Falling down an infinitely deep well

[Somerschool]

Back in 1921, Theodor Kaluza published his idea of a very tiny curled up fifth dimension, which I will refer to as the "w" dimension. In a space consisting solely of the x and w dimensions, space is cylindrical, and a ray traveling at the speed of light along x looks like a light wave, while the same ray traveling in the w direction looks like a particle. Changing the direction the ray travels produces all the distortions of length and time we recognize as the Lorentz transformations. Einstein was intrigued by the possibilities of such a curled-up dimension, but the idea never really caught on in his lifetime.

I'm trying to think through the mathematics of Kaluza-space, and need some mathematical help. I've worked out the formula for a ray traveling at the speed of light through an xw space where the "w" dimension gets larger as one travels along the x axis. (In such a case, xw forms a cone rather than a cylinder.) If the cone is long and skinny enough (i.e., about a light year from the vertex to a diameter much smaller than an electron) the formula for the observable motion of a ray is indistinguishable from d=1/2 a*t^2 (for small elapsed time "t") and from v=c (for very large t).

I'd like to compare my formula for displacement as a function of time with what the basic theory of relativity would predict, but it turns out to be hard to find the formula for the motion of a particle subjected to a constant force like that of gravity. I am NOT looking for the formula for a rocket that "accelerates" at a constant rate (within its own frame of reference), but for the path of a rock thrown down an "infinitely deep well" which just happens to have a physically impossible constant gravitational pull all the way down.

Does anybody happen to know this formula off hand? If not, can someone derive it for me?

 

[Somerschool]

I posted the same question to "Ask the Physicist." Here's his answer:

"This problem is not as simple as it seems. I have consulted some friends who are good with relativity and may have an answer for you later."

Since there aren't that many infinitely deep wells around, let me spell out an alternate scenario that produces the same formula:

"Suppose we could build a gravity generator that let's us create a powerful gravitational field without using mass. Now build a spaceship and mount the gravity generator on a long pole out in front of it. Turn the gravity generator 'on' and the spaceship 'falls' towards the generator... but the pole pushes the generator along as it goes. The ship travels as if it were falling down an infinitely deep hole where the gravity never changes. What formula describes the ship's motion?"

 

[DrGreg]

...which just happens to have a physically impossible constant gravitational pull all the way down.

Suppose we could build a gravity generator that let's us create a powerful gravitational field without using mass.

If you ask about things that are impossible, no-one can answer.

 

[Somerschool]

Perhaps I should have gone to a math forum. They deal with impossible things for a living.

But, seriously, folks--I'm trying to use a word picture to describe a straightforward physics question. The fact that we can't (yet) build a gravity generator doesn't change the physics or the math.

What formula describes displacement (d) as a function of time (t) for a body subjected to an unvarying force?

If this was on your graduate school physics final you'd at least TRY to answer it! If it can't be answered, you'd explain why. I can't believe a one-sentence question can stump this forum.

 

[Somerschool]

I don't need to tell the relativity experts that an "unvarying force" will produce varying acceleration as mass increases near the speed of light, but folks who don't do Lorentz transformation on a regular basis might need that clue.

 

[Cyosis]

You want to find the relativistic kinematic equations, in particular displacement, for an object in a constant gravitational field? Just solve F=dp/dt, with F a constant then solve for v and integrate once more.

 

[Al68]

Perhaps I should have gone to a math forum. They deal with impossible things for a living.

But, seriously, folks--I'm trying to use a word picture to describe a straightforward physics question. The fact that we can't (yet) build a gravity generator doesn't change the physics or the math.

What formula describes displacement (d) as a function of time (t) for a body subjected to an unvarying force?

If this was on your graduate school physics final you'd at least TRY to answer it! If it can't be answered, you'd explain why. I can't believe a one-sentence question can stump this forum.

Displacement relative to what? Displacement is the distance between two objects. Identifying the reference object or reference frame will make the answer easy.

The simple answer to your question is d = (c2/a) (sqrt[1 + (at/c)2] - 1), where a is proper acceleration, F/m.

Or d = .5at^2, where a is coordinate acceleration, or for velocities small relative to c.

Either way, d is the distance between two objects, there is no such thing as displacement for a single object without any reference to another.

 

[Somerschool]

Al, thank you! I THOUGHT it ought to be a simple question.

Here's a word problem that clarifies the frame of reference:

Identical twins Bill and Bob shake hands at their secret lab, then Bob gets into his UltraPod (TM) and zooms off into space. The UltraPod is subject to a constant force that always accelerates it in the same direction. What is the formula for the distance of the UltraPod from Bill (d) as a function of time (t) from Bill's frame of reference?

