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General relativity is a theory that tells us the geometry of spacetime. However it predicts that in some situations the geometry of spacetime is undefined - this happens when we get a singularity. There is a singularity at the centre of a black hole, and the Big Bang was also a singularity. So we have the odd situation that the theory of general relativity ...


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Black holes are black because everything that enters don't exists anymore. They are actually extremely red and you see things near them move slowly and the light from then looks very very very red. Like redder than red, like infrared then microwaves then even radio waves. And eventually it is too blurry to really make anything out and too faint (light ...


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Perhaps it's more illuminating to look at the whole thing in a spacetime diagram. we have the earth frame with coordinates $(t,x)$, and its trajectory through spacetime is the blue line. The trajectory of the spaceship is the red one. Straight worldlines are inertial frames of reference, curved or non-straight worldlines are non-inertial frames of ...


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The axis of simultaneity, or in other words, the set of events which are simultaneous as measured in the rest frame of the ship, does indeed change suddenly when we turn back. This is because it depends on your reference frame. There isn't a single inertial frame that stays with the ship for the whole journey; you can either accept that the frame is ...


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You can find some information about that on John D. Norton's website. Einstein thought of this at the age of sixteen. Here's another article: "If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating. There seems to be no such ...


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Imagine a space ship ran put of fuel and a second is going to help. They will do a docking maneuver to hand out some fuel. Now, the second ship has to adjust its speed when it's near the first (because the first can't change its speed). A passenger on the second ship feels the acceleration. But does the ship accelerate or decelerate? If the first ship ...


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Deceleration is a "special case" of acceleration. More precisely, acceleration is given by the vector $\vec a$ which has both a magnitude and a direction. Sometimes the same vector $\vec a$ increases the velocity $\vec v$ – when they are oriented in "mostly the same direction" – and we speak about "real acceleration" in the sense of an increasing speed. And ...


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SR is based upon two postulates: 1)The laws of physics are the same in all inertial frames of reference. 2)Speed of light $c$ is constant for all inertial frames of reference. To arrive at some results in relativity, one only needs to assume one of the postulates, other results are logical conclusions that require assuming the two postulates together. ...


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So Special Relativity states that for all non-accelerating objects of matter the laws of physics are the same. I think the point is just that the constants and the time and space derivatives that appear in a law of physics should not have to change the form of the equation if you measure the time and the space in two frames that move relative to each ...


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from the observers perspective information has travelled instantly, the firing of the first guillotine has turned on the light with no delay. Herein lies the flaw in your reasoning. The electrical signal doesn't instantly propagate, it propagates at the speed of light. From the perspective of every observer, the electrical signal from the sensor reaches ...


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Imagine a sensor on the front of the train which detects when a guillotine passes infront of it, this then sends an electrical signal down the train to a light at the back of the train which, I think, would turn on at the same time as the back guillotine drops, from the trains perspective That's not possible. Instead of a detector on the front ...


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To simplify notation let's denote $\gamma = \frac{1}{\sqrt{(1-0.99^2}}$, and let $m_{ball}, m_{Earth}$ be the rest (invariant) masses of the ball and the Earth. The way you write the gravitational force on the ball in the two FORs is: In the train: $F_{train} = g \frac{m_{ball}(m_{Earth}\gamma)}{d^2} = m_{ball} a_{train}$, hence $a_{train} = \gamma\;9.8$ ...


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Yes, the energy and the increase in energy all depend on your reference frame, but this is NOT special to relativity! The same thing happens in classical mechanics. I wrote a similar answer to the question, "Can you tell your absolute speed in space?" Consider the regular Newtonian mechanics equation, $\mathrm{Ke}=\frac{1}{2}mv^2$. If you weigh 50kg, are ...


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To answer question 2: If it fires a photon orthogonal to its heading, it will travel orthogonal to the source's heading from the POV of the source. From the POV of an outside observer, it will travel not orthogonal to the motion of the source, but slightly along its direction of motion. This can be easily intuited because in its own frame, the source is at ...


