Geometry of Space-Time, Mass & Speed: Higgs Boson

[nomisrosen]

How does moving at near the speed of light affect the geometry of space-time? How does an object increase in mass in relation to its speed? Does this have to with more collisions with the theoretical Higgs boson?

 

[JesseM]

Only the "relativistic mass" of an object increases as its speed increases, not the "rest mass", and the effect is frame-dependent (motion in relativity can only be defined relative to a reference frame, there is no objective sense in which one object is moving closer to the speed of light than another). Physicists nowadays tend to avoid talking about "relativistic mass" and just analyze things in terms of other concepts like momentum and energy, the relativistic mass idea has a lot of potential to mislead people, see this thread for a discussion.

 

[Naty1]

Movement tends to curve spacetime because movement implies energy and energy, along with mass, surves spacetime...that is, has gravitational effects. Even light which has no mass. Certain things curve spacetime according to GR...energy, momentum flow, mass, pressure, etc...

Some overview, no math, here:

http://en.wikipedia.org/wiki/Einstein_field_equations

Sometimes those things appear static in a given reference frame, other times they move. Since a faster object has more energy than a static one relative to a given frame, such a faster object would cause more gravity...more spacetime curvature.

You can get a rough idea from this diagram:
http://www.relativitet.se/spacetime1.html

Note the comment regarding slope...as you move more thru space, vertical, faster, you move slower through time, horizontal.

 

[Passionflower]

How does moving at near the speed of light affect the geometry of space-time?

That is a very complicated question and I do not believe there is a general answer to that question.

But moving masses definitely influence the geometry of spacetime.

 

[JesseM]

Sometimes those things appear static in a given reference frame, other times they move. Since a faster object has more energy than a static one relative to a given frame, such a faster object would cause more gravity...more spacetime curvature.

For more on the issue of the gravitational field felt from a moving mass, see [post=3294448]this post from pervect[/post]. I'm not actually sure it would be right to say "more spacetime curvature" though--how are you quantifying curvature, and what coordinate system are you using? In an asymptotically flat spacetime I imagine we'd be able to have some sort of "approximately" inertial coordinate systems, and in this case for what you're saying to be true, it would have to be the case that the spacetime curvature is in some sense "greater" in the frame where the object is moving than the one where it's at rest, even though we're talking about the exact same spacetime geometry expressed in different coordinate systems...it's quite possible that would be true under the appropriate definitions, but unless you know of a source that spells this out mathematically I don't think you can just trust intuitions here.

 

[PAllen]

Movement tends to curve spacetime because movement implies energy and energy, along with mass, surves spacetime...that is, has gravitational effects. Even light which has no mass. Certain things curve spacetime according to GR...energy, momentum flow, mass, pressure, etc...

Some overview, no math, here:

http://en.wikipedia.org/wiki/Einstein_field_equations

Sometimes those things appear static in a given reference frame, other times they move. Since a faster object has more energy than a static one relative to a given frame, such a faster object would cause more gravity...more spacetime curvature.

You can get an idea from this diagram:
http://www.relativitet.se/spacetime1.html

I partly disagree with this discussion. An object moving near the speed of light, inertially, has no different influence on spacetime geometry than a 'stationary' object. This is obvious because speed is frame dependent while geometry is not. The geometry can be analyzed in the object's frame, in which case it is stationary, and all geometric conclusions will hold in any other frame.

It is true that a system of fast moving particles will have more gravitational mass that a system of similar slow moving particles. However, I think the original post question is better covered by the prior paragraph. (The apparent contradiction is resolved by noting that for such a system if particles moving rapidly in different directions, all frames will show a system of rapidly moving particles; which is moving which way will change, but you can't transform away the fact that total KE of the system is greater than for a slow moving system of similar particles).

 

[nomisrosen]

Movement tends to curve spacetime because movement implies energy and energy, along with mass, surves spacetime...that is, has gravitational effects. Even light which has no mass. Certain things curve spacetime according to GR...energy, momentum flow, mass, pressure, etc...

What would happen if you tried to apply more energy to a photon? Would it develop a mass without having had one to begin with?

