time dilation problem

As I understand the formula, it is:

t = t'/sqrt(1-v^2/c^2)

Where t' is the amount of time that passes for the traveller, v is the velocity, c is the constant 'the speed of light', and t is the amount of time that passes for the observer of the traveller.

It's obvious that at V=C this question is undefined, but as v approaches c, the values for t approach infinity.

This brings me to a conundrum that thwarts my understanding.

If a ship is travelling at 299792416 meters per second, the relative time dilation is 1/1831.786 (For every one second for the traveller, 1831.786 seconds are being experienced by the observerers.) The same holds true for the traveller versus the observers, since to him, they are moving at 299792416 meters per second despite being 'stationary' to themselves.

This means, though, that while the traveller IS travelling 299,792,416 meters per second, to the people on the planet, he only seems to be moving at a mere 163661.266 meters per second, only a little under 1/1831 the speed of light. This stands to reason as the person's speed is measured in amount of distance travelled over X time. In his frame, he is always moving 299792416 meters per second, but since the observers on earth as subject to 1831 seconds for everyone one of his, they view the travel over that distance as taking longer.

This is what bugs me. Say we even go closer to the speed of light, infinitely close. Eventually, the apparent speed of the object approaches zero. Let's say we ramp it up to a time dilation of 1/149,896,229. The observed speed of the massive and flattened object (to the observers) would be 2 meters per second. A spacefaring earth could launch a rocket and get up in the path of the speeding object, and observe it crawl along with all it's relativistic effects at 2 meters per second without every solving the assymetry of the different time frames (their acceleration would keep them well below any significant relativistic effects). They never have to speed up to near lightspeed to go along with him. They just have to keep moving at 2 meters per second, and they'll be able to observe him. They can build a 2 meters/sec moving scaffold around the ship to study it. They could try to open a hatch on the ship, but probably violate the mechanical properties of that hatch, shearing it to pieces. If the inside of the ship was pressurized, it would take an enormous amount of time to leak out of the ship. I'm not even sure if the air could accomadate the speed at which the earthlings would be moving, if they managed to get aboard the speeding vessel.

I'm certain something must solve this certain case of assymetry, as it's completely intangible to me. Anybody with a more thorough understanding of the principles able to explain?

Say we even go closer to the speed of light, infinitely close. Eventually, the apparent speed of the object approaches zero.

You are confusing the time dilation aspect with velocity it seems.

The *object* has not *slowed* relativistically, the passage or rate of time on the object versus the observer has slowed.

Another thing is that as I understand it time essentially *stops* for the object if C is achieved because also you have reached infinite dilation and for all intents and purposes becomes a Schwarzschild metric, but as this is not a Black hole (insufficient mass) the object is essentially converted to a *photon* for the outside observer but it doesn't *slow down* because its velocity is C.

This may be helpful

Schwarzschild Geometry
http://casa.colorado...ajsh/schwp.html

Schwarzschild's Spacetime: The Light Cone
http://www.phy.syr.e...warzschild.html

Radial Paths in a Spherically Symmetrical Field (you might use these to make a combined proof)
http://www.mathpages.../s6-04/6-04.htm

Kerr spacetime
http://relativity.li...rticlesu25.html

5.8.0.2 Notes on Kerr naked singularities and on the Kerr-Newman spacetime.
 
The Kerr metric with a>m describes a naked singularity and is considered as unphysical by most authors. Its lightlike geodesics have been studied in [47, 49] (cf. [54], p. 375). The Kerr-Newman spacetime (charged Kerr spacetime) is usually thought to be of little astrophysical relevance because the net charge of celestial bodies is small. For the lightlike geodesics in this spacetime the reader may consult [48, 50].

I hope this helps and welcome to the inside/out curved spacetime aspects of GR. [lol]

I think there are two problems here. One is, you're mixing up your frames of reference. Two, you're forgetting length contraction (which is by the same factor as time dilation).

First of all, in the traveller's reference frame, he ISN'T MOVING! The universe is speeding past him at "299792416 meters per second".

But leaving that issue aside, let's look at it this way.