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The Theory of Special relativity tells us about the relativistic observations of the observer in an inertial frame (frames moving with constant velocity or experiencing no acceleration). The postulates of relativity drives the theory and make some amazing predictions about the relativistic observations of an inertial observer. Let's talk about a situation ...


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Is the concept of mass increasing as it approaches the speed of light based on the fact that this is only true relative to the observer? When you say mass increases what you mean is energy increases. You don't want to have a different mass for forces in different directions so the idea of mass changing with speed is now abandoned. You have energy ...


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How do we know the speed of light is constant and spacetime dilates rather than vice versa? We know that the speed of light is not constant. I'm afraid it's a popscience myth that the speed of light is constant. See Irwin Shapiro talking about it here: Some conspiracy nut was telling me that Einstein was BS and there's a giant conspiracy that he's ...


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You might be confusing some issues. In special relativity, space and time do not stretch or compress. It really comes down to measurements with clocks and rulers made by people that are moving uniformly with respect to each other. One option that is consistent with observations for SR is that there is one family whose clocks and rulers are right and ...


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I have no idea what you're on about, here. The one-sentence laymen's phrase for $p = \hbar~k$ is "Every moving object behaves like a wave, with wavelength equal to Planck's constant divided by that object's momentum." Usually you then add a one-sentence caveat, "The Planck constant is so tiny relative to most macroscopic masses that this wavelength is ...


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After some in depth discussion with the users, we found that there are a lot of assumptions in the question that does not hold water: "The commutation relation of $E$ and $t$ is not even a commutation relation. Time is not an operator in quantum mechanics." (thanks ACuriousMind for reminding again), it is a parameter thus there is no (straightforward) ...


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I believe John's answer is sufficient to guide you through the question. My answer will try to pinpoint your mistake in your working. Your working is mostly correct except the ratio $\frac{m_{rel}}{m_{rest}}$. To get the correct value for $\frac{m_{rel}}{m_{rest}}$, use Taylor expansion: $\frac{1}{\sqrt{1-\frac{2gh}{c^2}}} = ...


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This is a poor question because your rest mass does not increase when you climb 30m. However I can see how the examiner expects you to answer. If your mass is m then your energy when static at the bottom of the building is $mc^2$. To climb a distance h you have to hav work $mgh$ done on you, so your energy is now $mc^2 + mgh$. So the percentage change in ...


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By far the most convenient way for calculations is using four velocities. In lab reference frame components of four velocities are: $U_1=(\gamma,\gamma v_{lab},0,0)$ $U_2=(\gamma,-\gamma v_{lab},0,0)$ Where $v_{lab}$ is speed of particles as measured in lab, $\gamma=(1-v_{lab})^{-1/2}$. Now in rest frame of first particle those 4 velocities have ...


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There is a much easier way to solve the problem. Quick derivation: if something is moving past you at speed $u = \alpha~c$ then in a reference frame travelling at the speed $\beta ~c$ in that direction, the Lorentz boost puts its trajectory as:$$\gamma\begin{bmatrix}1&-\beta\\-\beta&1\end{bmatrix}\begin{bmatrix}c t\\ut\end{bmatrix} = \gamma ...


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Assuming that $0.890c$ is the approach speed of one particle in the frame of reference of the other, here's my suggestion. First of all, you've misinterpreted variables. If the convention of the symbols is typical then $u$ is not the numerical value you have been given. $u$ refers to the speed of one frame of reference relative to another frame of ...


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Does the accelerated rate of expansion of the Universe have any effect on the velocity of light? This is the subject of some debate. Check out the Wikipedia Variable Speed of Light article: "The idea from Moffat and the team Albrecht–Magueijo is that light propagated as much as 60 orders of magnitude faster in the early universe, thus distant regions ...


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The expansion of the Universe has no effect on the local speed of light. Any local measurement of $c$ will yield $c$, and $c$ won't change. There is one thing that often causes confusion about the speed of light or faster-than-light travel. A photon moving in an expanding space-time appears to move at an average speed faster than $c$. Consider a ...


