Shankar on Lorentz Transformation: Does x' = ct'?

[abrogard]

TL;DR Summary

x=ct, x1=ct1

On the Yale University Prof Shankar Youtube vid 'Lorentz Transformation' Prof Shankar writes up on the board that x = ct and then x prime = c t prime.

It is the basis of all that follows. But i don't understand.

at x = 0, t = 0 and x prime = 0 and t prime = 0. He's got that written up, too, and it all makes sense.

So when the event happens and is instantaneously known to have happened by all concerned at time 't' then that will also be time ' t prime' won't it?

't prime' is always the same as 't' is it not? In this scenario where the event is imagined to be immediately known as it happens at both x and x prime.

So it makes sense that distance x = speed of light time t but x prime cannot be the same surely? x prime is ct minus ut surely?

where am I wrong?

 

[Sagittarius A-Star]

Summary:: x=ct, x1=ct1

On the Yale University Prof Shankar Youtube vid 'Lorentz Transformation' Prof Shankar writes up on the board that x = ct and then x prime = c t prime.

It is the basis of all that follows. But i don't understand.

That is the 2nd postulate of SR. The Lorentz transformation follows from the two postulates of Special Relativty:

1. The laws of physics are the same for all observers in any inertial frame of reference relative to one another (principle of relativity).
2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source.

Source:
https://en.wikipedia.org/wiki/Theory_of_relativity#Special_relativity

't prime' is always the same as 't' is it not?

No. Frame $S$ and frame $S'$ have each their own time. That was verified by experiments. In daily life we don't observe this, because there we experience only relative velocities much smaller than $c$ and our watches are not accurate enough to measure relativistic effects at such small velocities.

 

[Ibix]

You should always provide a link to what you were studying, and a timestamp in the case of a video. That goes double when you know you didn't understand it. How are we supposed to know what it is you want explained when you don't link it and can't state it?

At a guess, Shankar is talking about events that are joined by a light ray. You can declare one event to be the origin of both frames, so is at $(x,t)=(0,0)$ and $(x',t')=(0,0)$ by definition. The second event happens at some time $t=\Delta t$ according to the unprimed frame, and the fact that light traveled there from the origin tells you that its coordinates are $(x,t)=(c\Delta t,\Delta t)$. In the primed frame the same event happens at a (possibly different) time $\Delta t'$. By the same argument its coordinates in the primed frame are $(x',t')=(c\Delta t',\Delta t')$. The Lorentz transforms must, therefore, map $(x,t)=(c\Delta t,\Delta t)$ to $(x',t')=(c\Delta t',\Delta t')$. This does not imply $\Delta t=\Delta t'$, merely that any event whose $x$ coordinate is $c$ times its $t$ coordinate must map to one whose $x'$ coordinate is $c$ times its $t'$ coordinate.

That's an educated guess, but there are other lines of argument Shankar could be taking. Provide a link and a timestamp and we can see.

 

[abrogard]

I thought I had effectively provided a link when I quoted youtube video and the name of it.

And I thought the actual math would be so widely known it would be familiar to all except perhaps beginners like me.

I am sorry if I have inconvenienced you. It certainly was not my intention.

Here is the link: shankar on lorentz

time about 6:00

He has another video 'introduction to relativity' where he does this same thing, too.

shankar intro to relativity

time in that one is 55:10 or close to it.

 

[Ibix]

I am sorry if I have inconvenienced you.

It's not about inconveniencing me, it's about knowing exactly which part of what source you were attempting to understand. Even if I'd gone to look for the video it's over an hour long, which is a long time to consider investing in searching.

The part you linked is not (yet) talking about the Lorentz transforms. He's talking about the Galilean transforms, which assume a shared notion of time and a shared notion of distance (although not position) for all observers. But the point about relativity is that it does not assume these things.

Historically, Einstein hypothesised that the speed of light was the same for all observers and deduced that this had implications, that neither notions of distances nor times nor simultaneity were shared between observers. So the Galilean transforms can't be generally true - although they do emerge as a low speed limit of the Lorentz transforms, which is how we didn't notice for so long.

 

[PeroK]

I thought I had effectively provided a link when I quoted youtube video and the name of it.

And I thought the actual math would be so widely known it would be familiar to all except perhaps beginners like me.

I am sorry if I have inconvenienced you. It certainly was not my intention.

Here is the link: shankar on lorentz

correction. I better go back to the other vid. I'll find that link and time for you, then it will all be there.