 

[PAllen]

For more on the issue of the gravitational field felt from a moving mass, see [post=3294448]this post from pervect[/post]. I'm not actually sure it would be right to say "more spacetime curvature" though--how are you quantifying curvature, and what coordinate system are you using? In an asymptotically flat spacetime I imagine we'd be able to have some sort of "approximately" inertial coordinate systems, and in this case for what you're saying to be true, it would have to be the case that the spacetime curvature is in some sense "greater" in the frame where the object is moving than the one where it's at rest, even though we're talking about the exact same spacetime geometry expressed in different coordinate systems...it's quite possible that would be true under the appropriate definitions, but unless you know of a source that spells this out mathematically I don't think you can just trust intuitions here.

Note that the invariant thing here is two bodies in very rapid motion relative to each versus two similar bodies with slower relative motion. If one is asking about the the effect of rapidly moving large body on a small 'test body', it suffices to analyze near-null geodesics in the large body's static geometry. Then transform to any other coordinates you want.

 

[PAllen]

What would happen if you tried to apply more energy to a photon? Would it develop a mass without having had one to begin with?

A photon can have any energy, while still having rest mass of zero. If you move rapidly toward a light source, you will measure higher photon energies than an observer comoving with the source. This is just the quantum statement of the doppler effect - the light will look blue shifted, and higher frequency light (bluer), has higher energy photons.

Further, a 'box of blue light' will, indeed, have more effective gravitational mass than a 'box of red light' (with the same number of photons of each).

 

[Naty1]

PAllen:

I partly disagree with this discussion...This is obvious because speed is frame dependent while geometry is not.

(It's a small miracle I got even part right!)

how to describe the curvature?

Jesse:

...how are you quantifying curvature, and what coordinate system are you using?

that's too advaced for me to address...any one you like!

but I get what you are saying...

Can you and PAllen maybe address the original question referencing the diagram I posted: Can that be better used to answer the curvature question, even if only basically, than I attempted?

http://www.relativitet.se/spacetime1.html

Using that diagram: As one moves faster in space (velocity) one moves slower in time, right? That's one change in geometry. Also, there has got to be some frame(s) in which gravity increases since energy does. So somehow curvature would seem to need to change: in the simple diagram, extra velocity increase the slope...the green line goes up faster than the red...does the geometry remain fixed?? Can we "visualize" somehow the gravittational (curvature) change?

will read pervect's post and article tomorrow...thanks, had not seen that...

 

[Passionflower]

I partly disagree with this discussion. An object moving near the speed of light, inertially, has no different influence on spacetime geometry than a 'stationary' object. This is obvious because speed is frame dependent while geometry is not. The geometry can be analyzed in the object's frame, in which case it is stationary, and all geometric conclusions will hold in any other frame.

I beg to differ.

For instance two masses moving relative to each other create gravitational waves, and one factor is the relative speed between the two masses.

 

[PAllen]

I beg to differ.

For instance two masses moving relative to each other create gravitational waves, and one factor is the relative speed between the two masses.

In no way does that disagree with what I said. The OP referred to a single object in isolation. In response to JesseM I noted that what is invariant is the relative motion of two bodies, *not* the the purported speed of either object in 'empty space'.

 

[PAllen]

PAllen:

Using that diagram: As one moves faster in space (velocity) one moves slower in time, right? That's one change in geometry. Also, there has got to be some frame(s) in which gravity increases since energy does. So somehow curvature would seem to need to change: in the simple diagram, extra velocity increase the slope...the green line goes up faster than the red...does the geometry remain fixed?? Can we "visualize" somehow the gravittational (curvature) change?

By definition, the geometry of spacetime is independent of coordinates or the motion of 'small test bodies' (defined as objects too small to significantly perturb the background geometry).

The expression of curvature is coordinate dependent, but the curvature tensor as a geometric object is coordinate independent.

Moving faster does not change geometry. Do you think moving near lighspeed in flat Minkowski space changes geometry from flat to something else? Well the same goes for moving fast in some non-flat background geometry.

If you transform the Schwarzschild metric to the Fermi-normal coordinates of a small body moving very fast in the Schwarzschild coordinates, it will look very different. But that is just a coordinate change - there is no change in geometry.