In the observer's reference frame, let's say there are two "mile markers", although of course they aren't "mile" markers. So let's call them LSM's, for light-second markers. They are spaced 299792458 meters apart (by the way, what constant did you use for lightspeed? Backtracking from your calculations, I get 299792460.67...).

Now, for the traveller, space is contracted by a factor of 1889.16862, so he sees the LSM's as being about 158690.153 meters apart, and he traverses that distance in about 1/1889.16836 seconds. So he calculates that the universe is flying past him at a speed of about 158690.153/(1/1889.16836), or about 299792416 meters per second. So the speed measured in the traveller's reference frame is correct.

But in the observer's reference frame, the distance between the LSM's was 299792458 meters. Now as you noted, the time the traveller measures is dilated in the observer's reference frame. But the time we're dilating is the 1/1889.16836 seconds, not 1.0 seconds.

In the observer's reference frame, the time it took the traveller to go between the two LSM's is (1/1889.16836), dilated by a factor of 1889.16862. Here, we can see why so many sig-figs were required on that dilation factor. The dilated time is 1.00000014 seconds.

Now, armed with that information, we can see that in the observer's reference frame, the traveller goes 299792458 meters in 1.00000014 seconds, which computes to a speed of 299792416 meters per second.

Hooray! You see, it works out after all.

Isn't velocity the measurement of distance you travel through time, though?
To put it in perspective, let'say you are walking past me and have a time dilation of 1/3. For every one second in your very brisk walk, I would experience three. If I were to measure your position and speed, it seems I would percieve you as moving 1/3rd as slow as you really were, since to me you are moving slower in time.
Maybe I'm being unclear. It's just, if you are moving 2 meters per second and have a time dilation of 1/3, then wouldn't observe you as taking 3 seconds to move 2 meters? Thus, it would seem you had a slower velocity.

Thanks for those links, I'll read through when I get home from work,. They look very interesting.

Edit: Now I understand. Thanks.

In the observer's reference frame, let's say there are two "mile markers", although of course they aren't "mile" markers. So let's call them LSM's, for light-second markers.  They are spaced 299792458 meters apart (by the way, what constant did you use for lightspeed? Backtracking from your calculations, I get 299792460.67...).

Yeah, I just noticed that Pike was doing some nasty things with the numbers while converting them into floats... sorry about that. Converting it to use the arbitrary precision library right now.

I didn't forget that the traveller isn't moving to himself, I just looked at the issue solely from the side of the observer.

This is right though, the contraction of space solves this assymetry. Excellent. Thank you for the reading materials.

You're forgetting length contraction. By whatever factor time is dilated, length is contracted by the same factor.

So if I am walking past you with a time dilation of 3, then for every second I measure, you measure three. But for every meter of sidewalk that you see me walk, I only see 1/3. Taken the other way, for every meter of sidewalk that I see myself walk, you see me walk three meters. So if I see myself walk 2 meters in 1 second, you see me walk 6 meters in 3 seconds, which is still 2 meters per second.

If you don't do the length contraction, then obviously there's a paradox. But the length contraction and the time dilation cancel each other, so to speak.

Edit: I wrote this post before armrha's last post had been submitted. So I wasn't trying to patronize him, I just didn't see that he had already worked it out...

Edit: I wrote this post before armrha's last post had been submitted. So I wasn't trying to patronize him, I just didn't see that he had already worked it out...

Not a problem at all Jaydfox. I am a total amateur at physics, heh. I appreciate the excellent dissemination of the information I wanted. : )

Is the force of gravity something exerts also a function of time? I.e., does a time and length-dilated object by 1/3rd exert 1/3rd less gravitational pull then it's mass-increased form would normally?

Heh, gravity is a whole other ball of wax! I'm not too shabby at the math of special relativity, having spent a lot of time analyzing Minkowskian space and its implications for energy and momentum, including such fun topics as mass shells and potential energy.

However, gravity introduces mathematical challenges that I haven't yet mastered. I actually spent weeks studying tensor math, in the vain hope of being able to understand the math of general relativity. While I got a decent enough understanding of tensors to write algorithms for converting coordinates and velocities on an arbitrary 2- or 3-dimensional surface (manifold) in 3- or 4-space into coordinates and velocities in 2-space or 3-space (for a video game), it wasn't nearly enough knowledge to understand general relativity's finer points.

