Help wanted with integration along a world line

[JohnFromAus]

Am doing problems at the end of a chapter on "Vector Analysis in SR"

Got stuck when asked to find the "elapsed proper time for the body as a function of t (Integrate d[math]\tau[/math] along its world line)"

 

The body concerned here is "uniformly accelerated" ie "its acceleration four vector has constant spacial direction and magnitude"

 

Ignoring y and z for the moment is the world line given by x = 1/2at^2? If so how do I get from there to [math]\tau[/math] as a function of t?

 

As v is increasing the [math]\gamma[/math] is of course a function of time too - hence the integration I think. Have already found the relationships for speed and distance as functions of time getting answers that agree with the book.

 

Am not really sure if this is the correct forum for this type of question but any help appreciated.

 

John

[swansont]

As v is increasing the [math]\gamma[/math] is of course a function of time too - hence the integration I think. Have already found the relationships for speed and distance as functions of time getting answers that agree with the book.

 

Right. For small v, you can use v = at.

 

If you analyze this in terms of the fractional change in frequency, you should end up with an expression that's ad/c^2 (acceleration x distance) which is exactly the same form as the gravitational term of gh/c^2. (Again, for small v, so you can use the binomial expansion for gamma)

[JohnFromAus]

Swansont - you are way ahead of me here.

I am still at the beginning of my GR self studies and wrestling with concepts.

What I need is :-

1 the ability to write down the equation of a world line for a "uniformly accelerated" body or any other world line come to that - is it just x = f(t)?

2 to understand what is meant by "integrate d[math]\tau[/math] along its world line"

 

Perhaps Im not even making sense with these questions! The next chapter I have is "Tensor Analysis in SR" and I dont think there is much point in trying to understand that if I am still mixed up with vectors etc.

 

Thanks for your reply - Im sure you have the answer I want if I can ever ask the right question!

John

[timo]

Do you know what dtau is? What expressions for dtau (as a function of other parameters) do you know?

[Obelix]

Am doing problems at the end of a chapter on "Vector Analysis in SR"

Got stuck when asked to find the "elapsed proper time for the body as a function of t (Integrate d[math]\tau[/math] along its world line)"

 

The body concerned here is "uniformly accelerated" ie "its acceleration four vector has constant spacial direction and magnitude"

 

Ignoring y and z for the moment is the world line given by x = 1/2at^2? If so how do I get from there to [math]\tau[/math] as a function of t?

 

As v is increasing the [math]\gamma[/math] is of course a function of time too - hence the integration I think. Have already found the relationships for speed and distance as functions of time getting answers that agree with the book.

 

Am not really sure if this is the correct forum for this type of question but any help appreciated.

 

John

 

The body's proper time is given by Minkowski's metric:

 

[math]d\tau^2 = dt^2 - \frac{1}{c^2}(dx^2+dy^2+dz^2)[/math]

 

The latter yields:

 

[math]d\tau = \sqrt{1 - \frac{1}{c^2}(\frac{dx^2}{dt^2}+\frac{dy^2}{dt^2}+\frac{dz^2}{dt^2})}dt=\sqrt{1 - \frac{\upsilon^2}{c^2}}dt[/math]

 

I understand you already have [math]\upsilon[/math] as a function of [math]t[/math], so all you need to do is integrate the above expression.

 

...The next chapter I have is "Tensor Analysis in SR" and I dont think there is much point in trying to understand that if I am still mixed up with vectors etc.

 

Thanks for your reply - Im sure you have the answer I want if I can ever ask the right question!

John

 

What book are you using? Who is the author?

[JohnFromAus]

Thanks Obelix - I just needed to go back to the definition of proper time and work from there! Will do that integration and see what comes of it.

 

But I am still confused by the "integrate [math]d\tau[/math] along its world line" Does it just mean "integrate [math]d\tau[/math] "? ( for that particular v )

 

I am working from "A first course in General Relativity" by B F Schutz. I have a couple of other books on order which have not arrived yet.

Relativity Demystified by D McMahon

Introducing Einsteins Relativity by D'Inverno

Vector and Tensor Amalysis by Borisenko and Tarapov

 

Am always happy to buy books - have got quite good at building bookshelves! So if anyone one has suggestions which they believe could help in my situation - self study - engineering background - B Sc - Dip Ed - maths lover ( mathamatophile?) - 350 kms from nearest uni ... I would be grateful.

 

John

[kevinalm]

Have you ever been to this website. Lots of free downloadable lecture notes you might enjoy.

