On 2/23/2014 8:41 AM, Jesse Mazer wrote:
On Sat, Feb 22, 2014 at 7:31 PM, meekerdb <[email protected] <mailto:[email protected]>> wrote:On 2/22/2014 3:43 PM, Jesse Mazer wrote:On Sat, Feb 22, 2014 at 6:34 PM, meekerdb <[email protected] <mailto:[email protected]>> wrote: On 2/22/2014 3:22 PM, Jesse Mazer wrote:On Sat, Feb 22, 2014 at 3:37 PM, Edgar L. Owen <[email protected] <mailto:[email protected]>> wrote: Jesse, But from the links you yourself provide: http://adsabs.harvard.edu/abs/1985AmJPh..53..661O To quote from the 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 increase 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=γ(1+β^2)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not γM but is approximately 2γM. So this reference from the Harvard physics dept. says that the active gravitational mass of a relativistically moving particle DOES INCREASE and has a stronger gravitational attraction to what it is moving relative to. So that seems to contradict your own conclusion. How so? Clearly from Harvard, the increased mass (relativistic mass) of a moving object DOES have an increased gravitational attraction. So since gravitational attraction is due to curvature of spacetime one can say that from the POV (the frame) of the stationary observer, the moving object must be curving spacetime more. I don't believe there is any rule which says that "gravitational attraction" as they quantify it in the paper is proportional to any simple measure of the "amount" of spacetime curvature, and if there isn't then you can't say that a greater attraction in this sense implies "curving spacetime more". I imagine the the attraction depends on the way in which the curvature tensor varies at different points along the object's path through spacetime.Note that the Schwarzschild metric (or any other metric) around a moving gravitating body becomes shortened in the direction of travel by the Lorentz contraction. So from the standpoint of a stationary test mass the field is stronger but of shorter duration as the gravitating body moves past, so it curves spacetime more. What do you mean by "curves spacetime more", though? Isn't the curvature of spacetime defined in a coordinate-invariant way in general relativity, in terms of the metric which gives the proper time or proper length of arbitrary timelike or spacelike paths through that spacetime? Are you talking about some specific coordinate-dependent quantity, and if so is it a scalar or a tensor?It would be coordinate frame dependent, like clock rates in SR. The tensor curvature is an invariant.OK, so when you said "from the standpoint of a stationary test mass the field is stronger but of shorter duration as the gravitating body moves past, so it curves spacetime more", were you thinking of a specific coordinate-dependent quantity that would in some sense measure the degree of curvature, such that this quantity would be larger measured in some coordinate system where the gravitating body is moving than it is in a coordinate system where it's at rest? If not I don't really understand what it could mean to say that the body "curves spacetime more" from one observer's perspective than another's.
You're right, that's rather lose language. I should have said something like the peak strength of the apparent gravitational field would be higher, but it's duration shorter. In the moving frame some components of the curvature tensor would larger and some smaller, but it would just be a Lorentz transform so the tensor would be the same.
This does raise a question in my mind though. If we had two oppositely charged classical particles orbiting one another, they have more mass-energy than if they were stationary. So the extra mass-energy due to their motion must show up as gravitating mass. Is this just because there's no Lorentz frame in which the motion can be zeroed?
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