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Author Topic: Experts in GR: ways of calculating acceleration without tensors?  (Read 7503 times)

Lamprey5

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Experts in GR: ways of calculating acceleration without tensors?
« on: 13/12/2011 02:26:22 »
I'm doing a physics project about the following: according to special relativity, what appears as momentum to one observer can be inertial mass to another; i.e. mass is relative. But mass and energy curve spacetime. I've read (in Gravitation by Thorne, Wheeler, et al.) that the energy-momentum tensor is the source of gravity, and this determines the differential geometry of the spacetime region around mass and/or energy. However I'm in high school and would like to make calculations based on general relativity with some data I have collected , but I'm not sure what approximations I should make. I've tried to understand the "Newtonian Approximation", and Post-Newtonian formalism, but have not been successful (yet).

So my issue is this: When very massive objects speed up, their masses and gravitational fields increase. I want to calculate the effect of  their increasing mass due to kinetic energy increases on an object for which I have the precise trajectory data and distance from the massive objects as a function of time. I know that substituting the relativistic masses into the Newtonian gravity formula would be terribly wrong. Is my only option to learn tensors?

yor_on

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Experts in GR: ways of calculating acceleration without tensors?
« Reply #1 on: 13/12/2011 05:36:08 »
Nowadays physics don't define mass that way. As I understands it mass should not be defined relatively. It is either invariant mass, defined as being at rest, invariantly the same if moved to any other 'frame of reference', although its weight may change. Or it is 'moving' in which case you have the invariant mass, and its momentum.

There used to be another definition though, in which physics referred to a moving mass as 'relativistic mass' which then seems similar to what you refer to as inertial mass, invariant mass plus its momentum. In particle physics referred to as the moving particles 'total energy' (rest energy plus kinetic energy E = KE + mc2)

I think you will be well on your way if you read this Special Relativity and SLAC. It's quite nice.

And using total energy to define it makes sense.
=

You might want to look at What is relativistic mass? to see the discussion.
« Last Edit: 13/12/2011 06:00:30 by yor_on »

yor_on

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Experts in GR: ways of calculating acceleration without tensors?
« Reply #2 on: 14/12/2011 00:56:49 »
Another thing.

"in general relativity gravity is not really a `force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial. If you toss a ball, it follows a parabolic path. This is far from being a geodesic in space: space is curved by the Earth's gravitational field, but it is certainly not so curved as all that!

The point is that while the ball moves a short distance in space, it moves an enormous distance in time, since one second equals about 300,000 kilometers in units where c = 1. This allows a slight amount of spacetime curvature to have a noticeable effect."

That one takes some time adjusting too, and it kind of hurts my head to see what that means, but it makes sense if you think of it as SpaceTime. Then you need all four parameters (dimensions) to define a event, as well as a coordinate system tailored to your needs, usually from locality which then should be the local clock you use to define a time. So not only mass and 'energy', but 'time' too.

Pmb

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Re: Experts in GR: ways of calculating acceleration without tensors?
« Reply #3 on: 05/01/2012 21:17:43 »
I'm doing a physics project about the following: according to special relativity, what appears as momentum to one observer can be inertial mass to another; i.e. mass is relative.
That is related to the 4-momentum, P defined as follows

P = (mc, p)

where m is defined as the m = p/v where p is defined as p= |p| and v = |v| . You can find the justification for this definition in Concepts of Mass in Contemporary Physics and Philosophy, by Mas Jammer, Princeton University Press[/b] , 2000. The chapter 2 is Relativistic Mass. The derivation starts on page 49 wth Eq. (2.27) ends on page 50 with Eq. (2.33). For those interested I Can scan these pages into a PDF file and post it on my web site.
But mass and energy curve spacetime.
To be precise it is stress, energy and momentum which curve spacetime. Mass is hidden in there somewhere. You'd have to see the derivation and the results. E.g. with a star, in the equation of gravity, the active gravitational mass (source of gravity) is the sum of an energy desity term and a pressure term.
I've read (in Gravitation by Thorne, Wheeler, et al.) that the energy-momentum tensor is the source of gravity, and this determines the differential geometry of the spacetime region around mass and/or energy.
Take a closer look at the definition of that tensor. See http://home.comcast.net/~peter.m.brown/ref/relativistic_mass.htm uder Specific Instances. The first entry is a derivation of the T^0j terms. The relation

T^0j = (mass density) x (mean velocity of mass flow )j

where by mass density the authors mean relativistic mass density.

