Derive Lorentz Transformations

let's imagine that we have two inertial frames of reference so in this case I have taken two rectangular Cartesian coordinate frames of reference and I've named them s and s - so one has the coordinates X Y Z and time and the other has a coordinate X dash y dash Z dash and for the sake of simplicity I have taken it in such a manner so that XYZ axis are parallel to X dash y dash and Z dash axis respectively and also the as - frame of reference is moving with a certain constant velocity V in the x axis only there is no motion happening in the Y and the y dash axis okay now a frame of reference is any kind of place from which you are making a measurement so for example if you're standing on the ground and you're looking at some kind of a physical event that ground is going to be your frame of reference if you're traveling inside a train and you're looking outside at some physical event your train is going to be your frame of reference and what we mean by physical event anything that happens at any given point in space and time so for example the explosion of a firecracker or the collision of two balls if I drop a ball it falls to the ground or the switching of a going of a light bulb they can all represent different kind of physical events so in this case if there are two observers let's suppose on both these two frames of reference they said there is one observer in the ass room of reference and there's another observer in the ass - tone of reference and they both look at some common physical event taking place then their measurements of the distances or the time of that physical event is not necessarily going to be the same but they are going to be related to each other by a simple very simple transformation equation which is simply given by in this particular case X so for as you're going to have X Y Z and time as the coordinates of the physical event and for AZ - you're going to have x - y - Z - and tea - as the coordinates of other physical event from this person's perspective so the transformation that exists between s and AZ - is given by X - is equal to X minus VT y dash is equal to Y because there is no motion taking place along y axis Z dash is equal to Z and T dash is equal to T now these transformation equations are known as Galilean transformation equations okay they simply give you a relationship between the coordinates of some physical event from two different observers perspective if you have seen some of my previous videos on special theory of relativity you must be familiar with the fact that these Galilean transformation equations are not compatible with the postulates of special theory of relativity more specifically the fact that the speed of light is a constant or the second portion at of Str which says that the speed of light is a constant in all inertial frames of reference is something that is not compatible with the Galilean transformations also and to previous videos I showed how the wash rates of special theory of relativity can predict phenomena like length contraction and time dilation however the Galilean transformations do not predict any sort of a phenomena so therefore when we are dealing with special theory of relativity we have to sort of get rid of the Galilean transformations because they are not compatible with special relativity and we have to obtain a new set of transformation equations that exists between X Y Z T and X dash y dash Z dash T - which is going to be compatible with the postulates of special relativity and which can also predict the relativistic phenomena like length contraction and time dilation so in this video I am going to simply derive a new set of transformation equations between two different observers frames of reference in this kind of a scenario where there is a motion happening only along one axis and I am going to do that simply from the postulates of special theory of relativity now before we do that there is a simple requirement that has to be satisfied by these kind of transformation equations and the requirement is that these transformation equations which relates X Y Z T and X dash y dash - must be linear in nature now why do they have to be linear the they have to be linear because we want first of all a one-to-one relationships and second of all we do not want accelerations creeping out due to the transformation equations themselves so for example if I take a nonlinear relationship if I say that a nonlinear relationship exists between X and X - let's suppose X Dash is equal to X square X suppose you have X Dash is equal to 4 centimeters okay just an example then X is going to be equal to plus minus 2 centimeters right now it means that for one person the physical event is happening at one location for the other person is happening at two different locations which is not possible we do not want a many to one relationship we want one to one relationship also we do not want any kind of an acceleration creeping up due to the transformation equations themselves so for example if you look at this particular equation X dash is equal to X square if I take a time derivative of this then DX dash by DT dash is equal to 2 X DX by DT right so for let's just assume that T is equal to T - in this case because I'm just showing an example here and if I take another time derivative this is going to be D - X dash by DT - 2 is equal to 2 DX by DT whole square plus 2x D 2 X by DT 2 now in this case this first term is nothing but the acceleration in the ass - remove reference this is - multiplying the velocity in the ashram of reference and let's say some velocity is U ok and this is simply 2x + acceleration in the ashram of reference now what happens is that if let's suppose the acceleration in the ass - of reference is equal to zero it does not necessarily imply that a is also going to be equal to zero are you understanding if they're serration of the particle or the physical event where it's happening the acceleration is zero in one frame of reference it