The relationship between space and time

[studiot]

Remember I am trying to keep it simple to bring out the important points.

So I am only going to consider frames that are not in relative motion with each other.

So any frame has an origin.

The the coordinates of the particles m₁ and m₂ are x₁ and x₂ in our original frame, call it the x frame.

If we now refer to a second frame, call it the x' (x prime) frame, whose origin is at x₀ in the original frame then

x₁ = x'₁ +x₀   and   x₂ = x'₂ +x₀ 

Now we have the force = mass x acceleration

and acceleration is a function of the x coordinate

So Force is proportional to the x coordinate in our original frame

F = F(x1 , x2)

Which is two equations

[math]{m_1}\frac{{{d^2}x}}{{d{t^2}}} = F = F\left( {{x_1},{x_2}} \right)[/math]

and

[math]{m_2}\frac{{{d^2}x}}{{d{t^2}}} =  - F =  - F\left( {{x_1},{x_2}} \right)[/math]

Which expresses the proportionality of force with acceleration in our original frame (the x frame)

But if we substitute the transformations to cahnge to our second frame (the x' frame) we have

[math]{m_1}\frac{{{d^2}x'}}{{d{t^2}}} = F = F\left( {x{'_1} + {x_0},x{'_2} + {x_0}} \right)[/math]

and

[math]{m_2}\frac{{{d^2}x'}}{{d{t^2}}} =  - F =  - F\left( {x{'_1} + {x_0},x{'_2} + {x_0}} \right)[/math]

 

Now we see two things.

1) The equations now depend upon the origin of the coordinate system loosing homogenity along the x axis.

2) The equations violate the Principle of Relativity since this law of Physics no longer has the same form in both coordinate systems.

And all we have done is transported (translated) the whole system along the x axis, nothing is actually in motion.

 

How we fix this is the subject of the next installment.

And yes, thus far we are still Newtonian mechanics.
 

Even in his day they were aware of the problem, but sidestepped it.

 

It was the second postulate of Einstein that was revolutionary.

[studiot]

To help you make sense of things try this.

(1) Consider f₁(x) = mx where m is a constant.

So f(x)          (are you comfortable with this notation?)

is directly proportional to x and the plot is a straight line through the origin.

(2) Now consider f₂(x) = mx + c where c is another constant.
This plot is also a straight line, but not through the origin.

Comparing

For (1) if we multiply x by another constant say a then

f₁(ax) = a f₁(x)

for example, if m = 3

then at the point x = 2

f₁(x) = mx = 3*2 = 6

and if we now double x so a = 2, at the point x = 4

f₁(x) = mx = 3*4 = 12

So f₁(2x) = 2f₁(x)

 

But compare this with f₂(x) with c = 5 and m still = 3

at the point x = 2

f₂(x) = f₂(2) = 3x + 5 = 3*2 + 5 = 6 +5 = 11

and at the point x = 4

f₂(2x) = f₂(4) = 3*4 + 5 = 12 + 5 = 17

So f₂(2x) is not equal to 2f₂(x)

 

Mathematically we say that So f₁(x) is a linear function and So f₁(2x) = 2f₁(x) is the condition for it to be linear.

 

The function f₂(x) is called an affine function.

[geordief]

37 minutes ago, studiot said:

 

 

Yes,I am completely OK with all that.I have also come across the term affine and ,whilst it is apparently a very simple  mathematical concept (far simpler than the name suggests)   I have not yet come across any circumstances where it is important (well not circumstances that I understood ,perhaps it was in connection with the dual space/tensors that I was trying to understand a few months ago iirc) .  

 

I have also looked at your previous post but I still need to go over it once or twice again (I have to do this when I learn something new)

 

Actually I find it a little hard to understand "The equations now depend upon the origin of the coordinate system loosing homogenity along the x axis."

 

Perhaps ,you might say the same thing with a different form of words?

Edited August 29, 2019 by geordief

[michel123456]

Very loosely, since space & time are transforming the one into the other it means they are "made up" of the same stuff. It also means that when you look at something in space you also look at that thing in time. It is evident for the stars: the farthest means also the oldest (in regard to us) and the youngest (in regard to the universe). It is less evident for things around you: the wall in your room is also at a distance in time.

In fact, what you see at a distance is truly another image of time, different from the sensation given by the tic-tac of the clock.