-
Posts
1111 -
Joined
-
Last visited
Content Type
Profiles
Forums
Events
Everything posted by elfmotat
-
In GR gravity isn't actually a force. Geodesic motion is "natural" motion, i.e. the motion that something exhibits if there are no forces around to mess with it. You can find things like "the normal force required to keep an object on the surface of the Earth" through a slightly tedious calculation of proper acceleration by taking the magnitude of four-acceleration. But "force of gravity" has no real meaning in GR.
-
Are you being asked to actually integrate the function, or to approximate it with rectangles or trapezoids? I ask because you have a discrete set of points there, not a continuous domain.
-
bjarne, you appear to be under the impression that if two observers disagree about length or time then there is no meaningful way to compare measurements. That's simply not true. A meter is a meter is a meter. In general the speed of light will not be the same for two observers in circular motion or near a gravitating body, because their clocks and rulers will differ. Locally (which means "over a small enough scale for a short enough time") the speed of light is always c.
-
Technically "locally" actually means "at an infinitely small scale." There is, in general no notion of simultaneity in GR for two different points. For short distances you can talk approximately about simultaneity, and at everyday scales/speeds/gravity "short" covers a lot. There's no cutoff or "barrier" as you say, it just depends on how accurate you're trying to be in your calculations and measurements.
-
"Approximately" meaning virtually undetectable given our locational/gravitational/velocity differences. Sure some things may be happening simultaneously in your approximate inertial frame (I say approximate because of course any frame at rest on Earth's surface is by definition not inertial). That doesn't mean that other observers will agree with you, and in general they won't. There's no "correct" frame, as is the essence of relativity. It's not our "perception" of the universe (whatever that means), so much as the way the universe works. As I said above, there's no "correct" frame in SR. In fact, in GR the notion of a correct frame becomes even more meaningless. Two events separated by vast distances cannot be meaningfully compared in terms of "when" they happened without choosing a particular coordinate system to work in, which usually entails considering the Earth to be at rest w.r.t. whatever we're comparing it to (which obviously can't be true in general), and then comparing proper times.
-
Global nonlocal simultaneity has literally no meaning at all. Though two events may feel simultaneous (or even be exactly in your rest frame), there's no true absolute simultaneity - though they are approximately simultaneous for anyone moving at everyday speeds relative to you, so much so that you don't notice. This is a clear example of when intuition completely breaks down at large distances and speeds.
-
As you suspected, surrounding yourself with a bunch of mass will make you age more slowly relative to someone far away from that mass. If you refer to my above post (#33), you'll see that the relevant number is: [math]k=-\frac{3}{2} GM \left ( \frac{b^2-a^2}{b^3-a^3} \right )[/math] Lets say, for example, that you surround yourself will a shell of mass that is 70,000,000 meters thick (roughly the radius of Jupiter), with a small hole in the middle for you to live in. The amount of mass in the shell required to roughly double your life span with respect to someone on Earth is approximately 16,000 times the mass of the Sun. This amount of mass compressed into a space the size of Jupiter (by far the largest planet in our solar system) is well over enough to ensure that it collapses into a black hole. So surrounding yourself with a bunch of matter is not a very likely or efficient way to travel to the future.
-
I argued that you can set the potential to any value you like, via a redefinition, without changing any of the physics. Unless I'm interpreting you wrong (which might be what's going on), you said that the convention of setting the potential to zero at infinity is somehow implied when determining the potential inside a spherical mass shell. That's simply not true. I think you're taking coordinates themselves too seriously. The metric is not an observable, and there's a tremendous amount of gauge freedom that comes along with it. You can shift to a coordinate system that is Minkowskian inside the sphere without changing any of the physics. I don't know whether or not an exact solution for this situation exists, but you can get a good feel for it by using the weak field approximation. We'll use the convention that [math]\phi (r= \infty)=0[/math]. If we have a shell of density [math]\rho = \frac{3M}{4 \pi (b^3-a^3)}[/math] with its inner wall at the coordinate r=a and outer wall at r=b, then: [math]\phi ® = \begin{cases} k= -2 \pi G \rho (b^2-a^2) & \text{ if } r < a \\ -GM/r & \text{ if } r> b \end{cases}[/math] (When a<r<b the potential will be some linear function of r which keeps the potential function smooth, but I can't be bothered to work it out right now.) Inside the shell the metric is given by: [math]ds^2=-(1+2k)dt^2+(1-2k)dr^2+d \Omega^2[/math] Outside the shell: [math]ds^2=-\left (1-\frac{2GM}{r} \right )dt^2 + \left (1+\frac{2GM}{r} \right )dr^2+d \Omega^2[/math] Now it's obvious from this that a person inside of the shell will age more slowly than a person outside of the shell. However, this does NOT mean that the spacetime inside of the shell isn't Minkowskian - it most certainly is. There are two ways of showing this: you can redefine the potential so that [math]\phi (r=0)=0[/math], which will make the potential outside of the shell positive everywhere (and asymptotically approaching [math]-k[/math] as you go to infinity). No physics has been changed - observers outside of the shell still age faster than those inside the shell, but now the metric is explicitly Minkowskian inside. The second way of making the inner metric explicitly Minkowskian is what I alluded to in my previous post: we change to a new convenient coordinate system. We define the coordinates: [math]T=t \sqrt{1+2k}[/math] [math]R=r \sqrt{1-2k}[/math] Inside the shell the metric is clearly explicitly Minkowkian, and outside of the shell we have: [math]ds^2=-\left (\frac{1-2GM/r}{1+2k} \right )dT^2 +...[/math] The 00-component of the metric still clearly has a greater magnitude outside of the shell, so again we find that observers inside the shell age more slowly. In general if the metric is proportional to [math]\eta_{\mu \nu}[/math] then you can trivially find a coordinate system where it is exactly the Minkowski metric. I.e. the physics of the two are equivalent. Coordinate transformations don't change any of the physics, but they may make certain things more or less apparent.
