[Colin2B]

I am not convinced about the veracity of the above equation.

Now much of the deadwood has been pruned it’s easier to see the trees.

Question I have about this    is what do you do about which changes with velocity and hence with time so do you need to include  on the rhs ?

[jeffreyH (OP)]

If you look at the equation for the Lorentz factor above then I would say yes that is likely to be the case.

[jeffreyH (OP)]

I am not convinced about the veracity of the above equation.

Now much of the deadwood has been pruned it’s easier to see the trees.

Question I have about this    is what do you do about which changes with velocity and hence with time so do you need to include  on the rhs ?

That results in which can be rearranged as . Now we are into homogeneous first order differential equation territory.

[jeffreyH (OP)]

Now I've just got to lift and separate. Er, sorry, separate and integrate.

[PmbPhy]

I think not but it would be interesting to get other views on the subject.

Yes. All forms of energy can create a gravitational field. Its whatever the T^00 component of the stress-energy-momentum consists of. Didn't I ever mention that the energy of a photon or a directed beam of light is all kinetic energy and that light can create a gravitational  field? This was done way back in 1931 by Tolman, Ehrenfest and Podolsky in the article

On The Gravitational Field Produced by Light, Tolman, Ehrenfest and Podolsky, Physical Review, Vol. (37), March 1, 1931, pg 602-615.

I placed it online in my own words at:
http://www.newenglandphysics.org/physics_world/gr/grav_light.htm

[jeffreyH (OP)]

Thanks Pete. I'll have a read through.

[jeffreyH (OP)]

Putting the equation through Wolfram Alpha gives the following 2 solutions.

[jeffreyH (OP)]

Determining c₁?

[jeffreyH (OP)]

Start with

[jeffreyH (OP)]

I haven't even checked the dimensions!

[Colin2B]

That results in which can be rearranged as . Now we are into homogeneous first order differential equation territory.

What I was thinking was to move  from under F and move it to rhs
So - 

What do you think?

[jeffreyH (OP)]

That results in which can be rearranged as . Now we are into homogeneous first order differential equation territory.

What I was thinking was to move  from under F and move it to rhs
So - 

What do you think?

I can give that a go.

[jeffreyH (OP)]

The conclusion I am coming to is that what is needed is a non homogeneous differential equation with a periodic force. So we basically introduce a wave function and apply the Lorentz factor. I have seen this method used without gamma. I have to think about how to introduce gamma. This would relate very well to both force and time dilation via the wave function.

[jeffreyH (OP)]

I have found one discussion on differentiating the Lorentz factor with respect to time. Pretty short.
https://www.physicsforums.com/threads/differentiating-lorentz-factor-with-respect-to-time.153702/

[Colin2B]

Sorry, what I posted doesn’t make sense, I was too focused on gamma vs time.
Try

[jeffreyH (OP)]

Well if we use the product rule on (dp)/(dt) we get m*(dv/dt)+v*(dm/dt). Then it is just a case of applying this to F=d(pγ)/(dt).

[jeffreyH (OP)]

The Lorentz factor is a nightmare here. I am looking at things like the four gradient at the moment. The gradient dγ/dβ works out as c²/[vc-v²]. This may work for the changing force. I have to double check. Here gamma is the Lorentz factor and beta is v/c.

[jeffreyH (OP)]

We can see that mass flow rate is also part of the overall solution via dm/dt.
https://en.wikipedia.org/wiki/Mass_flow_rate

[jeffreyH (OP)]

So what do we have?

[jeffreyH (OP)]

Now all I have to do is find a transform to convert the force in the primed frame to the force seen be the stationary observer. Then I can think about the possible relationship to gravity.