However, I can at least answer the question from a special relativity point of view. In special relativity, when time dilates, and space contracts, mass also increases, and again, it's by the same factor.

So an object travelling fast enough to have time dilated to three times slower, and lengths contracted to three times shorter, will appear to an observer to be three times more massive. For the curious, since the object now takes up three times less volume, it's density is nine times greater in the observer's reference frame!

But I digress. That's not rest mass, it's inertial mass, and there's a subtle difference.

The inertial mass will be three times higher, meaning that if the object bumped into another object, it would be as though an object three times more massive actually did the bumping.

As for gravity? Well, this is a tricky one without resorting to general relativity. I'll have to think about it more and come back with an answer later.

Is the force of gravity something exerts also a function of time? I.e., does a time and length-dilated object by 1/3rd exert 1/3rd less gravitational pull then it's mass-increased form would normally?

Yes as evidenced by the argument that time essentially *stops* (ceasing all *apparant* forward motion) when passing the event horizon for a Black hole.

I suspect Brian Wowk can fill in more details on this one.

Laz, depending on what you were referring to when you said "Yes", I'd be inclined to disagree.

If we're referring to special relativity and rapidly moving objects, then my answer is I don't know for sure. If we're talking about time dilation in the presense of a large gravitional potential (i.e., near a massive object) under general relativy, then I think the answer is No, not Yes.

The reason is rather simple: a black hole could exert no gravitational field if gravity effectively stopped at the event horizon. Moving down a notch in mass, a neutron star's gravitational field would be far weaker if its gravity were modulated by the time dilation factor at its surface.

Massive objects are one of those cases where it's more useful to think of gravity as bent spacetime, rather than as an emission of gravitons. The rate of emission of gravitons would (at first blush) be modulated by the time dilation factor, so that an object near a very massive object (e.g., neutron star, or hovering well above a black hole), experiencing a time dilation factor of 3 relative to distant observers, would only exert 1/3 the gravititational force on distant objects. This doesn't jive with intution or theory, as far as I'm aware.

Of course, theories of quantum gravity probably account for the time dilation problem by either increasing the rate of emission or increasing the effective force of each graviton, or both. I haven't read any quantum gravity theories, so I can only guess how they account for the time dilation problem without resorting to curved space-time and geodesics (or maybe they do a bit of both?).

As for the first case, a rapid object under special relativity, I'm inclined to think that an object experiencing a time dilation of 3, with a length contraction factor of 1/3, and a perceived inertial mass 3 times normal, would also exert 3 times the gravitational pull. The reason is simple: at such a high speed, the mass is 3 times normal due to the object having twice its mass-equivalent in kinetic energy, and energy exerts gravitional influence just as much as mass. If this were not so, then pumping a bunch of energy into an object to push it near lightspeed would make a bunch of "mass" disappear, gravitationally speaking, and I don't think this is possible.

But this is just a guess.

Hmm, I thought of something more related to the gravity issue in the presense of a large mass.

Consider a black hole, and a massive object (a white dwarf, for instance, just so we're talking about something small) falling into the black hole. As the white dwarf approaches the black hole, the gravitational pull on distant objects should not change, because the total mass/energy of the system is not changing (other than radiated gravitational waves, which are negligible up to the last minute, especially if the white dward isn't very massive).

As the white dward accelerates into the black hole, it's kinetic energy increases, but its gravitational potential energy decreases, resulting in no net increase in energy, and also no net increase in gravity. So the increased mass (due to kinetic energy) of the white dwarf does not exert increased gravitational pull. But this is because the increase in gravity due to the white dwarf's travelling near c is balanced by the decreasing gravitational potential energy, which means less gravitational pull due to the lost mass-equivalent of said energy. So it all cancels out, as we'd expect.

So, going back to my earlier statement:

an object near a very massive object (e.g., neutron star, or hovering well above a black hole), experiencing a time dilation factor of 3 relative to distant observers, would only exert 1/3 the gravititational force on distant objects. This doesn't jive with intution or theory, as far as I'm aware.