 

[Dale]

Hi Somerschool,

I agree with Al68's answer, but you should realize that it is simply the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html" equation, which you rejected as an answer a priori. If you really see some difference in what you are asking and a relativistic rocket then you will have to be more specific. Perhaps it would help if you specified the metric.

 

[Somerschool]

I'm asking questions here because my math is too rusty to answer them on my own. I need all the help I can get.

It seems to me like the relativistic rocket formula should be different from the "falling down an infinitely deep hole" formula. The force applied to the relativistic rocket changes (from the perspective of the observer at rest) over time, because time dilates as velocity increases. I'm looking for the formula for motion where the force remains constant, from the perspective of the observer at rest.

Which formula did you just give me, Al68?

If it makes things easier (which it shouldn't), the formula I'm seeking should also describe the path of a photon inside the event horizon of a black hole. Like that helps.

 

[Cyosis]

The differential equation to solve was listed in post #6, with p being relativistic momentum. Secondly for a relativistic rocket F is simply equal to ma, which is constant as well. You've to remember that for occupants in the rocket there is no such thing as increasing mass when speeding up. As for the black hole question, I don't think think gravity is constant within the event horizon.

 

[Dale]

I believe that you want the motion of a particle under the influence of a constant coordinate force rather than a constant proper force, and it sounds like you actually want it in flat spacetime.

Re the black hole and photon. You are talking about the Schwarzschild metric, which has no "infinitely deep" solution. All geodesics that cross the event horizon hit the singularity in a finite amount of proper time. That said, here is a nice paper on http://arxiv.org/abs/gr-qc/0311038" . The equations are not at all the same.

 

[Somerschool]

I only doggie paddle!

It's not so much that I WANT the answer in "flat spacetime." What I really want is to compare the "infinitely deep well" function the formula you get when you treat gravity as a tightly coiled up Kaluza dimension, which is:

d^2 + 2*d*R -c^2*t^2 = 0

Where R is a REALLY big distance constant (on the order of 1 light year).

 

[Dale]

IWhat I really want is to compare the "infinitely deep well" function the formula you get when you treat gravity as a tightly coiled up Kaluza dimension

The problem is that AFAIK there is no "infinitely deep well" metric, so you should probably take Cyosis' approach in flat spacetime.

 

[Al68]

I'm asking questions here because my math is too rusty to answer them on my own. I need all the help I can get.

It seems to me like the relativistic rocket formula should be different from the "falling down an infinitely deep hole" formula. The force applied to the relativistic rocket changes (from the perspective of the observer at rest) over time, because time dilates as velocity increases. I'm looking for the formula for motion where the force remains constant, from the perspective of the observer at rest.

Which formula did you just give me, Al68?

I gave you the formula for a constant force applied. An object in freefall has zero applied force and zero proper acceleration. Its distance relative to a local inertial reference will remain constant, neglecting gravitational non-uniformity.

The coordinate acceleration of an object in freefall depends on the reference frame, but is normally relative to a gravitational source, and so cannot be constant over an indefinite time period.

In your example of a ship with a gravity generator on a pole the coordinate acceleration of the ship relative to the gravity source would be zero, and the displacement of the ship would remain constant relative to the gravity source. That's the extent of the answer unless there are other reference objects defined.

 

[DrGreg]

Somerschool

I'm sorry my previous answer (#3) was a little flippant. I suppose my point was that you had already conceded that both your scenarios were impossible, and the short answer is that there are no equations to describe impossible situations.

The answer to the question you asked in post #8 is a lot easier. It's exactly as Al68's first answer:

$$\left( d - \frac{c^2}{a} \right)^2 - c^2t^2 = \left(\frac{c^2}{a} \right)^2$$
(d − c²/a)² − c²t² = (c²/a)² for the benefit of IE6 users who can't read the above equation​

What confused me was that in your original post #1, you said "I am NOT looking for the formula for a rocket that "accelerates" at a constant rate (within its own frame of reference),..,". But that is the answer to your restated question. I suppose the missing piece of the jigsaw is that a constant force applied to Bob's constant mass moving in a straight line does produce constant proper acceleration (i.e. as measured by Bob), but not constant coordinate acceleration (as measured by Bill). Under these circumstances, Bob and Bill both measure the same force, which is a bit surprising given that Bob and Bill disagree over almost every other measurement. There's no obvious explanation for this (that I know of), it's just something that drops out of the mathematics.

Note that the equation you gave in post #13 can be rewritten as

$$(d + R)^2 - c^2t^2 = R^2$$​

which is the same as mine/Al68's with R = −c²/a. (About one light year if −a is the "acceleration due to gravity" on the Earth's surface.)