However I'm in high school and would like to make calculations based on general relativity with some data I have collected , but I'm not sure what approximations I should make. I've tried to understand the "Newtonian Approximation", and Post-Newtonian formalism, but have not been successful (yet).
So my issue is this: When very massive objects speed up, their masses and gravitational fields increase. I want to calculate the effect of  their increasing mass due to kinetic energy increases on an object for which I have the precise trajectory data and distance from the massive objects as a function of time.
That is an extremely difficult task to accoplish. It has to be taken one instance at a time. Here is an example of such a calculation. The article is

Measuring the active gravitational mass of a moving object, D.W. Olson and R.C. Guarino, Am. J. Phys. 53(7), July 1985

Quote
Abstract

If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that Mrel = g(1 + b)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not gM  but is 2gM .
The symbols are messed up due to me writing this in text. Look at the AJP article for details and correct abstract. Here is the URL to that article
http://home.comcast.net/~peter.m.brown/mass_articles/Olson_Guarino_1985.pdf
I know that substituting the relativistic masses into the Newtonian gravity formula would be terribly wrong. Is my only option to learn tensors?
I'd have to say yes. Sorry. Here are some tutorials I wrote on the subject of tensors.

http://home.comcast.net/~peter.m.brown/math_phy/tensors_via_analytic.htm
http://home.comcast.net/~peter.m.brown/math_phy/tensor_via_geometric.htm

See http://home.comcast.net/~peter.m.brown/math_phy/math_phy.htm if you have trouble.

Pmb

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Re: Experts in GR: ways of calculating acceleration without tensors?
« Reply #4 on: 07/01/2012 13:23:26 »
Nowadays physics don't define mass that way.
I disagree (of course :) ). It is incorrect to say that physicists don't define mass that way when it's clear that many do. This conclusion is based on these two papers and many more texts too

On the abuse and use of relativistic mass, Gary Oas, physics/0504110 v2, 21 Oct
2005

On the Use of Relativistic Mass in Various Published Works, Gary Oas, :physics/0504111 v1, February 2008

It may be you're opinion that it should never be used. However that is quite different than saying that it is never used.

Note: If you don't have a degree in physics, or an equivalent (e.g. self taught), then you might only be able to follow the abstract, introduction and summary/conclusion(s).

Here are three relativity texts which use rel-mass

Relativity: Special, General and Cosmological, Rindler, Oxford Univ., Press, (2001)

From Introducing Einstein's Relativity, Ray D'Inverno, Oxford Univ. Press, (1992),

Gravitation, Misner, Thorne and Wheeler, W.H. Freeman & Co., (1973), page 141

See textual published examples at http://home.comcast.net/~peter.m.brown/ref/relativistic_mass.htm

Pete
« Last Edit: 07/01/2012 14:53:31 by Pmb »

yor_on

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Re: Experts in GR: ways of calculating acceleration without tensors?
« Reply #5 on: 11/01/2012 16:45:29 »
Yeah, I know Pete :)

And it's all about definitions as it seems to me. And, I'm not saying that you can't use it your way, but to me it becomes simpler when just relating to the idea of a 'universally' defined 'invariant mass' 'rest mass' whatever. But the kinetic energy will be there in a collision, no matter how you define it..

So maybe, a matter of taste? I think that is the conclusion 'What is relativistic mass?' comes to too, as I remember it.

I'm actually not sure what matter is either :) which upps the difficulties enormously of defining 'anything' moving :)

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Re: Experts in GR: ways of calculating acceleration without tensors?
« Reply #5 on: 11/01/2012 16:45:29 »