What's the physical reason that all (inertial) observers agree on the value of the spacetime interval $$ds^2 = (c dt)^2 - dx^2 - dy^2 -dz^2 \, ?$$  What would be the physical implications if different (inertial) observers wouldn't find different values of $ds^2$, analogous to how they find different time intervals and different distances?  EDIT: none of the linked questions (to which this question supposedly is a duplicate) explain why a constant speed of light implies that all observers agree on the value of the spacetime interval.
7253
This question is about why we have a universal speed limit (the speed of light in vacuum). Is there a more fundamental law that tells us why this is? I'm not asking why the speed limit is equal to $c$ and not something else, but why there is a limit at all. EDIT: Answers like &quot;if it was not..&quot; and answers explaining the consequences of having or not having a speed limit are not--in my opinion-- giving an answer to whether there is a more fundamental way to derive it from a law to explain this limit.
11274
asks why and how the speed of light is constant. I would like to ask a related, almost converse question:  Given that the speed of light is constant, how could I explain to a young-earth-creationist who is still a proponent of the disgraced Barry Setterfield  that the $c$-decay theory contradicts observed physical reality, as described and understood by all scientists, including those in the Physics and Astronomy fields?  Surely a variable speed of light would throw a monkey wrench into all sorts of fundamental laws or systems in physics? Surely it would be measurable on instruments today, if there was even a miniscule change in the  speed of light relative to a continuous cold caesium fountain atomic clock?  The fact that there is no observable change in any recent trustworthy data, is the beginning, middle, and end of the scientific argument. Am I correct?  My physics education ended at Grade 12. So I'm asking for a high-school level explanation.  I also wonder if $c$-decay theorists are contradicting themselves, or have sawn off the branch upon which they themselves sit; For example,  I wonder this; Would we have to modify our expectations of what the doppler effect (red shift) would look like, as light comes towards us from distant stars, if the speed of light is variable? Would we not have to throw all our doppler data out, and all the distances themselves, if the speed of light is variable in some undefined way, and thus unknown and unknowable?  The $c$-decay theory was spawned because redshift data indicates that the stars are too far away for their light to have reached us in the mere 6000 years the universe has existed.  In short, if $c$-decay theory exists to explain distant stars to young earth creationists and help them breathe a sigh of relief, wouldn't it be slightly-less-scientifically-bogus, to just dispute the doppler effect and the distance of those stars?
66800
There is a problem with my logic and I cannot seem to point out where. There's a rocket ship travelling at close-to-c speed v without any acceleration (hypothetically), and there is an observer AA with a clock A on Earth, and there's another observer on the rocket BB with a clock B and these two clocks were initially in sync when the rocket was at rest with a FoR (frame of reference) attached to Earth. Now, this rocket's moving and AA tells us that B is running slower than A by a factor of$$ \gamma = 1 /(1-v^2/c^2)^{1/2} $$ $$ t_A/t_B = \gamma $$ where v is the relative 1d- velocity between the two i.e the earth and the rocket! That would mean the time elapsed on A is greater than that on B but this will happen only in the FoR of AA? So $t_B$ in this equation must be the time on B as observed by AA? Is this correct? What do the terms mean in the equations? If the symmetry holds and BB doesn't accelerate, then BB could say that $$ t_B/t_A = \gamma $$ right? where $t_B$ and $t_A$ are the times on B and A with respect to FoR of BB? But I was solving this problem and I took the earth FoR of A, but the prof took the rocket FoR of B? Like how will I know which FoR to solve the problem from? It'd greatly help if the terms in all the above equations were laid down neatly! DO we even need these FoRs?? Because in all the solved problems the prof is't specifying any and is using random ones! Please help!!!! This is the question where I messed up. The first rocket bound for Alpha Centauri leaves Earth at a velocity (3/5)c. To commemorate the ten year anniversary of the launch, the nations of Earth hold a grand celebration in which they shoot a powerful laser, shaped like a peace sign, toward the ship.  According to Earth clocks, how long after the launch (of the rocket) does the rocket crew first see the celebratory laser light?  This must be 25 years. My reasoning is: If v = 3c/5 10v + vt = ct where t is time taken by the light to reach the rocket from Earth as calculated from Earth.. and I solved that for t. and added 10 years to that because the time starts at the launch of the rocket!  According to clocks on the rocket, how long after the launch does the rocket crew first see the celebratory laser light?  This is 20 years. Here, I say: If it takes 25 years as observed by clocks on earth for the laser to reach the rocket, what should be the corresponding time as seen on a clock on the rocket? Using the formula: 25 = $\gamma$t where $\gamma$= 5/4 solved for t!  According to the rocket crew, how many years had elapsed on the rocket's clocks when the nations of Earth held the celebration? That is, based on the rocket crews' post-processing to determine when the events responsible for their observations took place, how many years have passed on the rocket's clocks when the nations of Earth hold the celebration?  For this, I did the following: 10 years on earth = T years on rocket ship where T must be lesser than 10 as observed from Earth FoR! Therefore, T= 4(10)/5 years = 8 years! But, prof says, 10 years on Earth = T years in rocket ship where T must be GREATER than 10 as observed from the rocket FoR??? Therefore, T = 10(5/4) years = 12.5 years!! What does this question actually want?
