Problem Solving Strategies

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Kavin Ishwaran

07:16

@JohnRennie Hi !

John Rennie

@KavinIshwaran Hi :-)

Kavin Ishwaran

I was reading your answer in this Q and A

A: What is time dilation really?

Introduction This answer will use ideas discussed in the answers to What is time, does it flow, and if so what defines its direction?, so you really need to read the answers to that question before tackling this one. The key concept you need in order to understand time dilation is that a clock ...

John Rennie

OK ... ?

Kavin Ishwaran

In this You have given two graphs

saying that proper time is invariant

John Rennie

Yes

Kavin Ishwaran

07:20

In that graph we can clearly see that the two red arrows (a straight and curved one) are different in length, but they are actually the same. how they can be invariant

I suppose it is the way our universe built on ?

and how do you actually define proper time

John Rennie

In regular geometry we can measure the length of the line using Pythagoras' theorem. If we have a line from (0,0) to (x,y) then the length of that line is s² = x² + y². Yes?

Kavin Ishwaran

yes

John Rennie

And this extends to three dimensions s² = x² + y² + z², and in principle to any number of dimensions.

Kavin Ishwaran

I am fully clear on the Euclidean metric part

you were explaining this a while ago :-) but, I cant understand the invariant thing

John Rennie

Well this is the weird thing about SR. We have four dimensions (t, x, y, z) but the distances are not given by the Euclidean metric but rather by: s² = -c²t² + x² + y² + z²

@KavinIshwaran Didn't we talk about this? I remember explaining using the example of an arrow. If you rotate or move the arrow it doesn't change length.

Kavin Ishwaran

07:25

yes but what about a curved path ?

John Rennie

A curved path is just a collection of infinitesimally small straight paths.

Kavin Ishwaran

Yes

John Rennie

So you can treat the curved path as a sequence of infinitesimally small arrows arranged head to tail, and as we discussed the length of those arrows doesn't change just because we changed our coordinate system, and hence the total length of the curve doesn't change.

Kavin Ishwaran

lets take the same example

you are being stationary

and I am moving at velocity v and both space time graph have the common point 0,0

along x axis*

John Rennie

OK. So we zero our clocks as we pass each other so we both started from the same point (0, 0)

Kavin Ishwaran

07:37

yes

John Rennie

And you move until you reach some point that we have labelled P. e.g. we could literally stick a flag in the ground with "P" written on it to mark that point.

And we both record the time shown by our clocks when you reach that point P.

Yes ?

Kavin Ishwaran

yes, so when I reach point p in time t and after some time t you have moved a distance ct in space time and distance between 0 and point p is root (-c2t2 + v2t2) yes ?

@JohnRennie yes

John Rennie

In my rest frame I am stationary at the origin, and you are moving with velocity v. You travel for a time on my clock t, and because you are moving at velocity v the distance you move (as observed by me) is x = vt. Yes?

Kavin Ishwaran

yes

John Rennie

So in my frame the proper time for your motion is s² = -c²t² + v²t²

Which is what you said above :-)

I just wanted to be clear exactly what we were calculating.

So, is this all OK so far?

Kavin Ishwaran

07:44

yes and

this is where I am having a doubt

how do we define proper time

John Rennie

You mean how do we relate times to this quantity "s²" ?

Kavin Ishwaran

yes

John Rennie

OK, suppose I am at rest in my frame and I am holding a clock - so the clock is also at rest.

After some time T I have reached the point (T, 0) i.e. I moved T in time and 0 in space.
OK so far?

Kavin Ishwaran

yes

John Rennie

So if we substitute this into the metric we get: s² = -c²T²

where T is the time shown on my clock.

And c is a constant, so s² ∝ T² i.e. the "distance" s and the time T are related by a constant.

We call s the proper distance and T the proper time. They are basically the same thing because they differ only by a constant.

In fact there are two ways of writing the metric:
s² = -c²t² + x²
c²T² = c²t² - x²

Kavin Ishwaran

07:51

@JohnRennie and that constant is lorentz factor ?

John Rennie

No, s² = -c²T² and that constant c is the speed of light.

Kavin Ishwaran

ok, But at the end we see that t and T differ by a factor of the gamma (?)

John Rennie

Hang on. You asked me what the proper time means, and that' what I'm telling you. The proper time for an observer travelling between two points is the time recorded by that observer's clock.

The reason t and T differ in our example is because t is not a proper time while T is a proper time.

Kavin Ishwaran

@JohnRennie yes

then that is why we say that the length s is invariant ?

John Rennie

@KavinIshwaran I'm not sure what you are asking ...

The proper length, and the proper time since the two are proportional, is the length of a four-vector i.e. the length of a 4D arrow.

And it's constant because arrows don't change length.

Kavin Ishwaran

07:58

Ok.. Got it !

John Rennie

I need to drop out for a while, but I will be around later.

Kavin Ishwaran

Ok :-)

2 hours later…

user157860

10:29

@JohnRennie, Hi John, do you have a few minutes? :)

John Rennie

@user157860 I'm busy answering a question in another room at the moment, but I shouldn't be long.

user157860

ok, I'll wait

John Rennie

10:41

@user157860 Hi, I'm free now :-)

user157860

Hi, recently you told me that the electron in H atom has no orbit but it's spread in a cloud all around the nucleus, can you explain how that is compatible with classical laws that are still valid? Coulomb's law needs a well-defined distance and and so does centrifugal law: v^2/r besides a velocity. Or such laws are not valid?

John Rennie

The classical laws do not apply to quantum particles like electrons in atoms.

user157860

so the electron is not attracted to proton?

John Rennie

The electron is attracted to the proton, but the attractive force is more complicated than Coulomb's law.

user157860

OK, I see, thanks :)

Kavin Ishwaran

11:39

@JohnRennie Hi :)

Please tell me when you are free

cOnnectOrTR 12

Hi @JohnRennie.

I just have a simple question. If two masses one 5kg and 1kg are hanging in a Atwood machine and the pulley is hanging from a spring balance, then as the masses let go how much the spring balance meansure during acceleration?

On second thought let’s say there are two weights 5kg-f and 1kg-f.

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