-
I don't see how setting the potential to zero at infinity is required in the proof that the potential in a spherical mass distribution is constant. I don't believe you've made an error. After all, if you consider the weak field metric in the case of a constant potential then you can trivially change your coordinate system to be Minkowskian. I.e. if, for example: [math]ds^2=-(1+2 \phi)dt^2 + (1-2 \phi)d \mathbf{r}^2[/math] then you can simply define the new coordinates: [math]T=t\sqrt{1+2\phi}[/math] [math]\mathbf{R}=\mathbf{r} \sqrt{1-2\phi}[/math] so that: [math]ds^2=-dT^2 + d \mathbf{R}^2[/math]
-
I can honestly say I don't see the cross. Anyone mind circling it?
-
How's this: http://math.berkeley.edu/~kwray/papers/string_theory.pdf It's the easiest to follow I've seen for free on the internet. I've heard that Zwiebach's textbook is suitable for undergrads, and that it doesn't even require too much experience with GR or QFT. I can't confirm this myself because I don't own it. Anyway, if you really want to understand ST you're going to have a long road ahead of you. I would proceed roughly as follows: Newtonian Mech. --> Lagrangian/Hamiltonian Mech. --> EM --> SR --> QM --> GR --> QFT --> ST There are a few other things you should learn as well, such as Statistical Mechanics and Thermodynamics, when you see fit. Each new field will require a lot of new math as well. Newtonian Mechanics involves only basic calculus for the most part. Lagrangian/Hamiltonian Mechanics introduces calculus of variations, EM requires familiarity will vector calculus theorems, QM requires a good feel for linear algebra, GR requires differential geometry, and QFT requires group theory and a few other things. I'm not sure what new math ST requires because I haven't studied it yet.
-
Craft without Any propulsion Or engine- Possible?
elfmotat replied to SomethingToPonder's topic in Classical Physics
Mythbusters did an episode on "antigravity" devices, and (unsurprisingly) none of them produced any real antigravitational effects. If I remember correctly, they put the little tin-foil craft in a vacuum and it didn't work. Meaning that it uses air as its propulsion. -
Cosine identity for energy transmission of waves and speed of waves.
elfmotat replied to Vay's topic in Classical Physics
The easiest way to demonstrate this is to simply look at a graph of cos2(x). It's a periodic function, whose amplitude varies from 0 to 1. It's not unreasonable, therefore, to suspect that the average value of this function from 0 to 2pi is simply 1/2. If you do the integral, this turns out to be the case. -
Your new question is a bit like asking "how are 1 and 2 different numbers?"
-
To answer your question about EM waves, this is all explained by Maxwell's equations. Boiled down, what they say is that a changing electric field will produce a perpendicular magnetic field, and that a changing magnetic field will produce a perpendicular electric field. So basically what happens is that you start with an oscillating electric field, which makes a magnetic field, which makes an electric field, etc. You end up with sinusoidal electric and magnetic fields, which travel at precisely the speed of light. As far as your questions about QM: it's completely natural to be confused. Our brains simply didn't evolve to be able to visualize quantum effects. QM has nothing to do with consciousness, or "mental states" as you put it, so don't be thrown off by nonsense mysticism. As far as learning goes, you'd probably appreciate the Feynman Lectures.
-
My understanding is that tesseract images are "shadows" of 4D cubes in 3D space. Similar to how the following is a shadow of a 3D cube projected onto a 2D space, i.e. your computer screen or a piece of paper. Tesseract shadows, like the one below, are commonly rotated over time to illustrate its 3D properties, due to the limited nature of having a 2D space to draw on.
-
You're probably hearing something like this: Newtonian mechanics is our simplest, most intuitive picture of how the universe works. But the answers it yields are really just useful approximations that break down when we start talking about things moving very fast (we need Special Relativity), when we're near a really strong gravitational field (we need General Relativity), and when we're describing phenomena at the atomic/subatomic level (we need Quantum Mechanics). Newtonian mechanics works very well for everyday stuff, but its equations simply don't apply to more extreme phenomena. I.e. the approximation breaks down.
-
I think it's reasonable to ask here what exactly you mean.
-
The fast, easy version is that Kepler formulated his third law using the work/data of various people, relating the period of a planet's orbit to its orbital radius. Newton's law of gravitation was then a natural generalization which explained Kepler's law.
-
I don't really know what you're asking. What do you mean by "form?"
-
They have opposite charge, so they will move in different directions when placed in an electric field.
-
centripetal force vs centrifugal force.
elfmotat replied to casrip1@gmx.com's topic in Classical Physics
Could you give me an example of where the third law falls apart? Because I'm having trouble thinking of one.