD power terms.
So together they make up the power series for e^x

It's all very basic. e^x is the solution to the world's simplest differential equation y' = y
and it represents the simplest kind of growth---growth at a constant rate--exponential growth.
and sinh and cosh are simply the even and odd parts of e^x.

Ok I got it

 

[marcus]

Good! so the personal subjective impression I want to share with you is that these are NICE functions,

and the Qday time scale is the scale that puts the standard cosmic model (the socalled LambdaCDM) in a "nice" form.
where the scale grows as the 2/3 power of sinh(1.5x)

I think you already got that but I wanted to say it explicitly, maybe help other readers get it

 

[marcus]

Now since you know how to differentiate, you must know the power rule and the chain rule. Tell me if not.
f(x) = x^2/3

f'(x) = (2/3) x^-1/3

so what is derivative of u(x) = sinh^2/3(1.5x) ?

It must be (2/3)(sinh(1.5x))^-1/3 multiplied by the derivative of sinh(1.5x) which we know is 1.5cosh(1.5x) by the chain rule.

the (2/3) and the 1.5 cancel. Let me know if there is anything unclear.

$$u'(x) = \frac{cosh(1.5x)}{sinh^{1/3}(1.5x)}$$What I want to calculate is H(x) which is the derivative of u(x) divided by the function u(x) itself. the fractional growth.

 

[Quarlep]

I get tanh(1.5x) If I didnt make any mistake

 

[Quarlep]

or cosh(1.5x)/sin(1.5x)

 

[Quarlep]

u(x) = sinh2/3(1.5x)

Is it a function of time or distance I think time but I want to be sure.

And which equation did you use to make first graph I tried o find it but I get nothing and second graph ?

Thanks a lot

 

[marcus]

or cosh(1.5x)/sin(1.5x)

That is right! and that is called the hyperbolic cotangent, denoted "coth".
So the answer is that u'/u = coth(1.5x)

 

[Quarlep]

I made a simple mistake

 

[marcus]

No problem.
I wanted to answer another question, about sinh^2/3(1.5x)

Is it a function of time or distance I think time but I want to be sure.

And which equation did you use to make first graph I tried o find it but I get nothing and second graph ?

Thanks a lot

u(x) = sinh^2/3(1.5x) is a function of time (where time is measured in Qdays

For the first graph I used y = coth(1.5x)/17.3
I wanted a graph of H(x) as fractional growth rate PER BILLION YEARS and that is a fraction which is 1/17.3 smaller than the fractional growth per Qday, or per 17.3 billion years.
that is why I divided coth(1.5x) by 17.3

That should make a curve that levels out at 0.06 because the longterm growth rate H_∞ is 0.06 per billion years. The longterm growth rate is also 1 per 17.3 billion years, as you have noted. So one could also plot H(x) simply as coth(x) and it would level out at 1. It is just using a different scale on the y axis.

For the second graph I used y = |sinh(1.5x)|^2/3

It helps to have the vertical lines | ... |
to signify the positive value because the machine may get confused if it is told to take the 2/3 power of a negative number, and sometimes sinh is negative.

 

[marcus]

Q, are you using Desmos to plot curves?
https://www.desmos.com/calculator
If you are you don't have to type the "y ="
You just go to that link and type a vertical bar |
and it makes a space |...|
for you to type the sinh(1.5x)
then you go outside the |...| to type the exponent and type "^"
and it will move up and make a place to type the "2/3" as an exponent.

I think it is easy to use. I'd be interested to know if you find it useful. And if it is available online where you live.

it also has a "keypad" with a menu of functions to choose from. The icon for the keypad is at the lower right corner of the large empty screen.
I have used it both ways---sometimes with the keypad of symbols to click on and functions to select---sometimes without it, typing everything in.

Let me know if by any chance you are using a Mac, because it has a math utility called "grapher" which can plot things in somewhat the same way as Desmos.

 

[Quarlep]

I know desmos.Actually I used it when you told me equations of graph.But I hot nothing.First I didnt understand but know,as you mention it,I know the answer.I don't have mac

When you made your last post in my country time was 1 am so I have to go sleep.

 

[marcus]

I'm glad you can use Desmos. I think it is excellent. I will look for an online utility that can do integrals. Do you by any chance know of one.?