 

http://www.phys.uu.nl/~thooft/theorist.html

[JohnFromAus]

Kevinalm

Thanks - looks like some helpful stuff there - especially problems with solutions

John

[Royston]

Am always happy to buy books - have got quite good at building bookshelves! So if anyone one has suggestions which they believe could help in my situation

 

These books were suggested by ajb, from another thread, where I was asking about the math behind GR, hope it helps...

 

http://www.scienceforums.net/forum/showpost.php?p=397022&postcount=9

[timo]

But I am still confused by the "integrate [math]d\tau[/math] along its world line" Does it just mean "integrate [math]d\tau[/math] "? ( for that particular v )

It means to calculate the "length" of the curve in spacetime (where length is the common name but problematic due to the metric not being positive definite). You can see this as calculating the line of a curve in R^n which is a map from an interval I in R on R^n: [math]\phi : I \to R^n, \phi(s) = (x^1(s), \dots , x^n(s))[/math]. The length of such a curve in R^n between the points s=a and s=b>a is calculated as [math] L = \int_a^b \| \frac{d\vec x}{ds}(s) \| ds [/math], i.e. by summing up the lengths of small changes.

 

Not much changes in your problem except that you must find a parameterisation for the curve and that the magnitude of a vector is a pseudo-magnitude:

1) A practical approach is to take t as the curve parameter in which case [math]\phi(t) = (1t, \vec x(t)) \Rightarrow \Delta \tau = \int_{t_i}^{t_f} \| \frac{d\phi}{dt}(t) \| \, dt[/math]. And [math]\frac{d\phi}{dt}(t) = (1,\vec v(t))[/math] is a 4-vector, a tangential vector on the world line (sidenote: Strictly speaking (t,x) is not a vector but a position - in many cases in SR it can be treated as if it was a vector, though).

2) The relativistic squared pseudo-magnitude of a vector v is [math] v^2 = v^i v_i = g_{ij} v^i v^j[/math] (imho writing "v²" for that is nasty but it's common convention). If v is the tangential vector of the world line of a massive object, then v²>0 (or <0, depending on the signature choice of the metric - ajb is the only one on sfn that uses <0, I think). You could therefore write [math] \| v \| = \sqrt{v^2} [/math]. Plugging that into the equation for the length above yields [math] \Delta \tau = \int_{t_i}^{t_f} \sqrt{1- \vec v^2/c^2} \, dt[/math].

 

So much for a practical calculation of the given problem. Now why is that called "integrate dtau"? Tau is the eigentime of an object. The eigentime of an object is simply the length of its world curve. Had you chosen tau as parameter for the curve then you had had [math] \Delta \tau = \int_{\tau_i}^{\tau_f} 1 \, dt[/math] which is quite a no-brainer but useless for the calculation here.

 

Please note that factors of c in above are not guaranteed to follow some consistent usage, especially not the convention in your book. The reason is that those factors do not carry any "structure" but only give a scale. At least in theoretical physics it is common to completely omit them and only add them at the very end of a calculation in case a number is desired as a result (and only if that number shall be SI compatible).

[JohnFromAus]

Well I am really grateful for all the help - perhaps the worlds not such a bad place after all!

Snail - thanks for the list - I will certainly investigate the suggestions

 

Atheist - am trying to follow your reply. I got the same result but possibly my reasoning is either wrong or at least not rigorous.

I did the following:-

 

[math] \delta t = \gamma\delta\tau \Rightarrow \delta\tau = \delta t / \gamma \Rightarrow = \sqrt{1- \vec v^2} \, dt[/math]

 

taking c as 1.

 

I already had [math] v = \alpha t /\sqrt{1+ \alpha^2t^2} [/math]

 

which I then substituted in the above and integrated giving me [math]\tau = tan^{-1} \alpha t [/math] which is not the given answer.

 

Its is years since I did any serious integration so I could have got that wrong - I am much more concerned with being sure my method is Ok than getting the "right" answer!

 

John

Looked at a table of integrals - now have [math] \tau = \alpha^{-1}sinh^{-1}(\alpha t)[/math]

[timo]

From x=at²/2 => v = at (assuming your v means dx/dt - mine does), not v = at/sqrt(1+a²t²).

[JohnFromAus]

Ummm - I think I may have confused the issue some what. I have attached a copy of the problem from the book.

The answers given are as follows

(b) [math]v = \alpha t /\sqrt{1+ \alpha^2t^2}[/math]

[math] \alpha = 1.1\ast10^{-16} m^{-1}[/math]

[math] t = 2.0\ast10^{17} m = 22 years[/math]

all of which I got and so assumed I was on the right track

 

© [math] v = tanh(\alpha\tau) [/math] ????