As others have said, an "infinitely deep well" doesn't seem to make much sense in general relativity (GR). You also should realize that something falling under gravity isn't being accelerated by a force, according to GR. It's not accelerating at all relative to itself. But something that is hovering at a constant altitude is accelerating upwards, relative to itself, and experiencing a constant upward force.

 

[Somerschool]

I had been using an "R" of c^2/a, but I see you're right about the sign. If d is increasing with time, R should be a negative number.

 

[Somerschool]

All my formula does is trace the path of a ray along a cone. The "cone" I have in mind is produced by a very tiny curled up "w" dimension with a diameter that gets larger as you move along the "x" axis.

Kaluza showed that a ray traveling along an x-w CYLINDER went through all the Lorentz transformations. If the ray travels parallel to the x axis, it acts just like a photon. If it travels perpendicular to the x-axis (traveling solely in "w"), it acts just like a particle. As far as I can tell, all the counterintuitive features of special relativity can be neatly explained by tracing a ray on an x-w cylinder.

If the formula for the path of a ray on an x-w cone is really identical to the path of a particle subjected to an unvarying force, I THINK Kaluza space also matches up to what I know of general relativity. Everything I know about gravity (which isn't much) could be explained as the path of ray traveling along an x-w cone with a "bulge" at the center of the Earth (or other center of mass).

There's only one thing about my model that clashes with what I know of special relativity. In my model, a particle falling into a gravitational field would increase in velocity but not in mass. I expect relativistic particles to get heavier as they speed up. I don't suppose anybody has ever measured the change in mass of gravitationally accelerated particles?

 

[A.T.]

All my formula does is trace the path of a ray along a cone. The "cone" I have in mind is produced by a very tiny curled up "w" dimension with a diameter that gets larger as you move along the "x" axis.

How about calling the 'w'-dimension the elapsed proper-time for the moving particle? Is this what you mean?:
http://www.adamtoons.de/physics/relativity.swf
Set initial speed (red) to zero, and increase gravity (green) to get the unrolled cone.

Everything I know about gravity (which isn't much) could be explained as the path of ray traveling along an x-w cone with a "bulge" at the center of the Earth (or other center of mass).

The cylinder with a bulge:
http://www.adamtoons.de/physics/gravitation.swf

 

[Somerschool]

Yes, that's it exactly. "Proper time" is a new term to me, but in my tinkering around with the implications of vectors in a w-x cylinder or cone I had concluded that an observer's "experience" of time would be the same as his/her distance traveled in the w dimension.

The "cylinder with a bulge" graph is EXACTLY what I've been trying to figure out!

So--where do I find out more about "proper time," and has anybody established that it is NOT an actual very small curled-up physical dimension ("w")?

I have one empirical test for the very-small-coiled-up physical hypothesis. To the best of my knowledge, research data proves I'm wrong, but here's the test anyway just in case somebody can find the article to prove the point.

TEST: IF there was a very small w dimension, a photon traveling down the x-axis would "spiral" around in w with a measurable wavelength and frequency. Photons with slightly different frequencies would be traveling at slightly different angles in x-w space. A tight burst of photons from an extremely distant source would arrive at Earth slightly phase-shifted, so that the redder wavelengths arrived before the bluer waves. A pulsar produces tight bursts of photons, so it should be possible to see whether all the photons arrive at the same time (disproving my simple-minded physical curled-up dimension hypothesis) or arrive slightly "staggered."

The phase shift is very small--a function of the diameter of the w dimension, which is REALLY tiny. There's a way to compute that diameter, which I can't recall at the moment. Last time I looked into this, however, it seemed like one should be able to detect a difference in a carefully conducted observation of waves from a pulsar, and no such difference had been detected.

Does anybody know how to research pulsars?

 

[A.T.]

Yes, that's it exactly. "Proper time" is a new term to me, but in my tinkering around with the implications of vectors in a w-x cylinder or cone I had concluded that an observer's "experience" of time would be the same as his/her distance traveled in the w dimension.

Not so fast. The observer is not traveling and experiences coordinate time(=the observers proper-time, what a clock at rest to him shows). A moving clock experiences a proper-time, which is smaller than coordinate time, because the moving clock advances less along that dimension. This is time dilation.

So--where do I find out more about "proper time," and has anybody established that it is NOT an actual very small curled-up physical dimension ("w")?