does not necessarily imply the acceleration is also going to be zero in the other frame of reference which does not make sense because accelerations are usually the result of forces they are not the result of transformation equations so whenever forces are involving some kind of a physical phenomena the acceleration takes place and these forces are the same for all kinds of observers and therefore the accelerations are also going to be the same in all kinds of observers so therefore we do not want these transformation equations to end up giving us situations where the accelerations are arising due to the transformation equations themselves second point and the first point we want to want to one relationship and therefore because of these reasons we want to impose the condition that these transformation equations are going to be linear in nature so we will keep this in mind now let's take the first transform a Galilean transformation which said that X dash is equal to X minus VT right this is the first Galilean transformation equation here you can see that X Dash is dependent not only on X but it is also dependent on time ok so let's assume that there is a dependence of both X and T time on X - here and if you want to make that assumption if we make that assumption and write a general linear relationship between these coordinates then we can write that X Dash is equal to ax plus B T ok so X and T are the coordinates and the astrum of reference X Dash is a coordinate in the as - from reference and a and B here are nothing but constants okay this is a very simple general relationship that we're looking towards now to find the value of these constants a and B we are going to use a few examples in the first example I'm going to assume then there is some kind of a physical event which takes place at the origin of - frame of reference so a physical event could be let's suppose an explosion of a firecracker let's say okay so there is a firecracker which gets exploded at the origin of s - frame of reference so what is going to happen if I if this person here in the astronaut reference is looking at that explosion of firecracker so that explosion of firecracker which is of the origin of s - term of reference is moving at a velocity V in the x axis for this person isn't it so for this particular observer that origin of s - non of reference is moving with a certain velocity V and therefore his measurement of the distance is going to be equal to X is equal to V T now what is going to be the displacement according to this person this person's this spaceman is simply X - which is nothing but zero because it is his in the origin of his own frame of reference so if I apply both of these equations in the first equation let's say that this is equation number one so using the above in equation one so this is the case where physically when happens at the origin of as - no reference okay so using these two in the equation we get X dash is equal to ax plus B T or X dash is equal to zero a X is equal to V T so if I put V T here plus B T or a V T - is equal to B T or B is equal to minus a V quite simply let's suppose this is point number two so I have found a relationship between the constants involved if I substitute this equation du in equation number one I get X dash is equal to ax minus a VT or X Dash is equal to a X minus three so X dash is equal to a X minus VT is what we get what we have actually done here is that we have simplified using this particular example and we have reduced from two constant equation to a one constant equation right now we only have one constant in this equation so this is going to be much easier to solve now let's suppose this is point number three okay now you can also obtain the reverse transformation so if you imagine that the physical event happens at the origin of a stream of reference and this person is looking at that and that physical event is traveling in the opposite direction or in the negative x dash axis with respect to him with a velocity V or the velocity minus V according to this person so using that you can also obtain the reverse transformation reverse transformation equation the reverse transformation equation is simply going to be X is equal to a X dash plus VT so you simply replace V with plus V and you will end up getting the reverse transformation equation which let's suppose is equation number four now let's take this a step further we're going to use the Einsteins second postulate so the Einsteins second postulate tells us that the speed of light is constant for all inertial observers so it doesn't matter if you are at rest with respect to the source of light or if you are moving with a certain velocity with respect to the source of light you will measure the velocity of light to be exactly the same now this postulate comes up with certain very peculiar sort of a phenomena now let's look how this is going to affect our equations in this case so let's imagine that there is some kind if a bulb at the origin of these two frames of reference okay so let me clarify what kind of a thing is happening so let's say that at time T is equal to zero both these two frames okay have their origins which are coinciding with each other so at time T is equal to zero I am going to define the problem in such a manner that both these two frames have their origins which coincide at time T is equal to zero and after certain time of kind of time has elapsed one of the frames is moved ahead okay so time T is equal to zero both these two frames have their origins coinciding and after some time the as dash reference is moving ahead because it has a certain velocity now let's also say that at the origin when time T is equal to zero there is a bulb okay which goes off for an instant so it gives off a light in all directions so if there's a bulb which goes off at time T is equal to zero when these both these two frames of reference are coinciding what happens to that wavefront of light when both these two frames of reference has moved one