Geodesics, free fall and the equivalence principle - for dummies.

[Robittybob1]

Geodesics, free fall and the equivalence principle.

 

How can I relate my Newtonian understanding of physics with the principles of general relativity?

 

Is there no such thing as centripetal force, inertia and gravitational potential energy anymore?

 

When I'm jumping in the air am I on a geodesic?

 

[swansont]

You only need invoke GR in situations where Newtonian gravity fails to adequately explain the behavior.

[Robittybob1]

You only need invoke GR in situations where Newtonian gravity fails to adequately explain the behavior.

Is there a speed that is the cut off point where you need to change over?

[swansont]

Is there a speed that is the cut off point where you need to change over?

 

Not that I'm aware of, for GR, other than the part that we call SR.

[Mordred]

To expand on Swansorts reply, there is no hard and fast cross over point.

 

Every equation in physics is a reasonable approximation. For every day applications on Earth Newtons laws are of a high enough degree of accuracy that we don't need GR nor SR.

 

Until you get closer to Mercury Keplers laws are accurate to match observational evidence.

 

While GR is more accurate, Newtonian math works fine.

 

For example do you need to know the rate of time at your head is different than at your feet when you won't notice the effect over a thousand years?

 

GR and SR doesn't ignore Newtonian physics such as centrifugal acceleration.

 

Rather it modifies the observer coordinate changes.

 

an oversimplified way of thinking of this is Newton assumed time was a constant. SR doesn't.

 

Without time dilation effects spacetime is just space as time isn't used as a vector coordinate.

 

This is what is termed Euclidean coordinates (geometrically it's equivalent to Cartesian coordinates, visualize a flat map.)

 

When you add the time component as not being constant but light being constant to all observers. The geometry of the coordinates of the map must change, (length contraction). The map becomes curved like a globe (polar coordinates).

 

For the Euclidean to polar coordinate rules Google Lorentz transformations.

 

If you use a Lorentz factor calculator and play with the numbers, you can a better feel for when you may wish to use GR or Newton.

 

( This is what I did when I first started learning GR)

 

 

Now the reason I asked you to start this thread, from a previous thread you asked a question on geodesics and acceleration.

 

Geodesics are specifically falling objects or objects in freefall.

 

Think of it this way if your a rocket or object that generates its own acceleration you can choose whatever path you want.

 

However a freefall object must follow the spacetime curvature geodesic.

 

I find myself posting these metrics repeatedly, but if one can follow them the important transformation rules from Euclidean to polar coordinates are detailed. (Newtonian to spacetime curvature

 

 

Lorentz transformation.

 

First two postulates.

 

1) the results of movement in different frames must be identical

2) light travels by a constant speed c in a vacuum in all frames.

 

Consider 2 linear axes x (moving with constant velocity and [latex]\acute{x}[/latex] (at rest) with x moving in constant velocity v in the positive [latex]\acute{x}[/latex] direction.

 

Time increments measured as a coordinate as dt and [latex]d\acute{t}[/latex] using two identical clocks. Neither [latex]dt,d\acute{t}[/latex] or [latex]dx,d\acute{x}[/latex] are invariant. They do not obey postulate 1.

A linear transformation between primed and unprimed coordinates above

in space time ds between two events is

[latex]ds^2=c^2t^2=c^2dt-dx^2=c^2\acute{t}^2-d\acute{x}^2[/latex]

 

Invoking speed of light postulate 2.

 

[latex]d\acute{x}=\gamma(dx-vdt), cd\acute{t}=\gamma cdt-\frac{dx}{c}[/latex]

 

Where [latex]\gamma=\frac{1}{\sqrt{1-(\frac{v}{c})^2}}[/latex]

 

Time dilation

dt=proper time ds=line element

 

since [latex]d\acute{t}^2=dt^2[/latex] is invariant.

 

an observer at rest records consecutive clock ticks seperated by space time interval [latex]dt=d\acute{t}[/latex] she receives clock ticks from the x direction separated by the time interval dt and the space interval dx=vdt.