Proper time of something is just what a clock attached to it measures. I think you take that "curled-up dimension" stuff much too literary. It usually just means it is not visible like the space dimensions. In the diagram (cylinder with bump) the rolling into a cylinder doesn't have any physical meaning. It is done just to embed a curved 2d-manifold into a flat 3d-manifold for visualization. Rolling a diagram into a cylinder doesn't affect any distances in it. Only when you make a cone or create a bump it changes the straight lines(geodesics) in respect to the coordinates.

 

[Somerschool]

First of all, Adam, I LOVE the cartoons and animations on your site!

What I meant by "an observer's experience of time" is the same as what a moving clock would measure. (I can see that relativity requires extreme precision of expression--it is so hard to use everyday terms to describe relativistic situations!)

I may well be taking that "curled-up dimenstion stuff" way too literally, but it doesn't hurt to think it through. What if the curled up dimension was, literally, a spatial dimension that loops back on itself? In such a case, what you call "proper time" would have a physical meaning.

I'm not saying I'm right--but I am saying that the hypothesis is testable. If the "w" dimension is a physical space dimension, I would predict two counterintuitive results. (1) The mass of a particle accelerated by gravity would NOT increase as the particle approached the speed of light; and (2) photons of different wavelengths would travel at (very slightly) different speeds. Both of these effects would be very hard to measure, but not impossible.

I assume that people have looked at pulsars hard enough to prove, to my satisfaction, that photons of different wavelengths do, in fact, travel at identical speeds. I'd like to see the research that shows that, but I'll bet it's out there. (If it isn't, it really isn't that hard an experiment to conduct--you just need to compare the incoming pulses of different wavelengths from a suitably distant pulsar.)

 

[Somerschool]

I doubt this is the right thread for a discussion of pulsars. Any suggestions where I should take this conversation?

 

[A.T.]

I may well be taking that "curled-up dimenstion stuff" way too literally, but it doesn't hurt to think it through. What if the curled up dimension was, literally, a spatial dimension that loops back on itself? In such a case, what you call "proper time" would have a physical meaning.

"Proper time" has a physical meaning. But rolling the space-propertime diagram into a cylinder doesn't have any. It has nothing to do with propertime being curled or hidden. You could visualize the curvature of the ordinary space dimensions in similar way

The mass of a particle accelerated by gravity would NOT increase as the particle approached the speed of light;

Well but it does. It is observed in accelerators. You can draw a very similar diagram like http://www.adamtoons.de/physics/relativity.swf" for rest mass(y-axis), momentum(x-axis) and dynamic mass(line from the origin intersecting line of constant rest mass):

Rest mass is the projection of the dynamic mass onto the proper-time dimension.
Momentum is the projection of the dynamic mass onto the space dimension.

 

[Somerschool]

I'm aware that the mass of particles in accelerators increases with velocity. My counterintuitive prediction is that mass accelerated by gravity does NOT behave exactly the same as mass accelerated by other forces. In my model, a particle accelerated nearly to the speed of light solely by gravity would look just like a photon--virtually massless. By contrast, a particle accelerated to near the speed of light in a particle accelerator would be VERY heavy.

 

[Somerschool]

A.T., I assume, then, that one photon with twice the frequency of another has twice the rest mass?

 

[Dale]

Twice zero is still zero so sure.

 

[A.T.]

A.T., I assume, then, that one photon with twice the frequency of another has twice the rest mass?

What DaleSpam said. Photons do not advance in propertime, so the projection of their dynamic mass onto the propertime dimension (= rest mass) is zero. Their dynamic mass entirely goes into the spatial dimensions (= momentum).

 

[Somerschool]

I seem to vaguely recall that high-energy photons have more momentum than low-energy photons. In my "w" space, the frequency of a photon measures the number of times a photon "spirals" around the very-tiny-curled-up w dimension. In this model, the w component of the vector is identical to what we call "mass." Frequency is therefore directly proportional to the "mass" of the photon, which means that the momentum of a photon with twice the frequency should be twice the momentum of a photon with half the frequency (since the velocity is essentially the same--c--but the "mass" is doubled).

As I'm learning about "propertime," I see that what I have been crudely thinking about in terms of "mass" (with quotation marks!) may simply be what trained relativity folks know as "propertime." It seems as if, for a given particle, mass and propertime have many characteristics in common.

Has anybody ever proved that a photon isn't just an electron moving along the x-axis with almost no w component, and an electron isn't just a photon moving along the w axis with almost no x component?

 

[f todd baker]

see http://www.askthephysicist.com/ask_phys_q&a_old2.html#relativistic constant force

 

[Somerschool]

Thanks, Todd. The folks at "Ask the Physicist" have been very kind to respond to my questions, as has this board.

 

[f todd baker]

since the original post asked for x as a function of t, not v, I have added a calculation of x to the above link