of them has moved ahead with respect to the other so if you look at it from both these two observers perspectives you will get a very peculiar sort of thing happening so from the perspective of this observer for him the bulb went off and his origin right but after some time has eloped lapse the light moves out in the from form of a spherical wavefront which looks something like this okay which has a center at his own origin of his frame of reference from this person's person's perspective however the light went off from his own frame of reference origin and after some time has elapsed the light will move out in the form of a spherical wavefront something like this so both these two person will sort of see this kind of an event in a similar fashion that the light went off at their own origin of the frame of reference and after some time there is a spherical wavefront coming out of it now this happens simply because of the fact that the speed of light is a constant for both these two inertia inertial observers now if we're dealing with some other thing let's suppose sound waves coming out of a point now if we consider sound waves we are going to have a very different result so sound waves will have different velocities with respect to different observers which are moving with respect to the source so you will not get this kind of a situation with respect to sound waves but when it comes to light because of the second postulate of einstein's Str you are going to have this kind of a situation where a pulse of light which originates at the origin of their own frame of reference will come out in the form of a spherical wavefront and they will come out in such a manner the velocity of light is going to be C for both of these so we're going to use this particular example so in the second example let's suppose that the pulse of light goes off at the or at time T is equal to zero so in such a case for the person in the ash frame of reference the displacement of light in the x-axis will be given by X is equal to CT T is the time elapsed and C is the velocity of light and for the person in the ass - no reference the displacement in X - X is equal to CT - where C is the velocity of light which is same for both and T - is a type array which has elapsed so if I substitute these here in equation number three so let's suppose this is equation number 5a and 5b and I substitute 5a and 5b in this equation transformation equation that we have taken X Dash is equal to a X minus V T where X Dash is CT - is equal to a X is CT minus VT right or CT - is equal to a C minus V T or T by T - is equal to C by a C - V I am going to keep this here as point number 6 now again if I substitute 5a and 5b in the reverse transformation equation X is equal to ax - plus VT - then also I'll opting something similar X is equal to a X - plus V T - if I put here CT is equal to a CT - plus V T - or CT is equal to a c plus V T - T by T - is equal to a c plus V by C let's suppose this is point number seven now what point number seven and six are the same except for the right-hand term so equating 0.6 and seven so both these two terms are same so I can write a C + V by C is equal to C by a C - V right so I am doing this to find out the value of the and if you are guessing why I am doing all of this so if I multiply C and I take this right-hand side this becomes C square is equal to a square c plus v multiplied by C minus V so this is a square C square minus v square so if I reverse the signs I get a square is equal to C square by C square minus v square if I take C square below I end up getting 1 by 1 minus v square by C square right so just for the moment remember this expression okay X dash is equal to ax minus VT I am going to rub this from here so this expression therefore the constant finally becomes a is equal to 1 by root over 1 minus v square by C square this is the constant in the equation that we had written let's say this is point number a so this is the constant in our transformation equation that we are taking so in some cases this constant is given a very special symbol so let's call this constant as gamma ok gamma is nothing but 1 by root over 1 minus v square by C space so if I substitute this constant in the transformation equation that I just now rubbed so this constant in then is then going to become so we have the transformation equation X Dash is equal to a X minus V T so is this expression this therefore becomes X Dash is equal to X minus VT by root over 1 minus v square by C square that's it X and T are the coordinates V is the velocity with which this s as - frame of reference is moving and C is the speed of light then this is a transformation equation the first transformation that equation that we obtain similarly you can also substitute this constant in the reverse transformation equation the reverse transformation equation will be X is equal to a X - plus V T - so X is equal to X - plus V T - pi one minus v square by C square so let's write down these two transformation equations here so that we can remember it now it becomes much easier finding out the other set of transformation equations so let's use these two equations to obtain the other set of transformation equations okay so the first equation is X Dash is equal to gamma X minus VT okay and the other reverse one is X is equal to gamma X - plus V T - okay what I'm going to do is I'm going to substitute this X - from here to here so then it becomes X is equal to X - plus V T - root over 1 minus v square by C square since because of this gamma if I take this expression to the left hand side this becomes X root over 1 minus v square by C square if I substitute X - from here the next - is nothing but X minus VT by root over 1 minus v square by C square minus V ok if I add both of these two terms then I end up getting root over 1 minus v square by C square X minus VT minus V T root over 1 minus v square by C square if I take this term to the left hand side then it simply becomes X 1 minus v square by C square