[math] x = \alpha^{-1}[(\sqrt{1+\alpha^2t^2)} - 1][/math]

10 years.

 

The first © answer does not seem to tie up with the question but I agreed with the formula for x.

I cannot find from where the formula for v came or get the final answer of 10 years.

 

I hope this puts my questions in a better context and allows you to get some idea of just how much of a beginner I am. The question interested me as it seemed to show that a body can have constant acceleration for ever and yet never exceed c.

 

John

[Obelix]

Thanks Obelix - I just needed to go back to the definition of proper time and work from there! Will do that integration and see what comes of it.

 

But I am still confused by the "integrate [math]d\tau[/math] along its world line" Does it just mean "integrate [math]d\tau[/math] "? ( for that particular v )

 

The worldline is likely to be given as a function of proper time: [math]\gamma(\tau)=(ct(\tau),x(\tau),y(\tau),z(\tau))[/math].

 

Likewise with 4 - velocity: [math]V(\tau)=(c\frac{dt}{d\tau},\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau})=(c,\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt})\frac{dt}{d\tau}=(c,\vec{\upsilon})\sqrt{1-\frac{\upsilon^2(t)}{c^2}},[/math] we throughout assume the coordinates and their derivatives as functions of proper time, i.e. as evolving along the specific world line - otherwise differentiating with respect to [math]\tau[/math] would be meaningless! The specific form of [math]\vec{\upsilon}(t)[/math] and [math]\upsilon^2(t),[/math] above, result accordingly from this specific functional form of [math]t(\tau),x(\tau), y(\tau), z(\tau)[/math]

 

Working with the Minkowski metric we obtain:

 

[math]d\tau^2=dt^2-\frac{1}{c^2}(dx^2+dy^2+dz^2)\Rightarrow\frac{d\tau}{dt}=\sqrt{1-\frac{\upsilon^2(t)}{c^2}}\Leftrightarrow\tau=\int\sqrt{1-\frac{\upsilon^2(t)}{c^2}}dt[/math]

 

Integration with respct to [math]\tau[/math] is thus performed along this specific world line, since [math]\upsilon(t)[/math] has resulted as above.

 

I am working from "A first course in General Relativity" by B F Schutz. I have a couple of other books on order which have not arrived yet.

Relativity Demystified by D McMahon

Introducing Einsteins Relativity by D'Inverno

Vector and Tensor Amalysis by Borisenko and Tarapov

 

Am always happy to buy books - have got quite good at building bookshelves! So if anyone one has suggestions which they believe could help in my situation - self study - engineering background - B Sc - Dip Ed - maths lover ( mathamatophile?) - 350 kms from nearest uni ... I would be grateful.

 

John

 

D' Inverno's book is good, and was recomended as an introductory text by prof. David Robinson of King's College/London, in his General Relativity class, spring semester, 1993. (Can you guess when I was there?) I haven't read the other two.

 

A good introductory text with a lot of examples could also be Wolfgang Rindler's "Relativity: Special General and Cosmological", Oxford University Press, 2nd. edition (2006), as well as his "Introduction to Special Relativity", Oxford University Press, 1991 (not sure if available now, but a good part of it is contained in the former).

 

For a more detailed study one can go on with D. F. Lawden's "An introduction to Tensor Calculus, Relativity and Cosmology", John Wiley (I have the 1986 reprinting, but there has been a quite recent edition) L. P. Hughston's and K. P. Tod's "An introduction to General Relativity", Cambridge University Press (1990 - not sure if available) and eventually, for an acquaintance with higher formalism: Hans Stephani's "Relativity", Cambridge University Press, 3rd edition (2004).

 

Finally, Robert Wald's "General Relativity", University of Chicago Press, 1984 (an excellent book, still widely availbale) can serve as an introduction to the highest mathematical formalism of General Relativity, like rigorous Differential Geometry and Topology.

 

I believe you can order any of that online, with a credit card (350 km from nearest Uni! Where are you man, middle of the desert???) or get information about titles in general.

 

Of course all the above titles are only indicative. The bibliography is VAST (I suppose there must be Australian editions too, unknown in Europe or America, just like there are some Greek books on the subject, this end of the line) and it is much better if you do your own digging, gaining experience!

 

Most authors include their institute, so you can contact them via e - mail in case of unanswered questions and/or doubts. Do not hesitate to do so: Many of them are happy to offer help. Not all though, and least of all the famous ones: It's practically impossible to get a reply, e.g., from Stephen Hawking or Roger Penrose. Quite understandable: They receive thousands of e - mails each day...