113489
In all the sources I’ve been able to find, the Minkowski metric appears ad hoc, or is defined analogously to the euclidean metric. I’d love to see an argument why this metric (time coordinates positive, space coordinates negative) must follow from the constancy of the speed of light. It is clear that the Minkowski metric is preserved under the hyperbolic transformation of space-time, but likely others are as well. Why this particular metric and not something else.  Consider the determinant function of an n by n matrix. It has a god awful mathematical form involving the sum of n ! terms. Yet all you need to get the (unique) formula are a few postulates — the determinant of the identity matrix is 1, the determinant is a linear function of its rows (or its columns), interchanging any two rows of the determinant reverses the sign of the determinant, etc. etc. This basically determines the (unique) formula of the determinant. I’d really like to see the Minkowski metric come out of something like that.
86945
Can anyone provide me with a complete mathematical proof about why an observer frame  cannot surpass the vacuum speed of light? I have looked for answers in Quora and FB groups but no one is really convincing. In fact the majority assume that the constancy of vacuum speed of light in all referentials without any proof, other people &quot;only&quot; say that it can be proved using the causality principle but without any proof, which is really frustrating! I hope my curiosity will be satisfied in PhysicsStackexchange, have a nice day!
56665
My question is not about why speed of light has a particular constant value which has been sufficiently addressed in other questions on SE already. I want to know whether the fact that speed of light has to be constant is a consequence of other fundamental laws of Universe, or is it just a fundamental law itself.  My motivation for asking this question is thinking about Special Theory of Relativity. One way of understanding why time dilation and length contraction happen is to look at it from the perspective of constancy of speed of light. Alice who is on the moving source of light and Bob, a fixed observer see the speed of light as constant. In order for Alice to see the speed of light as a constant value, time slows down and length expands (in forward direction) thus keeping the speed constant.   In the  I'm doing, the Professor tells that this constancy of speed of light could be more generally referred to as, "Principle of wave constancy". That is, the speed of any wave is a property of the medium (and thus it remains constant). So, the speed of a wave in water, for example, remains constant even if the source of the waves (say a paddle keeps moving). Frequency and wavelength may change but the speed remains constant. In a similar fashion, light is an electromagnetic wave and thus its speed is constant. And the constant speed is defined by the medium (which happens to be vaccum i.e., no medium for light).   My questions are twofold now -    Is the above reasoning correct to explain why speed of light is constant? (i.e., thinking of it as constancy of any wave in general) Theoretically, could there be other waves that could travel through vaccum at a different speed? If yes, time dilation and length contraction will have to be different.   I'd appreciate it if the answer is kept non-technical (essential math is okay). More importantly, I request you to try and explain without using General Theory of Relativity. Because I'm thinking how the scientific understanding by 1905 had already knew that speed of light has to be constant.
95353
I have never seen a “causality operator” in physics. When people invoke the informal concept of causality aren’t they really talking about consistency (perhaps in a temporal context)?  For example, if you allow material object velocities > c in SR you will be able to prove that at a definite space-time location the physical state of an object is undefined (for example, a light might be shown to be both on and off).  This merely shows that SR is formally inconsistent if the v &lt;= c boundary condition is violated, doesn’t it; despite there being a narrative saying FTL travel violates causality?   Note: this is a spinoff from the question:  .