Now, taking into account lost energy due to lost gravitational potential energy, we can see that this hypothetical object actually would exert less gravitational pull. The reason is that if the object had fallen from a great distance to the height at which the time dilation factor due to gravity for a stationary object is 3, then it would still be exerting its full gravitational field. If this rapidly falling object was then stopped, by transferring its kinetic energy to something else that then leaves the system, then it would exert a much smaller gravitational force on distant objects.

Which then makes me wonder if this reconciles the issue with QG that I mentioned.

However, now I'm wondering how black holes continue to exert their gravitational pull. However, consider that as a black hole falls in on itself, it doesn't lose kinetic energy. So the total kinetic/potential energy of the collapsing star remains in balance, and hence the gravitational pull does not decrease. As the star's matter approaches the event horizon, it's effectively approaching a speed of c, as far as distant observers could tell (measuring local distance travelled versus local time, up to the limit of observation capabilities), so the kinetic energy increases enough to offset the drop in potential energy due to time dilation, sufficient for the mass to never effectively reach zero. Of course, what happens at the event horizon? Normal math no longer applies, and one must use the tensor math to get an answer.

If this rapidly falling object was then stopped, by transferring its kinetic energy to something else that then leaves the system, then it would exert a much smaller gravitational force on distant objects.

Which would make the answer to the question a qualified Yes. I'll assume therefore that this is what Laz meant by his Yes.

The reason is rather simple: a black hole could exert no gravitational field if gravity effectively stopped at the event horizon.

We are at the space/time duality issue. Gravity is not a force of space over time even though it *propagates* as such (inverse square law) but gravity is a property OF space/time in the sense of the overlap of the two.

According to the math of the physics governing an event horizon time essentially comes to a standstill as gravity equals (or tries to exceed) the speed of light. Can gravity create a field wherein the escape velocity exceeds C?

Of course, that is why it is a black hole.

If escape velocity exceeds C what happens to time for a relative body within that field?

Is it even meaningful to discuss a relative body with respect to a black hole once inside the event horizon?

Material within a black hole exists at or above an *acceleration* (with respect to the *force* of gravity) equivalent to *C*. You can't ignore how this alters time for the properties of the matter (physical chemistry) within the black hole and the math says it becomes infinite, or zero; take your choice depending on how you run the variables. The point is that with respect to both extreme solutions time *stops* in relationto the rest of the the coexitent universe, which is where we are sitting outside the event horizon.

Moving down a notch in mass, a neutron star's gravitational field would be far weaker if its gravity were modulated by the time dilation factor at its surface.

I believe the time dilation aspect is accounted for at the scale of a neutron star.

If I remember correctly time even *moves* at a slightly different pace on Jupiter than Earth but the difference is still infinitesimal with respect to mass. You would age slower in a significantly larger gravitational field (Jupiter) with respect to Earth time but your experience of Jupiter time would become the *referent norm* and you would notice no difference experientially (existentially). However as the respective fields become more powerful existence on such a world would itself be physically fatal to the forms we possess.

Are black holes a type of portal into the multiverse?

Once matter is within a black hole could that create a relative microverse (space/time existence) with respect to our *universe*?

If escape velocity exceeds C what happens to time for a relative body within that field?

Is it even meaningful to discuss a relative body with respect to a black hole once inside the event horizon?

I think we're talking past each other here. My point is how gravity is felt at a great distance (effectively with no time dilation/space warping effects of any significance).

After a black hole collapses to/inside its own event horizon, distant objects will continue to feel the gravitational effects of the mass that is no longer visible to us by other means. The gravitational effect of that mass is not zero, even if time stopped for all intents and purposes from the point of view of a distant observer.

Moving down a notch in mass, a neutron star's gravitational field would be far weaker if its gravity were modulated by the time dilation factor at its surface.

I believe the time dilation aspect is accounted for at the scale of a neutron star.

You probably wrote this before seeing my reply to myself, where I realized this:

Now, taking into account lost energy due to lost gravitational potential energy, we can see that this hypothetical object actually would exert less gravitational pull. The reason is that if the object had fallen from a great distance to the height at which the time dilation factor due to gravity for a stationary object is 3, then it would still be exerting its full gravitational field. If this rapidly falling object was then stopped, by transferring its kinetic energy to something else that then leaves the system, then it would exert a much smaller gravitational force on distant objects.

