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For the same reason as in Newtonian gravity, which after all is an emergent framework from General Relativity: the equations have to be solved. In Newtonian gravity the orbits are solutions of conic sections, circles , elipses, parabolas and hyperbolas. In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach ...

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Proper time is defined for an arbitrary time-like trajectory $x^\mu(t)$ and an arbitrary spacetime metric $g_{\mu\nu}(x)$ by the formula $$\tau=\int\sqrt{g_{\mu\nu}(x)\dot{x}^\mu\dot{x}^\nu} dt.$$ I would recommend to compare this with the general formula for the length of a curved path to get a feeling that this is something invariant, and does not depend ...

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Let us start from the notion of affine space next focussing on the Euclidean $3$-dimensional physical space and finally coming to Minkowski spacetime. An affine (real) $n$-dimensional space is a triple $(\mathbb A,\vec{\cdot}, V)$, where $\mathbb A$ is a set whose elements are called points, $V$ is a real $n$-dimensional vector space and $\vec{\cdot} : ... 0 The concept of a line is totally independent of the inner product and makes sense for a general vector space; mathematically a line is an affine one-dimensional subspace. So yes, two points do determine a unique line, as long as you're in a flat vector space. If you introduce curvature as in General Relativity, things change. The only difference with ... 1 Let me answer this question systematically. 1)The first is how to think of causality and locality. Locality (sometimes people use locality to talk about microcausality but that's not very important) is the statement that two events cannot communicate with each other if they are separated by spacelike distances. So what does this mean? Suppose you have a ... 2 The true meaning of all the trouble is that coordinates are just tools for calculations and don't have any intrinsic meaning in GR. You should never interpret the coordinates alone. These can be very misleading and in particular become singular in some regions of spacetime (because coordinates are local). And never trust the labels of coordinates: Fact that ... 0 There is a similarity. In quantum mechanics the uncertainty relation is determined as an anticommutator. There is also non commutativity in Lorentz transformation boosts so there is an order dependency of (some) repeated Lorentz transformations. Understood in a similar way as quantum mechanics this could be seen as an uncertainty relation with associated ... 0 Timelike is when an event is inside the lightcone (as you have mentioned) and as a result, one event CAN affect the other event (there can exist a causality between the two events. E.g. lets say there are two events, where I shoot a laser and another event where someone gets hit by a laser. If they are timelike seperated then the laser that hit the dud could ... 5 The spacetime interval is a relativistic invariant, and is proportional to the travelers proper time. So in a since you are traveling one second per second, per your own wrist-watch. Every other measurement would be the speed of some other inertial reference system, measured with your clock. Let$s^2 = x^2 + y^2 +z^2- (ct)^2$, where$x$,$y$,$z$are ... 0 The question is whether time is an operator in the sense of${\hat T}|t\rangle~=~t|t\rangle$. This at first glance would seem to make sense because we do have a position operator${\hat X}|x\rangle~=~x|x\rangle$. However, this does not work. This is a subtle question in many ways. Quantum mechanics is unitary. Consider a state vector$|\psi(t)\rangle$... 5 Is time made into one observable? No. It is known that an operator$T$that satisfies$[H,T]=i\hbar$is either self-adjoing and$H$unbounded below or anti-self-adjoint. Therefore, the theory is either intrinsically flawed (arbitrary negative energy) or$T$is not observable (anti-self-adjoint$\Rightarrow$imaginary eigenvalues). 0 The theorem that time is not an observable is quite general, Unruh W., Wald R. prove this in "Time and the interpretation of canonical quantum gravity, Physical Review D Volume 40 issue 8 1989" in the following form: "... in the context of ordinary Schrodinger quantum mechanics, no dynamical variable in a system with Hamiltonian bounded from below can act ... 0 In that sense, in Relativistic Quantum Mechanics based on Dirac's Equation, is time made into one observable? No, but it is treated on equal footing to the space coordinates. Dirac's equation is $$(i \hbar \gamma^\mu \partial_\mu - mc )\psi$$ where$\psi$is not the classical wavefunction, but a four-component spinor. The components of the spinor are ... 1 Invariants are useful, in general, because they represent something that all observers can agree upon. Relativity showed us that the concept of time-intervals, spatial-distances, and even sequences of events can be drastically different from different observers. So how can one observer 'relate' to another? I.e. how could I, standing still, figure out what ... 1 The spacetime interval invariance property allows us to, for example, compare the rate of time passing for two observers moving at relative velocities to each other. Although no observer in the universe is at complete rest, the interval is a benchmark for comparison of the physical effects of differences in velocity, or indeed location. Say one observer is ... 