 

[Passionflower]

By definition, the geometry of spacetime is independent of coordinates or the motion of 'small test bodies' (defined as objects too small to significantly perturb the background geometry).

Perhaps we first should ask the poster if he is talking about test bodies with insignificant mass or massive bodies.

 

[nomisrosen]

Perhaps we first should ask the poster if he is talking about test bodies with insignificant mass or massive bodies.

Let's say a body with insignificant mass, like an electron, for example.
Could you theoretically apply so much energy to the speeding electron that it would become heavy enough to warp space-time infinitely, like a miniature black hole?

 

[PAllen]

Perhaps we first should ask the poster if he is talking about test bodies with insignificant mass or massive bodies.

The original post mentioned only one object moving fast. The question was whether the 'relativistic' mass viewed by an observer for whom the object was moving fast resulted in more curvature. There is no ambiguity here. If the OP wanted to talk about some massive observer, they would have mentioned that. As for a single massive object, its relativistic mass is a coordinate effect and has no impact at all on the geometry - you can just switch to a frame where the massive object is stationary.

 

[PAllen]

Let's say a body with insignificant mass, like an electron, for example.
Could you theoretically apply so much energy to the speeding electron that it would become heavy enough to warp space-time infinitely, like a miniature black hole?

Actually, to get any complication you need two bodies of 'similar' mass moving at near lightspeed relative to each other; this will be different from similar bodies moving at e.g. 25% lightspeed relative to each other. For one body, however massive, viewed by some low mass instrument, its speed will be not affect how it perturbs the geometry. As I keep saying, you can do all the analysis in the 'moving' body's rest frame, noting that geometry is coordinate and frame independent. You can never get a black hole by virtue of near lightspeed motion.

Now it is true that a co-moving instrument will make different measurements that one for which the body is moving very fast. However, a better way to look at this is again in the frame at which the body is at rest, where, for spherically symmetric non-rotating objects you just have Schwarzschild geometry. Then, instruments moving slowly compared to the object in its rest frame will make different measurements than instruments moving near lightspeed. But never will they see a black hole. This way of looking at it helps to see that the intrinsic curvature produced by a body is independent of its state of motion, as long is we are talking about uniform motion.

 

[nomisrosen]

I appreciate the reply but you lost me after a few words! I'm guessing the answer to my question is no :) Thanks anyways!

 

[PAllen]

For more on the issue of the gravitational field felt from a moving mass, see [post=3294448]this post from pervect[/post]. I'm not actually sure it would be right to say "more spacetime curvature" though--how are you quantifying curvature, and what coordinate system are you using? In an asymptotically flat spacetime I imagine we'd be able to have some sort of "approximately" inertial coordinate systems, and in this case for what you're saying to be true, it would have to be the case that the spacetime curvature is in some sense "greater" in the frame where the object is moving than the one where it's at rest, even though we're talking about the exact same spacetime geometry expressed in different coordinate systems...it's quite possible that would be true under the appropriate definitions, but unless you know of a source that spells this out mathematically I don't think you can just trust intuitions here.

Looking over the references provided by Pervect, I want to comment on different ways of looking at this.

For comparison, note that relativistic mass arises from treating relativistic momentum with the Newtonian formula p = mv. This relativistic mass does not generalize to f = ma, demonstrating its limited utility (but it is not necessarily useless).

Similarly, the paper referenced takes a static vacuum solution, transforms it to coordinates representing high speed observer, and attempts to interpret the results in a Newtonian way. While I don't consider this endeavor useless, it seems to me that both the definition of the coordinates (especially their extension near the massive source), and the Newtonian interpretation are not unique. I would put the results on the same footing as relativistic mass - a sometimes useful Newtonian analog.

Meanwhile, none of this is necessary to compute any observable. For example, suppose you have a camera and a test body moving near lightspeed from the distance toward the center of a Schwarzschild geometry, such that the test body passes through significant curvature while the camera does not; and while distant, the camera and test body start out stationary relative to each other (as defined, e.g. by doppler observation). You can compute what the camera will observe in the simple Schwarzschild geometry directly.