I have to get back to a different task at the moment and you wrote the subsequent posts WHILE i was responding to the first.

I promise to return to this issue later. )

Material within a black hole exists at or above an *acceleration* (with respect to the *force* of gravity) equivalent to *C*. You can't ignore how this alters time for the properties of the matter (physical chemistry) within the black hole and the math says it becomes infinite, or zero; take your choice depending on how you run the variables.

Actually, my understanding of the situation is a little different. My favorite illustration comes courtesey of a book I read, by a (semi-)famous physicist (I hedged, because I don't remember which book or which author it was). I'm of course recalculating these, so this isn't a quote, or even so much a paraphrase:

Consider a black hole the with the mass of the sun. Such a black hole would:
- a radius of about 2.95 km (3.12E-13 light-years),
- a "density" of about 3.23E+20 kg/m^3 (3.23E+17 g/cm^3), and
- a gravitational pull at the horizon (under Newtonian physics) of about -1.52E+13 m/s^s (about 1.55E+12 earth g's).

A black hole with 3 solar masses would:
- a radius of about 8.86 km (9.37E-13 LY),
- a "density" of about 3.59E+19 kg/m^3 (3.59E+16 g/cm^3), and
- a gravitational pull at the horizon of about -5.07E+12 m/s^s (about 5.16E+11 earth g's).

A black hole with 100 solar masses would:
- a radius of about 2.95E+05 km (3.12E-11 LY),
- a "density" of about 3.23E+16 kg/m^3 (3.23E+13 g/cm^3), and
- a gravitational pull at the horizon of about -1.52E+11 m/s^s (about 1.55E+10 earth g's).

A black hole with a billion solar masses would have:
- a radius of about 2.95E+12 km (3.12E-04 LY),
- a "density" of about 3.23E+02 kg/m^3 (3.23E-01 g/cm^3), and
- a gravitational pull at the horizon of about -1.52E+04 m/s^s (about 1.55E+03 earth g's).

Notice this last black hole is about 25 times denser than air.

A black hole with 155 billion (1.55E+11) solar masses would have:
- a radius of about 4.58E+14 km (4.84E-02 LY),
- a "density" of about 1.35E-02 kg/m^3 (1.35E-05 g/cm^3), and
- a gravitational pull at the horizon of about -9.82 m/s^s (about 0.999, basically 1.0 earth g's).

Note that our Milky Way is approximately 400 to 1,000 billion solar masses, so this black hole would be the equivalent of the a significat portion of the Milky Way galaxy collapsed into a black hole just 1/20th of a light-year across.

Now, this huge black hole would have a gravitational pull at its surface of 1 g, under Newtonian physics. You'll notice I've been careful here to point out that these gravity levels are what would be predicted "under Newtonian physics". Near the event horizon, spacetime becomes a bit squirrelly. But the point is, gravity per se is quite weak near such a large black hole. It's the bent space-time and the time dilation that makes it seem impossible to escape from very near the event horizon. At the event horizon is another story altogether, of course.

You guys might find the following useful

http://antwrp.gsfc.n...bh_pub_faq.html

Needless to say, the kinds of issues you are discussing are very non-trivial. Understanding them at an intuitive level requires hitting one's head against a blackboard and cursing many times over a period of years.

---BrianW

Understanding them at an intuitive level requires hitting one's head against a blackboard and cursing many times over a period of years.

I'll second that! I don't have access to a blackboard, though, but would a whiteboard suffice?

Hmm... do all immortalists/transhumanists have a fascination with black holes and warped spacetime? (and relativity and quantum mechanics)..

Jay, if you have time why not plot the no. of earth solar masses versus gravitational pull at the event horizon. I just noticed your point, however, which is the inverse relationship between black hole mass and event horizon gravity. I did not know that. What equation are you using?

I'm using 2GM/c^2 to calculate the event horizon radius. I'm using -GM/r^2 to calculate gravitational pull at the event horizon (Newtonian physics). I got the constants for solar mass, G, c, and a light-year off google (just googled the relevant term, e.g., "gravitational constant").

As for density, just using M/V, where V = 4/3*pi*r^3, and r=2GM/c^s of course.

BTW, mind your units when working with numbers so big that you have to use scientific notation.