# Distortion of space following a moving body

Is it possible for space to be distorted in the region trailing a moving body through space (i.e. in a region that is greater in volume or at least different in shape than the expected gravity well in space)?

If so, would the volume and shape of such a region be velocity dependent? I understand that rotating bodies can cause frame-dragging, so intuitively one suspects that similarly a body moving through space distorts the trailing space behind it. I've seen reference to "linear frame dragging" as implied by Einstein's general relativity, but have not found papers that discuss this effect.

Can someone direct me to papers that might discuss linear frame dragging?

## 2 Answers

There is no such linear frame dragging effect, and we can show this by a simple argument.

With a rotating object the frame dragging means the acceleration of a test mass does not always point towards the object. Suppose we let our test mass fall inwards on an initially radial trajectory. If the acceleration always pointed towards the object then mass would simply fall radially inwards. But frame dragging means the object is dragged sideways as well, so the acceleration is not purely radial.

The analogous effect for linear motion would be that again the acceleration of our test mass would not point directly towards the object, but would have some component in the direction of motion. But we can use a Lorentz tranformation to switch to the rest frame of the moving object. In this frame the object is stationary so the test mass comes hurting in, passes the object and goes hurtling out again. Since in this frame the object is stationary there is no frame dragging and the acceleration of the test mass always points towards the object.

But the Lorentz transformation is a linear transformation and that means in the original frame the acceleration must also always point towards the object. Therefore is no frame dragging in the original frame either.

Just as a large body with angular momentum will "drag" the spacetime around it, so will a large moving body with linear momentum drag the spacetime with it. A test mass falling into this warped spacetime will see itself as accelerating in the direction of the moving body, and the rest of the universe will necessarily appear to be accelerating in the opposite direction.

I don't know of any papers on the subject, although I'd also be interested I reading up on the subject. I have heard of this being referenced as "induction inertia" as well but can't find any verification of this.

• Thanks Connor! That matches my suspicion. Countto10 notes the idea is likely incorrect, though. He/she suggests to test the idea by tracking Cassini's orbit around Saturn. Reasonable, although it would seem to depend on the orbital path. If the orbit does not cross Saturn's trajectory, it may be outside the wake region, so such a wake may not be easily detected. c.f: physics.usyd.edu.au/~laszlo/kepek/anderson2008.pdfpaper for the anomalous earth flybys; paper says "the Earth's rotation may be generating an effect much larger than the frame dragging effect of General Relativity". – H. Trenchard Jul 26 '17 at 21:40
• The issue with that though is the Earth's angular frame dragging wasn't experimentally verified until '06, and any linear frame dragging would be much weaker and thus harder to measure. I.e. it's not likely to be found anytime soon. Why does Counto10 say it's likely to be incorrect? I'm curious only because I'd like to learn! – Connor Hayward Jul 26 '17 at 22:35
• Hi Connor. In the rest frame of the object it is not moving so there is obviously no dragging. The spacetime geometry is just Schwarzschild or something similar. But the metric is covariant so in a moving frame there is also no dragging, and so the effect you describe does not exist. – John Rennie Jul 27 '17 at 11:16
• Sorry Connor, I did not mean to be mysterious in any way, mainly because I was just absolutely guessing :) Obviously I have heard of frame dragging, but your (ingenious) idea did not feel correct. Errors in our predictions of say, the Trojan satellites of Jupiter, even with a tiny trace of this effect, might have lead it's discovery, but I had never once heard of or read about it. It's a moot point now, I think John's reasoning sorts it out. – user163104 Jul 27 '17 at 13:42
• Well, John, your reasoning seems sound, I must admit I'm too rusty to work through the metrics without a brush up first. Thanks for correcting me and clarifying. – Connor Hayward Jul 27 '17 at 21:47