Let me know if any of my explanations in this thread are hard to follow, or confusing. I'll try to make them clearer.
I live in California and the time is called "pacific time" Your time, I think, is about 10 hours ahead.

In my local time it is about 10:30 PM as I post this now.

 

[Quarlep]

Here time is 08:31 in the morning.I don't know integral online calculator but I can find. I read your new thread its really helped me to understand deeper.

 

[marcus]

Here time is 08:31 in the morning.I don't know integral online calculator but I can find. I read your new thread its really helped me to understand deeper.

Here it is 10:52 PM in the evening. That means you are 10 hours ahead.
Thanks for the encouragement. I found an online numerical integration website that does definite integrals between lower and upper limits
http://www.webmath.com/nintegrate.html
It is very minimal. The integrand must be defined with just a few characters

 

[marcus]

At that "web math" site I was able to integrate
(sinh(1.5x)^(-2/3) between limits .1 and .8
but they would not let me type in more characters so I could not integrate
1.311(sinh(1.5x)^(-2/3)
that was too long for them

 

[Quarlep]

http://www.integral-calculator.com look this maybe it helps I calculate 1.311sinh(1.5x)^(-2/3) and it gave me result but indefinite integral I guesss you can make definite to click options

 

[marcus]

that's interesting. I'll go look at it.
I found something that does definite integrals just now that seems good:
http://www.numberempire.com/definiteintegralcalculator.php

It has plenty of room to input the integrand, no unreasonable character limit.

as a test, I did 1.311(sinh(1.5x))^(-2/3) from 0.1 to 0.797 and it gave 1.33100...
It seems easy to use and simple. Not a lot of extras

EDIT (at 11:42 PM pacific time). It's late now, so I'm turning in.

 

[marcus]

Now 6AM pacific, I want to check out that "numberempire" numerical integrator a bit more
Wondering if it has other hyper-trig functions like hyperbolic secant ("sech(x)")
*************************
"numberempire" is actually pretty nice.
It goes way beyond the small job we have for it which is basically integrating the stretch factor S(x) = 1/a(x) = 1.311/sinh(1.5x)^2/3

In trig you may have encountered a special name for 1/sin, it is called "cosecant" and abbreviated "csc".
When I was in school it seemed to me needlessly fussy to have a special name for 1/sin.
I think it goes back to the days before computers when people worked with TRIG TABLES and it might save you a step in the calculation to have 1/sin already tabulated.

Anyway numberempire knows "csch" the hyperbolic cosecant so even though it may seem silly or unnecessarily fancy we can write the stretch factor S(x) as
S(x) = 1.311 csch(1.5x)^2/3

If I integrate that from .1 to .797 it is the distance NOW a flash of light can have traveled if emitted at time x = 0.1 (if not scattered or absorbed, if allowed to travel freely)
And I can multiply by 17.3 to get the answer in billions of light years. Here is how it looks when the Empire does the integral for us.
This page is live so you can put in x=0.2 and calculate for that, and put in 0.3 and calculate, and so on.
http://www.numberempire.com/definiteintegralcalculator.php?function=17.3*1.311*csch(1.5*x)^(2/3)&var=x&a=.1&b=.797&answers=

Here it is for x=0.2:
http://www.numberempire.com/definiteintegralcalculator.php?function=17.3*1.311*csch(1.5*x)^(2/3)&var=x&a=.2&b=.797&answers=

 

[Quarlep]

I am looking through my phone I ll look as soon as possible

 

[marcus]

I am looking through my phone I ll look as soon as possible

It may be that the search is over and it isn;t necessary to look further. I have had some experience with "Number Empire" numerical definite integral and I really like it. It is a simply presented powerful tool. My impression is that it is reliable. Of course I could be mistaken.
http://www.numberempire.com/definiteintegralcalculator.php
When you have time, check to see if you can get it, where you live. If you find it useful we don't have to look any more.

The place where you type in the function to be integrated, and the upper and lower limits, is right at the top of the page. It is direct, they don't distract the user with other stuff.

 

[Quarlep]

Yeah I got my questions answer.Again thank you.I look at it and its good integral calculater thanks for that too.