0 Proper time is a directed scalar. As of this writing, there is no mathematical terminology or analysis technique that does justice to defining it, including and especially as proper time was defined by Minkowski, hyperbolic / rotationally bound to something else defined as space, which is the same directed scalar. For a directed virtual or real EM wave of ... 1 Any object with mass that accelerates (is it linear or angular acceleration) produces gravitational waves, though in most occasions those will be much too small to be detected. As @CuriousOne pointed out, same happens with electromagnetic waves and accelerating charges. The gravitational waves that can be detected usually come from very massive objects (such ... 1 our aging is directly proportional to metabolism of body and division,growth and death of cells of body and if gravity has some effect on the rate of above aspects astronauts will be definitely younger 2 The first answer has all the results, but I will try to show some calculations, cause I have been writing them since there was no answer. It is known from General Theory of Relativity (GTR) that the closer you are to a massive object - the slower the time goes. On the other hand Special Theory of Relativity (STR) gives us the next statement: the faster you ... 4 This is anwered in Gravity on the International Space Station - General Relativity perspective, where we learn that time dilation in the ISS with respect to Earth equator is 1.00000000028655. So after 17 years for us, the astronauts would come back younger by about 0.15 seconds than if they have stayed on the ground. Note that a full GR treatment is ... 2 Although I do not quite follow your analogy, there are a few things that can be said about mutilating spacetime in general. To be very non-technical and a little casual... Firstly, the fundamental idea of spacetime is that it is the arena of physics: within a spacetime effects propagate from place to place through time and typically if two places A & B ... 2 In relativity, gravity is the word we use to describe curved space. Don't think of curved space as simply 'no longer flat', also consider that it includes a gravitational gradient that will cause matter to move across that gradient. As an example, Earth bends the space around where you are right now, with a vertical gravitational gradient, causing you and ... 0 Physically, the question of what happens precisely at the corner points isn't especially meaningful. Instead, you could think about phenomena in a neighborhood of the points (e.g. flux through a circle or tube enclosing the singularity), and use that to characterize the point itself. By definition, a manifold should be locally Euclidean. Any open ... 0 You are using the wrong scientific terminology but you are right in a sense. The speed of light is indeed determined by the environment it travels in. We don't call it medium in relativity or physics, we call it spacetime. Spacetime has a geometry, a 4 dimensional (ignoring quantum gravity) manifold that has 1 time dimension and 3 spatial ones. The simplest ... 0 As described in this article, whenever we are talking about gravitational waves in cosmology, we actually assume the underlying spacetime to be flat (so just Minkowski metric$\eta^{\mu\nu}$) and a small perturbation$h^{\mu\nu}$causing ripples in the metric is the graviton field: $$g^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu}+O(h^2)$$ Truncating this expansion ... 2 Gravity, in general, couples very weakly to matter --- that's why it is often called the 'weakest force'. You can see this by examining the 'coupling constants'---where gravity is$10^{37}$times weaker than electromagnetism, or comparing how much 'stuff' you need to get equivalent forces---where gravity is about$10^{32}$times weaker. Your second ... 4 The Gödel universe is homogeneous and every observer anywhere in the universe observes the universe to be rotating around them. So a Gödel universe has no centre. 1 When you sit in your F1 car and turn right with 1g of lateral acceleration, you feel a centrifugal force (acceleration) of 1g pulling you in the opposite direction (left, in the car's frame). That is, there is a fictitious force caused by your curving reference frame. This is a Galilean point of view--an accelerated reference frame causes fictitious forces. ... 1 Welcome here. From your profile I see that you are at the beginning stages of learning physics. This is an arduous process that needs a lot of elbow grease in solving problems and/or doing experiments in order to get a basic intuition for the subject. Here is a simplified answer to your questions: Why do most theories about what Dark energy and Dark ... 1 Physics does not answer "why" questions at the level of basic laws and postulates, i.e. "why" this postulate. Newtons gravitational law : Newton's law of universal gravitation states that a particle attracts every other particle in the universe using a force that is directly proportional to the product of their masses but also inversely proportional to ... 4 A better way to think of it is "speed of causality". That's the fastest any cause-and-effect will spread over space. With nothing to cause it to go slower, changes to electric and magnetic fields will occur at that speed. No coincidence that changes to spacetime (causing gravity) propigate at the same speed. You really need to show how Minkowski spacetime ... 2 Yes, I agree with your interpretation of Minkowski's statement. You might be interested in reading the answers to What is time, does it flow, and if so what defines its direction? as I discuss exactly this point in the first part of my answer. The key point is that there is no observer independent way of separating the time and spatial coordinates so ... 