Relativistic Mass thought exercise with approaching bodies

[Egregious]

"Relativistic Mass" thought exercise with approaching bodies

I understand that the concept of 'relativistic mass' is somewhat controversial, but for reasons that are unclear to me. Nevertheless, I humbly pose the following thought exercise/inquiry:

For starters, I understand the formula for 'relativistic mass' to be:
M=m₀(1 - v²/c²)^-1/2

Knowing that there is no such thing as a 'resting' inertial frame, I understand the above equation to be a calculation of the mass of an object moving at velocity v relative to another frame.

My thought exercise is as follows:

I am sitting in a spacecraft that is parked 10 meters away and parallel to a conveniently placed length of massless red yarn that extends in a straight line ahead and behind my location for a distance of several light years. As I am parked in this location, I notice an identical craft in the distance approaching my direction in a path parallel to the yarn and 10 meters on the other side of it at a velocity of say: c*(1-10^-10). I know from the relativistic equation above that the approaching craft has tremendous mass and due to its proportional gravitational attraction, I will surely accelerate across the yarn to collide with it. The occupant of the oncoming craft perceives that he is parked and that I am approaching at the velocity stated.

My question is this: Which craft will accelerate across the yarn to collide with the other? If the other craft should drift toward me, I will be perplexed that this massive object would change course as a result of the gravitational force of my relatively tiny mass. The occupant of the other craft would be equally puzzled if I were to accelerate in his direction across the yarn.

Many thanks for your thoughts...

 

[sweet springs]

Hi. Thanks for an interesting problem.

As the third point of view or reference system, the both spacecraft are moving at the same speed but opposite direction, so they are mutually approaching and then leaving.

I do not think they collide because the relative velocity is much larger than the escape velocity. In case they are at rest initially, they will collide.

Regards.

 

[elfmotat]

Mixing relativistic mass with gravity isn't a good idea. It's a recipe for misconceptions.

Anyway, due to the symmetry of the problem both ships must accelerate toward each other equally.

 

[sweet springs]

Hi, Egregious.

If the other craft should drift toward me, I will be perplexed that this massive object would change course as a result of the gravitational force of my relatively tiny mass.

So when you do sky-diving, you are perplexed that this massive object, Earth, would fall on you as a result of the gravitational force of your relatively tiny mass, aren't you?

 

[PeterDonis]

My question is this: Which craft will accelerate across the yarn to collide with the other?

Before even considering this question, you need to first answer another question: if the two objects were sitting at rest relative to each other, would they exert any significant gravitational force on each other? In other words, would they accelerate appreciably towards each other as a result of each other's gravity?

I suspect that in your scenario you intended the answer to this question to be "no". If it is, then the answer to the question quoted above is "Neither craft will accelerate significantly at all." If two objects do not exert appreciable gravitational attraction on each other when at mutual rest, then they still won't if they are given a large relative velocity.

 

[bahamagreen]

This problem needs a few things sorted out before looking at what happens.

The massless red yarn seems to be serving to mark an absolute line in space shared in common by the different reference frames of both craft...
If the yarn is red, it it already emitting light... which would be subject to gravitational influence; maybe you could just replace it with a laser beam?

How would you notice the other craft in the distance approaching at that speed? We might assume that you and the other craft are making this maneuver based on prearranged flight plans, but be aware that because of mutual time dilation from each other's frame of reference, this flight plan may have to have been pretty complicated. Or the agreement could be for the other craft to just plan to show up whenever and pass you by whenever at some non-predetermined time for you.

The mechanics of the collision appears to either assume that the gravitational influence due to the mass increase precedes the proximity of the other craft or that the duration of passage at proximity is long enough to establish sufficient influence to collide. Acceleration into a parabolic orbit (swinging around behind the other craft from each perspective) seems more likely.

 

[harrylin]

Hi welcome to physicsforums!
Just adding my 2cts:

I understand that the concept of 'relativistic mass' is somewhat controversial, but for reasons that are unclear to me.

That's normal as the reasons have to do with taste -which is personal- and usefulness, depending on the type of problem. In particular, it's not so useful for problems that involve gravitation.

[..] I understand the above equation to be a calculation of the mass of an object moving at velocity v relative to another frame.

Relative to another inertial frame, yes.

[..] due to its proportional gravitational attraction, I will surely accelerate across the yarn to collide with it. [..] the gravitational force of my relatively tiny mass. [..]

That's just the kind of problem for which "relativistic mass" doesn't provide a useful shortcut; in general relativity the equations are more complex (even the approximate ones), with active and passive mass. However, the symmetry approach is of course valid (posts 2 and 3).

 

[jartsa]

Let's try to answer the question: why am I, with my relatively tiny mass, not accelerating very much?

Let's say I shoot a laser beam into the gravity field of the approaching spacecraft .

The deflection of light in a gravity field can be explained by saying that light chooses the fastest path, or by saying that one side of the EM-wave is slowed down more than the other, which causes the wave fronts to turn.

In this case the gravity field is Lorentz-contracted.

Let's say a glass ball is approaching at speed 0.99999999999999999999 c. It's like a window pane, because of length contraction. Light doesn't deflect much going through a window pane.

Length contraction of gravity field seems to be the answer to our question.

 

[Egregious]

Thanks all for the many replies.

This has been insightful and your feedback has given me much to ponder..

 

[pervect]

Mixing relativistic mass with gravity isn't a good idea. It's a recipe for misconceptions.

Specifically, people seem to think for some reason that you can use Newtonian formulas for gravity , and plug "relativistic mass" into them and come up with correct answers

The first thing you should understand about relativistic mass is that it's not the source of gravity in GR. In GR, the source of gravity is the stress energy tensor.

Anyway, due to the symmetry of the problem both ships must accelerate toward each other equally.

Yep. Optimistically, I like to think that the motivation for his question was the OP noticing that you didn't get sensisble results by plugging relativistic mass into Newtonian formulas. In which case it's a simpe matter of agreeing "don't do that".

I"m a bit concerned that they'll flail around trying to continue to re-invent gravity n terms of outdated concepts, rather than take the effort to study GR and find out what it's REALLY about and how it is actually formulated, rather than trying to invent something fictional based mostly on half-remembered high school physics.

 

[zonde]

My thought exercise is as follows:

I am sitting in a spacecraft that is parked 10 meters away and parallel to a conveniently placed length of massless red yarn that extends in a straight line ahead and behind my location for a distance of several light years. As I am parked in this location, I notice an identical craft in the distance approaching my direction in a path parallel to the yarn and 10 meters on the other side of it at a velocity of say: c*(1-10^-10). I know from the relativistic equation above that the approaching craft has tremendous mass and due to its proportional gravitational attraction, I will surely accelerate across the yarn to collide with it. The occupant of the oncoming craft perceives that he is parked and that I am approaching at the velocity stated.

There is some ambiguity in your formulation of exercise. Just because yarn is massless does not mean that it is not attracted toward gravitating mass. So it would be nice if you would state explicitly the speed of yarn. Assuming that yarn is at rest relative to you then it does not seem very realistic that approaching craft can move parallel to the yarn as it will be attracted toward craft.