That video is one hour long. You have to transcribe your question here.

Note that you must be careful to distinguish between general coordinates and a specific trajectory. To take an example. We have the relationship between Cartesian and Polar coordinates:
$$x = r\cos \theta, \ \ y = r \sin \theta$$
If you take any point in the plane, it has coordinates $(x, y)$ and $(r, \theta)$ with the above relationship.

Now, consider the line in the plane $y = x$. That transforms to the line $\theta = \frac{\pi}{4}$. Those equations only hold along that line and are not equations relating the coordinates in general.

In your example, the path of a light ray in the unprimed coordinates is $(t, ct)$, defined by the equation $x = ct$. And, the path in the unprimed coordinates is $(t', ct')$, defined by the equation $x' = ct'$.

That does not change the general relationship between the coordinate systems, which may be related by a Lorentz Transformation.

It's now a good exercise for you to show that if you transform the line $x = t$ into the primed coordiantes, using a Lorentz Transformation you get the line $x' = t'$.

 

[abrogard]

I have edited my post. It might be clearer now. Provided times, too.

 

[abrogard]

I think I get it.

Though Shankar does all this talking about this one and that one claiming distances are with and without ut in fact neither party is doing any of that.

Our stationary view is that x is ct because we know that a beam of light was sent from the origin at time 0 and the event occurred at time t. So we happily say that distance is therefore ct. Job done.

And the other bloke does exactly the same thing. He's ignorant of the fact that he's got this distance covered between the origin and where he is now because he considers himself to be stationary and if anyone is moving it must be us. Not him.

So he says the distance to the event is the time since t=0 multiplied by the speed of light. i.e. in our maths setup t prime times c.

That ut was always bugging me. Why would they claim to be measuring x prime when they're clearly measuring x prime plus ut ?

And that's it isn't it - they claim it because to them they're not moving and the time t prime represents to them the time from the common origin, O, to the event.

So his time value - t prime - represents the time taken by a light ray to cover ut and x prime together, the whole distance, at what's effectively a new speed of light, his own speed of light, not the same as ours, because his time is unique to him, affected by his speed.

So then Shankar goes ahead with the arithmetic where we look at these two frames of reference from our third frame of reference or something.

And we say 'he thinks this', 'he claims x is x prime plus ut' though of course he does no such thing.

and we get his xprime + ut and our x - ut and we multiply them together and whatnot and then inject that masterstroke where two different lengths are claimed to be the same... and that produces a value for the 'fudge factor' as Shankar calls it.

I think it could all probably be demonstrated much more simply by you mathematicians if you cared to, couldn't you?

I couldn't get over why would he claim to be measuring x prime when he's clearly measuring the whole distance. That was my problem.

But it's because he thinks he's the stationary one. That's about it, isn't it?

 

[PeroK]

I think it could all probably be demonstrated much more simply by you mathematicians if you cared to, couldn't you?

It's about being able to apply some elementary mathematics. We have our general relationship between coordinates:
$$t' = \gamma (t - \frac{vx}{c^2}), \ \ x' = \gamma(x - vt)$$
Now, consider a trajectory (line/path), defined by $x = ct$. That is a set of points expressed in $x, t$ coordinates. How can we describe that same set of points using $x', t'$ coordinates? Let's transform points on the line into the $x', t'$ coordinates:
$$t' = \gamma(t - \frac{v(ct)}{c^2}) = \gamma(1 - \frac{v}{c})t$$ $$x' = \gamma(ct - vt) = c\gamma(1 - \frac{v}{c})t = ct'$$
And, we see that we can describe the line in $x', t'$ coordinates using the equation $x' = ct'$.

And we interpret this as a light ray traveling at $c$ in coordinates $x, t$ is also traveling at $c$ in coordinates $x', t'$. I.e. the speed of light is invariant under a Lorentz Transformation.

 

[abrogard]

sorry. didn't understand that.

I meant those, equations coming up with the lorentz contraction that Shankar used.

the exercise is presented to people such as I - raw beginner students - as some sort of clever demonstration of how a basic relativistic phenomenon can be 'discovered' or revealed by simple high school maths.

you see? A kind of 'who would have thought it?' miracle moment. There's relativity hiding right there beneath our noses all this time. You didn't have to be Einstein to discover relativity - high school maths can do it.

So this exercise Shankar has put before us.