 

[latex]dt=d\acute{t}^2=\sqrt{dt^2-\frac{dx^2}{c^2}}=\sqrt{1-(\frac{v}{c})^2}dt[/latex]

 

so the two inertial coordinate systems are related by the lorentz transformation

 

[latex]dt=\frac{d\acute{t}}{\sqrt{1-(\frac{v}{c})^2}}=\gamma d\acute{t}[/latex]

 

So the time interval dt is longer than interval [latex]d\acute{t}[/latex]

 

transformation rules.

Here is relativity of simultaneaty coordinate transformation in Lorentz.

 

[latex]\acute{t}=\frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}[/latex]

 

[latex]\acute{x}=\frac{x-vt}{\sqrt{1-v^2/c^2}}[/latex]

 

[latex]\acute{y}=y[/latex]

[latex]\acute{z}=z[/latex]

Edited February 24, 2016 by Mordred

[Robittybob1]

I know I'm a dummy when it comes to all that math but in my mind I think how does matter work all this out and do it at the speed of light. It seems too complicated.

I watched a 2 hour (I tried) video https://www.youtube.com/watch?v=pES_tNZJm3Q on "for beginners - Einstein Field Equations explained" and I'd say I'm not much more wiser.

I'll try again tomorrow.

Thanks for you input Mordred but it is going to be difficult. It was a good idea to look at "use(ing) a Lorentz factor calculator and play with the numbers" I'll definitely try that. I'll have to try something simple to begin with.

If I wanted to use relativity more often where do you start? For example how would we use it to look at the Earth orbiting the Sun?

[Mordred]

Here try these gifs

 

http://casa.colorado.edu/~ajsh/sr/wheel.html

http://www.physics.nyu.edu/~ts2/Animation/special_relativity.html#

The math above on a graph would look like this gif on the last link

 

Space-Time Diagrams of Events in Primed and Unprimed Frames Showing Grid

[studiot]

Geodesics, free fall and the equivalence principle - for dummies.

 

Perhaps it would be easier to understand if you separated the disciplines, Mathematics and Physics (Mechanics).

 

The study of geodesics is very firmly set in the realms of Geometry and the latter concepts in Physics.

I’m sure Mordred knows considerably more of this Physics than I so I will leave that side to him.

 

After tallying, geometry was the earliest form of mathematics and developed to a high standard well before algebra.

Geodesics were known semi precisely to the ancients.

More modern detail came with the development of cartography and surveying following the rennaissance;

Newton actually wrote about the motion of a body “In its right line”, a clear reference to the ancient definition of a geodesic.

From Gauss to the late Victorians, the subject developed, with applications in manufacturing in naval architecture, sheet metal working, later extended to paper and cardboard working.

The idea of a geodesic as a particular path through space or on a surface was formalised and the ruled surface much studied in industry.

Differential Geometry was introduced as a subject, followed by non linear geometry and new more esoteric geodesics were studied.

 

In my opinion, the best way to study geodesics for understanding in a modern setting is to start with the simple classical geodesics on a sphere and cylinder.

Then move on to look at geodesics on a geoid.

Then introduce a modern manifold (which the surfaces of spheres, cylinders and geoids are).

This route leads to better analogies than the usual picture of a ball on a trampoline (and explains that one too).

 

We can follow this route if you are interested.

[Robittybob1]

Geodesics,...

 

We can follow this route if you are interested.

I have quite a lot of understanding on the origin of geodesics but a quick recap from your perspective would be OK. I can soon tell you If I know it all ready.

[studiot]

 

Robbitybob1

I have quite a lot of understanding on the origin of geodesics but a quick recap from your perspective would be OK. I can soon tell you If I know it all ready.

 

 

I thought I provided a recap?

 

How about you tell us your understanding of what a geodesic is?

Perhaps I can learn something.

Edited February 24, 2016 by studiot

[Robittybob1]

 

I thought I provided a recap?

 

How about you tell us your understanding of what a geodesic is?

Perhaps I can learn something.

What I do struggle with is that we seem to need Newtonian formulas to make quick calculations yet we don't use the terminology that Newton would have used.

We have to describe the physics in terms of the Einsteinian spacetime model but use Newtonian formulas to calculate the parameters.

 

In future when I use Newtonian formulas my description will be Newtonian.