I was wondering, if two cars (A and B) are going at different velocities let's say A=90km/h, B=80km/h then if we add the two speeds we get 170km/h meaning they are relative to each other.  But if one of the cars were going at the speed of light and the other at 90km/h it's impossible to add them because nothing can go faster than the speed of light meaning they are/aren't relative.?  So my question is: is light relative?
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.
Is there a way to separate variables between multiple notebooks? Why can't Mathematica integrate this? What's the preferred method of writing AJAX-enabled plugins? Proof of the First Isomorphism Theorem for Groups The statement is the First Isomorphism Theorem for Groups from Abstract Algebra by Dummit and Foote. This proof was left as a exercise, so I'd like to check if all is ok. In particular, I'm a bit worried about the (*) line. It looks a bit awkward, but is it too bad? Any suggestions?    Theorem: If $\varphi : G \rightarrow H$ is a homomorphism of groups, then $ker \varphi \trianglelefteq G$ and $G/ker \varphi \cong \varphi (G)$.  Proof: Let $\varphi:G \rightarrow H$ be a group homomorphism. If $g \in G$ and $h \in ker \varphi$, then we have that $ghg^{-1} \in ker \varphi$ because $$\varphi (ghg^{-1})= \varphi (g) \varphi (h) \varphi (g^{-1})= \varphi (g) \varphi (g^{-1}) = \varphi (gg^{-1}) = \varphi (1) =1,$$ so that $ker \varphi \trianglelefteq G$. Now, define $\phi: G/ker \varphi \rightarrow \varphi (G)$ by $$\phi (g(ker \varphi))=\varphi(g),$$ for each $g(ker \varphi) \in G/ker \varphi$, for some $g \in G$. This is well defined because if $g' \in g(ker \varphi),$ then $$\phi (g'(ker \varphi))= \varphi(g')= \varphi(g)= \phi (g(ker \varphi)).$$ Also, this is a group isomorphism because for each $\varphi (h) \in \varphi (G),$ for some $h \in G$, we have $$(*) \ \phi^{-1}\{\varphi(h)\}= \phi^{-1} \varphi[g(ker \varphi)]= \phi^{-1} \varphi(\pi^{-1}\{g(ker \varphi)\})= \{g(ker \varphi)\},$$ a set with a single element of $G/ker \varphi$, so that it is a bijection, and for every $g(ker \varphi), g'(ker \varphi) \in G/ker \varphi$, for some $g,g' \in G$, we have $$\phi [g(ker \varphi) g'(ker \varphi)]=\phi [(gg')(ker \varphi)]=\varphi(gg')=\varphi(g) \varphi(g')=\phi [g(ker \varphi)] \phi [g'(ker \varphi)],$$ so that it is a group homomorphism. How to check if Receiver is registered in Android? In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True? Provided we have this truth table where "$p\implies q$" means "if $p$ then $q$":  $$\begin{array}{|c|c|c|} \hline p&amp;q&amp;p\implies q\\ \hline T&amp;T&amp;T\\ T&amp;F&amp;F\\ F&amp;T&amp;T\\ F&amp;F&amp;T\\\hline \end{array}$$  My understanding is that "$p\implies q$" means "when there is $p$, there is q". The second row in the truth table where $p$ is true and $q$ is false would then contradict "$p\implies q$" because there is no $q$ when $p$ is present.  Why then, does the third row of the truth table not contradict "$p\implies q$"? If $q$ is true when $p$ is false, then $p$ is not a condition of $q$.  I have not taken any logic class so please explain it in layman's terms.       Administrative note. You may experience being directed here even though your question was actually about line 4 of the truth table instead. In that case, see the companion question  And even if your original worry was about line 3, it might be useful to skim the other question anyway; many of the answers to either question attempt to explain both lines. Why was Gotham City changed from a mostly fictional city to New York City in the Nolan trilogy? In Batman Begins, Gotham City has a futuristic/fictional and run down look: think of Oldtown, the trains built by the Wayne Corporation etc.   In The Dark Knight Rises, Gotham City looks like plain old real world New York City — we don't see the trains anymore. Oldtown is just a standard poor neighborhood. Lots of the locations used in the movie are easily identified as real world NYC locations: the stock exchange or East River bridges for example.   My question is: Why did Nolan change Gotham city from a shady fictional more cartoon-like town to plain old NYC?   Edit: The examples from Batman Begins and The Dark Knight Rises were just for illustration; I didn't intend to limit the question to those two movies.  Edit 2: I specifically want to know why Gotham City was changed from a cartoon-like city in Begins to an almost real-world NYC in The Dark Knight Rises (via not so cartoon-like but not yet real-world like Chicago in The Dark Knight Rises). Why it's NYC or Chicago is not so important for me. But why, e.g., were the trains which featured prominently in Begins gone in The Dark Knight Rises? There was only a very short in-universe time-span between the events of the two movies, yet Gotham changed drastically. How coherent light is created for single slit experiment? Deleting a comment (mine) shouldn't count as a comment vote I had an old question that I revisited in order to link it in a new answer. I noticed that in the comments Steven A Lowe had a funny comment that I -- at the time -- had responded to with "why can't I upvote comments?"  Well, of course, now I can.  So, I voted Steven's comment up then went to delete mine. I got the warning box that I can only submit comment votes once every 5 seconds. This seems like a bug to me. Deleting is not the same as voting. What algorithm is used by computers to calculate logarithms? Matrix with labels nested in braces I need to get UTF-8 working in my Java webapp (servlets + JSP, no framework used) to support äöå etc. for regular Finnish text and Cyrillic alphabets like ЦжФ for special cases.  My setup is the following:   Development environment: Windows XP Production environment: Debian   Database used: MySQL 5.x  Users mainly use Firefox2 but also Opera 9.x, FF3, IE7 and Google Chrome are used to access the site.  How to achieve this? What is charge? I know this isn't the right place for asking this question, but in other places the answers are so awfull.. I'm studying eletricity, so, I start seeing things like "charges", "electrons has negative charges",etc. But I didn't quite understand what charge is. I search a little bit on the internet and found it related to electromagnetic fields, then I thought "negative and positive may be associeted with the behaviour of the particle in the field, great!", but the articles about e.m. fields already presuppose "negative" and "positive" charges. In other places, I see answers relating charges to the amount of electrons/protons in an atom, but if that's right, the "negative" electron is an atom without any protons? What about the neutron? So, my questions are (1) What are charges; and (2) How a particle can "be" electrically charged. What does that really mean? Thanks for your time. Why can we always drop any constant term in a Lagrangian density in quantum field theory?  This issue is somehow related to the constant term being some kind of cosmological constant.  Can you please explain this issue? One of is followed by plural noun/pronoun and singular helping verb How do you set the expiry date or a maximum age in the HTTP headers for static resources in IIS Entries I can safely exclude doing backups Where did they go / do they went? Where did they go?   Where do they went?   Can you please advise which of above forms is correct?  I'm pretty sure that first one is fine but wondering about second one.   Can I use a second form as well or it's incorrect? In polar coordinate how we can get $dx\;dy=r\;dr\;d\theta$?  with these parameters:    $r=\sqrt{x^2+y^2}$ $x=r\cos\theta$ $y=r\sin\theta$  Tanks. Partial Fraction Decomposition?