0 You say: My research into gravity indicates that warped spacetime, with time as the major influence, is gravity then ask: why would a mass seek a location where time runs slower? But the problem is that few of us will recognise the qualifier with time as the major influence, and few of us would describe the motion in a curved spacetime as mass ... 2 Nonsense. Maxwell derived his electromagnetic equations, with$\epsilon_0$and$\mu_0$, and those quantities were known. The fact that his equations led to the speed of electromagnetic waves to be, in terms of$\epsilon_0$and$\mu_0$, equal to the approximately then known speed of light is a big part of what led Maxwell to conclude that light is ... 1 Physics is full of theories that to the outsider seem to pose a chicken and egg problem, but there is no issue: both "mass tells space how to curve" and "space tells mass how to more" are axioms of a conceptual structure whose validation is that it's predictions agree with reality. There is no reason to expect one to come "before" the other because they are ... 12 Not really. The "speed of light" has very little to do with light; it is built into the actual geometry of spacetime independent of what matter fills it. In particular,$\epsilon_0$and$\mu_0$don't tell us anything physical about the vacuum; looking at the (simplified) expressions$$E = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2}, \quad B = ... 2 I am adding this answer to respond to your last followup comment on L.Levrel's answer. Point One: Alice and Bob have to agree on how many times Alice's clock ticks per rotation, and they have to agree on how many times Bob's clock ticks per rotation. Therefore, if Alice says Bob's clock is slow, Bob must say Alice's clock is fast, and vice versa. Point ... 1 I assume that it is a time varying "bent" , which is a gravitational wave. Only charges radiate electromagnetic waves. At the elementary particle level photons will be produced only during the interaction time of changing space, i.e. graviton-charged_particle interaction. Classically, accelerated and decelerated charges radiate, so a classical ... 1 Only from charges. It's the quantum property that couples to photons. Gravity couples to anything, but there has to be some charge acccelerated to create photons. So gravity could accelerate it, but it also would depend on what is your frame of reference as to whether you'd see it. Another way to create photons from charges that normally don't exist is ... 0 Is important to realize time dilation and the Doppler effect are 2 frame-dependent parts of a single relativistic phenomenon: the constancy of the magnitude of four-velocity (which is always ||c||). Requiring that all inertial observers see a four-velocity with magnitude ||c|| leads to frame dependent observations of time dilation and Doppler effect. For a ... 1 It depends what exactly you mean by "coordinate". If your Lagrangian/Hamiltonian is time-independent, then you may consider time to be purely a parameter parametrizing e.g. the integral curves of the vector field associated to the Hamiltonian on phase space. If your Lagrangian/Hamiltonian is time-dependent, you should indeed properly consider your theory on ... 1 I think this issue is best clarified by closely looking at the way time is mixed into coordinate frame transformations in Classical Mechanics as opposed to Relativistic Mechanics. Let's take the case of an observer, Alice, moving at velocity$v$in the positive$x\$ direction away from her friend Bob. Both Alice and Bob are looking at an object situated at ...

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In special relativity there are two major assumptions: -the laws of physics are the same in all inertial frames -the speed of light that you observe is always the same, (thus independent of the relative motion between the light source and the observer). From this two assumptions follows the famous Lorentz transformations. In these Lorentz transformations ...

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The set of transformations that leaves the speed of light unchanged is the Lorentz group. Representation theory enables us to investigate the irreducible representations of the Lorentz group. The lowest-dimensional representations act on scalars four-vectors However, take note that usually we consider representations of the corresponding Lie algebra ...

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Yes, the video is an accurate description of the way that relativity describes motion in a gravitational field, and actually I think it's very well done. However you need to remember that in general relativity it is spacetime that is curved i.e. time is curved as well as space. It's impossible to describe curvature of time in any simple and intuitive way or ...

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If there are eleven dimensions as M-Theory asserts Let us suppose it is true would that mean that the majority of what we are made from exists in the seven other dimensions? Define dimension: In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to ...

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A simplistic answer is, 3/11ths of us exists within the normal three dimensions, and the other 8/11ths exists in the hypothetical extra dimensions. This answer does make string theory sound silly (which personally I think it is :) , but i can't see why it should not be true.

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