16430
There's another question on physics.SE whose , if I have understood it correctly, explains that the farther the points are in space the faster they are moving away from each other.     Actually, the apparent speed with which two of the points on the   circle in a distance $D$ of each other would move relative to each   other will be $v = H_0 D$ where $H_0$ is the speed the balloon itself   is expanding.   This means that even the ruler sitting on my desk is expanding, just that the expansion is very very slow owing to the ruler's small size relative to the cosmic scales. However, if everything is scaling then everything should apparently be of the same size to an observer in such an expanding space. The  quote from the show "The Big Bang Theory" comes to mind.   To be able to counter that, there're some phenomena that we can observe which are scale invariant. Red-shift would be one such phenomenon. But is red-shift the only way we can tell that the space is expanding? Is there a logical flaw in my thinking? I know that red-shift is a result of the Doppler effect - there's bound to be a decrease in the frequency of a signal, as perceived by an observer, emitted by a source which is moving away relative to the observer.
40940
When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(c^{2}dx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}   $$ where $ x^0,x^1,x^2,x^3 $ come from $ x^{\mu} : \mu = 0,1,2,3 $, and $ c $ is the speed of light. The first resource I had access to--I have to do a bit of digging for the exact paper--of course addressed that this invariant takes the tensor form: $$ ds^2 = g_{\mu \nu}dx^{\mu}dx^{\nu} $$ as well. This two things I've seen in all of my texts and online resources regarding resources. The element in question that varies between authors, is the time coordinate, $ x^0 $. When it was first explained to me, it was using the standard Cartesian representation for the spatial portion of the coordinates, and the time coordinate was labeled as $ x^0 = ict $. Squaring gives $ (x^0)^2 = -c^2t^2 $, and applying differential calculus, we get $ (dx^0)^2 = -c^2dt^2 $. Sensible and expected to have the tensor formula spit out the Minkowski Metric. The author then later explicitly states the coordinates are $ x^0 = ict, x^1=x,x^2=y,$ and $x^3 = z$.  My questions is then, why is it most authors on the subject omit the imaginary unit on the time coordinate? For example, .   The only reason I can fathom omission is if the author is using metric signature $ [+,-,-,-] $, where I started off learning the theory with signature $ [-,+,+,+] $ which may be the reason seeing the time coord with no imaginary unit seems dissonant to me. All help appreciated!  Edit: After reading the other answers, my questions is now one of why and how (mathematically) do we obtain the Minkowski Metric Signature. More specifically the one element with a different sign.
107594
In his 1905 , Einstein derives the Lorentz transformation using the two postulates of SR;constancy of $c$ for all inertial frames and the Invariance of the laws of physics for all inertial frames.   I'll summarize his mathematical derivation and then ask one specific question about it.  So we consider two frames $(x,y,z,t)$ and $(\xi,η, ζ,\tau)$ in relative motion along the x-axis with velocity $v$, and we're interested in finding a spacetime transformation that relates thier coordinates.   We consider some arbitrary point $x'=x-vt$. This point is at rest in  $(\xi,η, ζ,\tau)$ since it's moving with $v$, Therefore this point has $x',y,z$ coordinates that is independent of time, in other words the distance between that point and the origin of $(\xi,η, ζ,\tau)$ is constant.    We consider the following scenario: hit a beam of light from the origin of $(\xi,η, ζ,\tau)$ at $\tau_0$ arriving at the point $x'$ at $\tau_1$ and then being reflected and arrive at the origin of $(\xi,η, ζ,\tau)$ at $\tau_2$.  