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The Michelson-Morley experiment indicated (contrarily to it's original intent), that the speed of light perceived by an observer is not dependent on their speed relative to anything. Along these lines, our movement in space with respect to any arbitrary point has no impact on the speed of light measured in the earth system. Disclaimer: did not hear a ...


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A hyper cone exists in 4-dimensional Minkowski space. You may be confusing the curved geometric properties of a Euclidean solid in flat space with the warped properties of space-time in General Relativity. The equation of a hyper cone is: a^2 + y^2 + z^2 - w^2 = 0 There is no adjustment for relativistic speeds or gravitational fields. You might think of ...


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I understand the main question to be "What is the physical intuition/geometric consideration that led one to relate periodicity in space to momentum?". Or, in other words, why do we identify the physical property "momentum" with the mathematical operator $-i\hbar \partial/\partial x$? (This is essentially the same question, because the eigenfunctions of ...


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A physical quantity is introduced by its operational definition. Yes. Excellent. And physical quantities would include facts like whether an observer receives one signal between receiving two other signals or sees one mark between two other marks (clocks and rulers are designed on these principles) In general relativity we use a differential ...


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There isn't a simple answer to your question because, well, it depends. In the lab the only time I can think of that this issue arises is with gravitational time dilation e.g. experiments with atomic clocks or GPS satellites. In that case I would guess we'd use EM signals (light or radio) and correct for the known travel time. In astronomical contexts the ...


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Can you tell your absolute speed in space? Yes. You just look at the CMBR dipole anisotropy. This tells you how fast you're going relative to the universe, and that's as absolute as it gets. "From the CMB data it is seen that our local group of galaxies (the galactic cluster that includes the Solar System's Milky Way Galaxy) appears to be moving at ...


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The problem with questions like this is that they include many misunderstandings of physics! For example, you say "as one approaches light speed more energy is required to accelerate faster". What you may not be aware of is that in classical mechanics, it's also true that to an observer on the ground, the faster you are going, the more energy you need to ...


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That won't work, and here is why. It's subtle. Say that you are on a ship, leaving the solar system with some technology that is able to thrust you in such a way that you experience a constant acceleration of 1g, as measured on the ship. You can measure this by placing a 1kg weight on a scale. From the point of view of the passengers of the ship (where the ...


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I'm afraid this won't work. If for example you have a rocket motor capable of producing 1g of thrust then it will still produce 1g of thrust no matter how long you accelerate for (assuming you don't run out of fuel). From the perspective of the observer on Earth your acceleration will indeed slow down, but at the same time the Earth observer sees your time ...


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Andromeda and the Milky Way belong to a group of galaxies called the Local Group. The two galaxies are the largest galaxies in the group, so to a pretty good approximation their interaction can be treated as a two body problem, with the other galaxies in the group producing only minor perturbations to their motion. So as you suspected, it isn't the case ...


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Their mutual gravity will pull them towards each other, with the more massive galaxy causing more acceleration on the smaller. According to this article the more massive is the Milky Way, so it will cause Andromeda to accelerate more than the Milky Way, not that it really matters. Since interstellar space is mostly empty (i.e. there is a lot of distance ...


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Strictly speaking, the universe has no rest frame (that we know of). If you want to set the rest frame of the CMB to be stationary, (which is reasonable in many applications), you may compare the velocity of the Milky Way and the velocity of Andromeda with respect to the CMB rest frame. (I don't know if anyone has ever done this, however.) The CMB dipole ...


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The truth of the Lorentz transformation as an accurate description of the co-ordinate transformation between relatively uniformly moving observers needfully implies relativity of simultaneity. Contrapositively, the Lorentz transformation cannot be sound if simulteneity is not relative. So, in the sense that the soundness of the Lorentz transformation has a ...


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If you and your friends took some helicopters to the north pole and went up and then took off in different directions and flew at the same altitude you would feel like you were being bent towards each other, but yet as you all started to approach the south pole you would notice that you were all moving a way from other at first, you were all moving parallel ...



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