Relativistic Energy in an Inverse Square Field: The Impact of Velocity

[granpa]

for an inverse square field the force is proportional to 1/r^2

obviously we integrate over distance to get energy ≡ 1/r (where energy = 0 at infinity)

but what happens when velocity becomes relativistic?
is relativistic energy proportional to 1/r?if its any easier what I am really looking for is gamma as a function of r for an inverse square field and gamma = 1 at infinity

 

[BruceW]

Yes, if the force is proportional to 1/r^2, then the energy of the particle is proportional to 1/r. (Even in relativity).

 

[granpa]

wikipedia says that

KE = m₀(gamma - 1)

so

gamma = KE + 1 = 1/r + 1

 

[Dale]

gamma = KE + 1 = 1/r + 1

Huh? How did you get this? Gamma is not a function of position.

 

[granpa]

its for an inverse square field where gamma = 1 at infinity

I'm assuming that KE = 1/r

 

[BruceW]

wikipedia says that

KE = m₀(gamma - 1)

so

gamma = KE + 1 = 1/r + 1

Wikipedia are using the convention of setting c=1. If you want to include c, then the equation is:
$$KE = m_0c^2(\gamma - 1)$$
So then you'd get:
$$\gamma = \frac{KE}{m_0c^2} + 1$$
And if we say gamma=1 when r=infinity, then KE=0 at r=infinity.
We know 1/r gives the change in energy of the particle. And assuming the rest mass of the particle doesn't change, then 1/r is proportional to the KE.

So then we have
$$\gamma = \alpha \frac{1}{r} + 1$$
Where I've left in the alpha as a constant of proportionality, since there is the rest mass, strength of the force and the speed of light which are all constants that must be taken in.
(So yes, I agree, as long as the constants are kept in, which also provide the correct dimensions)

 

[Dale]

I'm assuming that KE = 1/r

Oh, ok.

 

[granpa]

I've been told that the gravitational time dilation at any point in a gravitational field is equal to the time dilation that a particle falling from infinity to that point would experience due to its velocity alone.

Is that right?

 

[BruceW]

I'm not sure, since I don't know much about general relativity.

According to wikipedia, the gravitational time dilation of an object at rest in the vicinity of a non-rotating massive spherically-symmetric object is:
$$\frac{1}{1-\frac{r_0}{r}}$$
Where r₀ is the Schwarzschild radius of the massive object.

 

[granpa]

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

where T is the time interval measured by a clock far away from the mass

since

 

[Parlyne]

I've been told that the gravitational time dilation at any point in a gravitational field is equal to the time dilation that a particle falling from infinity to that point would experience due to its velocity alone.

Is that right?

Note that, if you want to talk about gravity in a relativistic manner, the 1/r potential is no longer a correct description.