So that we have: $1/2(\tau_0+\tau_2)=\tau_1$. Since $\tau$ is a function of $(x,y,z,t)$ we have:  $\dfrac{1}{2}[\tau(0,0,0,t)+\tau(0,0,0,t+\dfrac{x'}{c-v}+\dfrac{x'}{c+v})]=\tau(x',0,0,t+\dfrac{x'}{c-v})$  Assuming that $x'$ is infinitely small then taylor expanding this equation and approximating it to first order we get:  $\dfrac{\partial \tau}{\partial x'}+\dfrac{v}{c^2-v^2}\dfrac{\partial\tau}{\partial t}=0$  Solving it then we have:   $\tau=a(t-\dfrac{v}{c^2-v^2}x')$  where $a$ is some unkown function of $v$ (in fact $a=1$).  Finally consider a beam of light emitted from $(\xi,η, ζ,\tau)$ at the origin, it's $\xi$ coordinate is given by  $\xi=c\tau=ca(t-\dfrac{v}{c^2-v^2}x')$  It's given by $\dfrac{x'}{c-v}=t$ in $(x,y,z,t)$, plugging in for $t$ we get:  $\xi=a\dfrac{c^2}{c^2-v^2}x'$  He then states:      Substituting for $x'$ its value, we obtain   $\xi=a\dfrac{1}{\sqrt{1-v^2/c^2}}(x-vt)$   ...   My question is:  1) the three equations $\tau=a(t-\dfrac{v}{c^2-v^2}x')$ and $\xi=c\tau$ and $x'=x-vt$ when combined together gives :  $\xi=c\tau=ca(t-\dfrac{v}{c^2-v^2}x')=a\dfrac{c^2}{c^2-v^2}x'$, since $x'=x-vt$ by plugging in we get:  $\xi=a\dfrac{c^2}{c^2-v^2}(x-vt)=a\dfrac{1}{1-v^2/c^2}(x-vt)$ not $a\dfrac{1}{\sqrt{1-v^2/c^2}}(x-vt)$ .  But He says     Substituting for $x'$ its value, we obtain   $\xi=a\dfrac{1}{\sqrt{1-v^2/c^2}}(x-vt)$   ...   All these equations are copied from Einstein's original paper, So what is wrong with my calculations that does not make it match up with that of Einstein?
99993
I (desperately) need help with the following:  What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$  I don't know how to transform a metric into a geodesic...! There is no need to start from the Lagrangian. I know that $$0=g_{ij}V^iV^j$$ where $V^i={dx^i\over d\lambda}$  where $\lambda$ is some parameter. But I don't know what that parameter is nor how to find the geodesic.  Many thanks!! Please help!
116739
In books the equation for length contraction is derived by supposing that the velocity of the spacecraft is the same for both observers. So the question is that, is the velocity really the same for both observers? Whereas there must be some trouble in the velocities noted by observers. I mean: If the observer on the spacecraft measures the velocity of the spacecraft to be $v$ then is it possible for the other observer to measure the same velocity from the Earth?
105833
I recently watched  by Veritasium where he talks about the One Way Speed of Light and talks about the limiting case where in one direction the speed of light is $c/2$ while it's instantaneous in the other. He also says this is perfectly fine according to our Physics theories. He also points at Einstein's assumption in his famous 1905 paper where he assumes that the speed of light is same in all directions. This made me ask this question is taking the speed of light same in all directions an axiom of some sort? As I've often read no information can be sent at more than the speed of light but here one-way taking the speed to infinite makes no difference. So are all of our physics theories based on the assumption and what would happen if light turns out to be moving at different speeds in different direction? Will that enable transfer of information faster than the speed of light and is there any way for us knowing that the transfer happens faster than the speed of light? The video takes a Earth Mars case where he says it isn't possible for us to every realize this discrepancy but is there a more general proof which says it isn't possible
32543
I am reading A Brief History of Time by Stephen Hawking, and in it he mentions that without compensating for relativity, GPS devices would be out by miles. Why is this? (I am not sure which relativity he means as I am several chapters ahead now and the question just came to me.)