But I'm surmising it could even be simpler than that. Given you present these variables supposedly linked in these ways but they contain this 'mystery ingredient' whatever it is, you'll always come up with this 'fudge factor' as Shankar calls it.

don't explain myself very well do I. sorry. and I might be completely wrong. I'm just trying to say I think there's an underlying mathematical principle underlying the maths he's presented which a mathematician would have identified immediately.

Like the one factor in all those figures that gives rise to the 'fudge factor'.

Not clear? Ah well, doesn't matter. I just don't have either the English powers nor the mathematics to explain myself.

Thanks for the interest. :)

 

[Sagittarius A-Star]

But it's because he thinks he's the stationary one. That's about it, isn't it?

• That's one correct part of the argument (SR postulate #1, the principle of relativity).
• The other would be, that for all observers, the vacuum speed of light is the same in their respective rest frame (SR postulate #2, the invariance of the speed of light).

 

[Sagittarius A-Star]

You didn't have to be Einstein to discover relativity - high school maths can do it.
..
I'm just trying to say I think there's an underlying mathematical principle underlying the maths

It's not only about math. Einstein "discovered" the two SR postulates, which include - experimentally verified - assumptions about physics. The math comes afterwards, when deriving the Lorentz transformation from those two postulates.

Prof Shankar skips details, how to calculate $\gamma$ at video time 06:30. You can find an example for a calculation of $\gamma$ there:
https://en.wikipedia.org/wiki/Deriv...termining_the_constants_of_the_first_equation

 

[abrogard]

No, his 'intro to relativity' provides the details of the first method at your link, which is a very good one, thank. Those are what I've been following. All the way to the second vid.

Let me try and be clear for I feel I have not been.

For me it is all about the math. My question is all about the math.

I'm not trying to learn or dispute or investigate relativity or any part of it here in this question.

The math is all I am talking about. The logic of the math.

Nothing else really except where it cannot be avoided.

And I've finally come to where I see the explanation of the confusing math is right there in 'what cannot be avoided'.

Why does 'he' say x^' = ct^' ? Because he thinks he is stationary and therefore the distance ut doesn't exist for him.

That's the 'non mathematical but unavoidable' bit I was not seeing.

I claim I was confused by Shankar's repeated iterations of 'you say this' and 'I say that' and 'I dispute your contention and supply a fudge factor' and so on.

Providing a narrative and personalising the math to where I had it clear in my head that 'he' saw distance x as ut + x^' - i.e. was constantly aware of the existing of the distance ut.

So then when the math states that he says the distance x^' is ct^' I'm thinking why would he say that?

My fault of course. But I never claimed the math was wrong. Only that I didn't understand.

For people like me Shankar would do better to omit all reference to 'he sees' and so on.

 

[PeroK]

For me it is all about the math. My question is all about the math.

The math is all I am talking about. The logic of the math.

On the contrary, there is no mathematics in any of your posts.

Others have posted mathematics in this thread, which you have totally ignored.

 

[abrogard]

That's the kind of thing that I find so inexplicable and so depressing in these forums.

That's an ad hominem response. Impertinent. Unnecessary. Unhelpful. And wrong.

I'm was not asking anything in my last post - I was clarifying, I was explaining that I was purely asking about the maths.

I know what I'm asking about. I know the contents of my own mind. I know my motivations, goals, aims, desires.

What arrogant presumption. I'm fairly speechless. I expect better on these forums because of the general level of intelligence implied but apparently mathematical ability does not necessarily equate with common sense, politeness, or even normal comprehension.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

I know what I'm asking about.

Do you? In your post #13, you first say:

The math is all I am talking about. The logic of the math.

But later on in the same post, you say:

That's the 'non mathematical but unavoidable' bit I was not seeing.

So the actual issue that you needed to understand better had nothing to do with the math itself. It had to do with a "non mathematical but unavoidable" bit that you were not seeing.

What arrogant presumption. I'm fairly speechless. I expect better on these forums because of the general level of intelligence implied but apparently mathematical ability does not necessarily equate with common sense, politeness, or even normal comprehension.

I am not going to issue a warning for this because I can see how @PeroK's response was not helpful, particularly since in your post #13 you appeared to have figured out your issue.

However, please note that the attitude displayed in the quote just above would have gotten the thread closed even if you had not figured out your issue. If you feel that someone else's post violates forum rules (and we have forum rules about civility), the proper response is to use the Report button to bring it to the attention of the moderators. Please do not respond in kind in the thread. That only exacerbates the problem because now the moderators have to review two potentially offending posts instead of one.

This thread will remain closed.