When visualizing the relativistic effects of accelerating to the speed of light, or c, I can picture a long rocket ship, accelerating faster and faster as it approaches c.

I can imagine its relative length, as seen from an outside observer, in the direction of its velocity shortening to “compensate” for the observers aboard the ship always measuring the speed of light to be 186,200 mps.

I also understand that their relativistic mass would increase, in such that if they achieved c, it would appear that they have infinite mass.

Would their density somehow “compensate” for this as well, not unlike the relativistic shortening? Wouldn’t the ship appear to become an increasingly smaller object, until it appears to become a singularity upon reaching c?

Relativistic mass has fallen out of favour, but if we do use the concept of relativistic mass, the density of an object would appear to increase, with relativistic foreshortening aggravating this increase.

Relativistic mass, falling out of favor as it is, is there any thnking along the lines of mass and GR when accelerating close to c, or is it not really considered in the same way foreshortening is?

I suppose I’m just looking for some general enlightenment along these lines.

The problem is that the Lorentz equations involve the inverse of a ratio which, at the speed of light, equals zero. The inverse of zero is undefined.

This is why, most often, they say that a physical object can be accelerated to close to the speed of light, but never all the way to the speed of light.

The descriptive equations simply fail.

Now, who knows? In the real world, maybe the equations do break down very near the speed of light. I always think of the spring-constant rules, which describe the behavior of springs…and which fail, in the real world. In reality, when you stretch a spring too far, it simply deforms and loses its springiness. MAYBE something like that happens when you accelerate a proton to within some velocidy preposterously close to c.

Nobel prize waiting for the team that accomplishes it!

No matter how fast you go, you won’t collapse into a black hole, and that’s a good thing, because anyone and anything is going as fast as you like, from the point of view of some reference frame or another. Whether or not something is a black hole is a global and invariant property: If a thing is a black hole in one reference frame, it’s a black hole in every reference frame.

I don’t think it’s a case of QM and GR bumping into each other; I would have said that it’s almost entirely a GR issue. (Heck, an SR issue!) We can keep pumping energy into those particles, and add a few more nines to the ratio…

(Oh, gawd, no, don’t anyone start talking about an infinite number of nines! We’re DOOMED!)

Well, I suppose that’s why I asked the question I did in the OP. I had always imagined the foreshortening effect in SR, but these latest patch of threads concerning spacetime, GR, and the expansion of the universe got me into thinking about matter approaching c, and its mass climbing toward infinity under SR.

From there, I started to think about its ramifications in GR, and what physics has to say about it in that regard.

In thinking about the equivalence between acceleration and gravity, what makes the singularity formed by the gravity(mass) of a black hole different than matter traveling at (or near?) c?

One of the origins of the idea of the “black hole” was pre-Einsteinian. Someone (blast, I can’t recall…was it LaPlace?) simply reasoned on the basis of Newtonian escape velocity. The escape velocity from the surface of the earth is 7 miles per second. From Jupiter, higher. From the sun, higher yet. Simply extrapolating, you can work out the mass from which the escape velocity is the speed of light.

Einstein’s equations point in roughly the same direction. There is (or can be) a concentration of mass so dense that the escape velocity is the speed of light.

But the GR equations go further, and describe a mathematical singularity. Then Hawking and Penrose and all got hold of it and showed other, deeper implications.

The easy answer is that the equations actually permit such a concentration of matter, and, now, astronomers are pretty darn sure they’ve seen convincing evidence of it. An object travelling at the speed of light isn’t permitted under the equations, and (as you noted) the equations hold true up to speeds incredibly close to the speed of light.

As was mentioned in the thread about the age of the universe, it all comes down to the math.

I just looked at the Wikipedia article on tensors, and, lemme tell ya…I’m now more ignorant than before! I’m terribly afraid that I’ll never comprehend tensors.

As John Cleese said in Silverado, “Today, my jurisdiction ends here.”

Yep! Michell and Laplace. In the late 1700s the Newtonian thinking that light was made of particles was still prevalent, and they were the first to wonder if there could exist a star so massive, its escape velocity could keep even light itself from emitting; rendering the star dark.

Anyhow, I’m in the same boat. Calculus is over my head, let alone tensors. But most things, as such, can be laid out in a metaphor or three, for some of us to at least get an idea.

My gut tells me that matter accelerated to c is different than matter forming a gravitational singularity due to some form of symmetry breaking or non-uniformity (?), but I can’t really find my way out of the dark on this one.

Well, the first thing to realize is that mass isn’t what causes gravity. Or rather, mass isn’t entirely what causes gravity. Mass is just one component of a tensor (you can think of it as a 4x4 matrix) called the stress-energy tensor, and that’s what causes gravity. Now, under ordinary circumstances, mass is by far the dominant component, so it’s usually good enough to just ignore all the others. But a particle flying along at .9999999c isn’t ordinary circumstances, and there, you have to pay attention to some of the other components, as well.

Because of the type of work I do, I am familiar somewhat with matrices and transformations thereof.

It looks like pseudotensors are used when working with GR and the equivalence principle (as the stress-energy of gravity reduces to zero, if not). Any way to expound?

Well, in just trying to familiarize myself with stress-energy tensors, I found this snippet in Wikipedia.

I work in pretty advanced CG/3D, so geometry and topology across pretty simplistic time have sort of become my window into seeing GR a bit better, despite my ignorance of 90% of the math involved (or so I’d like to think).

Sometimes it’s necessary to mess with local or global matrices of the scene’s Cartesian space, and/or transform them for certain purposes.

Also, I can even zero-out local transformations by what’s called “freezing” them, so it keeps two sets of matrices for an object, one with all the location and time information according to global space, and one I work with where all that information is set to zero, so it makes performing transformations much easier. I wonder if that’s akin to pseudotensors (or if all of this isn’t even in the same territory).

OK, I’m a bit out of practice, but I think that the reason that Wikipedia uses the term “pseudo-tensor” there is that in many contexts, we’re only able to do the calculations perturbatively, and hence can’t deal with the energy stored in the gravitational field itself. In a gravitational wave, for instance, the true geometry of space is changing, but since gravitational waves are in practice extremely weak unless you’re extremely close to the source, we generally treat them as if there were a real “background space” that’s just flat, through which the tensor field which makes up the wave is passing. But since there isn’t any real background, this perturbation tensor isn’t real, either, just a calculational convenience.