49355
I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me.  The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\gamma (x-vt), \gamma (t-v\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\Delta s_A^2$ is greater than Bob's $\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path.    According to Alice: $\Delta s_A^2=-c^2\Delta t^2 + \Delta x_A^2$ where $\Delta t=\dfrac{2d}{v}$ and $\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\Delta s_A^2=-\dfrac{4c^2d^2}{v^2}$  Still according to Alice, Bob's $\Delta s_B^2=-c^2\Delta t^2+\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\Delta x_B=2d$ ($\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\Delta s_B^2=-\dfrac{4c^2d^2}{v^2}+4d^2$.     The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\Delta s^2$ is conserved under Lorentz transformations.    According to Bob: $\Delta {s'}_A^2=-c^2\Delta t'^2 + \Delta {x'}_A^2$ where $\Delta t'=\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\gamma d'$ as lengths contract by $\gamma$ i.e. $d'$ is smaller than $d$ by factor $\gamma$. Also, $\Delta {x'}_A=2d'$, so we have $\Delta {s'}_A^2=-\dfrac{4c^2d'^2}{v^2}+4d'^2=-\dfrac{4c^2d^2}{\gamma ^2 v^2}+\dfrac{4d^2}{\gamma ^2}=4d^2\dfrac{v^2-c^2}{\gamma^2 v^2}\neq \Delta s_A^2$  Although I haven't included them here, my calculations for $\Delta {s'}_B^2$ and $\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated!
103439
I've often heard that Einstein shattered the notion of absolute motion (i.e. all things move relative to one another) and that he established the speed of light as being absolute. That sounds paradoxical to me; I cannot understand how the two concepts can be reconciled.  Before going further, I'd like to say: 1) Over the years, I've seen many layman's explanations on these topics (including the nice YouTube video by Vsauce, ). I understand everything that's said (or, at least, I think I do). Just nothing I've found seems to address this apparent contradiction. 2) More recently, I've tried to find the answer on my own. That includes searching the posts on this site. Some come close (), but nothing I've been able to find seems address specifically what I'm asking.  Back to the question: Relativity shows us that there is no universal frame of reference by which to judge motion, so object A might be reckoned as moving at 10&nbsp;m/s relative to object B or as stationary relative to object C. This is fine for me. I can grasp that the universe has no intrinsic coordinate system, that we only think that way on Earth because we have the ground to move over.  Then there's the speed of light (in a vacuum). The speed of light is the ultimate "speed limit," it's often said. But if there is no universal frame of reference, how can there be any such speed? The very idea only make sense if there is a universal frame.   If one object is moving (uniformly) at 60%&nbsp;c and another object is also moving at 60%&nbsp;c, but in the exact opposite direct, then from the perspective of either one (if they could still see each other) the other would appear to violate that speed limit. All these spacetime bending consequences used to explain why nothing can move past this speed only seems to enshrine the concept that there is some ultimate speed standard. If there is only relative speed, then the concept of light having a specific speed in the vacuum should be a nonsensical one since it having speed (x m/s) would only make sense when measured against some other body.   Since I was very young, it has always sounded to me like motion is only mostly relative, that until you get close to the speed of light, the effects of an absolute frame of reference are negligible. Perhaps that there is an actual fabric of space which everything moves relative to, which is why there is something to expand between galaxies (faster than light can propagate) in the metric expansion of space. Growing up, I always thought this would just start makes sense with time. Now I'm up to a first year (college) level in physics, I even know basic calculus, yet I'm still hopelessly confused.  EDIT:  Thank you to whoever suggested this may be a duplicate of . It and others are very much related and at least partially answer my question. Unfortunately, explaining that distances become shorter and time becomes slower as a way to stop you from exceeding the speed of light does not explain how that speed is not an absolute.  By my reckoning, if all speed is relative, then no matter how fast you go light should always race away from you at the same apparent speed. I.e. there should be no speed limit. For there to be a speed which you cannot exceed or you would catch up (and make time irrelevant) requires the very concept of some external speed by which light can travel and nothing else can reach - thus my logical paradox continues unabated.
14701
In Special Relativity, the spacetime interval between two events is $s^2 = -(c{\Delta}t)^2+({\Delta}x)^2+({\Delta}y)^2+({\Delta}z)^2$ giving the Minkowski metric $\eta_{\mu\nu}=\text{diag}(-1, 1, 1, 1)$. What is the justification for making time have a negative coefficient, and how closely is that related to the 2nd law of thermodynamics? Sure, by letting $\eta = \text{diag}(1, 1, 1, 1)$, we get a pretty boring spacetime, and the boosts in the Poincaré group become trig instead of hyperbolic functions, but what's the physical reasoning behind this?