is equal to X minus VT minus VT root over 1 minus v square by C square oh if I separate the terms this becomes X minus X v square by C square is equal to X minus VT minus VT root over 1 minus v square by C square ok I made a mistake that there should be a t - f this is T - okay because this is X - plus VT - and there's a plus sign again sorry so it is a plus sign and a t - symbol here so then X gets cancelled all so the V is also common in these expressions the V also gets cancelled you are left with this with equation becomes t minus V X by C square is equal to t - root over 1 minus v square by C square or this can be written as t - is equal to t minus V X by C square divided by root over 1 minus v square by C square that's it that's the equation so you have a equation relating t and t - which also involves the x-coordinate and V is the velocity relative velocity between the two frames and see this is the speed of light this is a summation equation for time so let's write the transformation equation here so what about 1 minus v square C square this is nothing but gamma so this can be written as T - is equal to gamma t minus V X by C square ok so this way I'm going to write here as T - is equal to gamma t minus VX by C square similarly by substituting X now X in this equation you can also obtain the reverse transformation equation which is simply minus there is change in sign here so this is going to be T is equal to gamma t - plus V X - plus C square so these are the two transformation equations what about the other two transformation equations relating y&z it's quite simple because in our case what is happening is that there is no motion taking place along y&z axis so there is no relative motion taking place between y&z axis so what's going to happen is so in the linear relationship that we had already taken there is a time dependence the time dependence comes simply from the fact that there is a relative motion taking place in this particular direction but in the Y and the z axis there is no relative motion however taking place so there is no going to be any kind of a time dependence so if we have some time if a relationship between y and y dash so let's suppose Y is equal to some constant k1 x y dash okay and Z is equal to some kind of a constant K 2 multiplied by Z dash so k1 and k2 or some kind of a constants ok now in relativity all kinds of frames of reference are sort of given in similar importance there is no absolute frame of reference both of these two frames of reference are inertial frames of reference and one is not necessarily preferential compared to the other so if you are this observer for you this entire system is moving in your right hand direction with relative velocity if you're this observer this entire system is moving in this direction in left-hand side velocity however if you're looking at the y&z components there is no relative motion taking place whatsoever so if you have this kind of a linear relationship that exists for y and z and y dash and y Z dash as trial reverse relationship should also exist for example Y - we also have some kind of a relationship y dash is equal to K 1 y and z dash is equal to K 2 Z now you might have seen that I use the same constants the reason I will use the same constants there because both of these two frames of reference are sort of symmetrical in that sense because there is no motion taking place in y&z axis so if there is a transformation between the coordinates of y&z - why - and Z - then a similar reverse transformation will also happen where one both of them will have similar sort of a constants okay and if this kind of a case is only possible where this constants k1 is equal to k2 is equal to 1 I understand because there is no motion place in the y-axis or in the z-axis so both sort of a transformation equation happening from Y to Z sorry from Y - y - and from Y - 2 y is going to be exactly the same and you will end up getting Y is equal to Y - and that is equal to Z - ok so finally I can write this entire set of transformation equations here so y is equal to y dash is equal to Y Z dash is equal to Z and Y is equal to y dash and Z dash is equal to Z - so this is our complete list of the transformation equations now that we have obtained which is going to be compatible with the postulates of special theory of relativity so if you look at the transformation equations here the gamma here contains that particular factor which deals with the velocity of light gamma is root over 1 by 1 by root over 1 minus v square by C square now this set of equations the original set of equations going from s to s - no reference and the reverse transformations going from AZ - trash term of reference are known as the Lorentz transformation equations and these equations are compatible with the second postulate of special theory of relativity which says that the speed of light is constant for all inertial frames of reference these transformation equations also can predict the existence of analyticity phenomena such as length contraction and time dilation for example in fact if you look at these equations this factor gamma which is root 1 by 2 o 1 minus v square by C square becomes almost equal to 1 for very small velocities and this factor V by C square becomes almost equal to 0 for very small velocities so you can also show that these transformation equations become the Galilean transformation equations for very low velocities however if you want a general picture where we only also not only deal with low velocities but we also deal with very high velocities comparable to the speed of light then these are going to give you a complete picture of a transformation going from one inertial frame of reference to another these are known as the Lorentz transformation equations and that's it for today

Channel: For the Love of Physics

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Keywords: derive lorentz transformation, derivation of lorentz transformation, lorentz transformation derivation, lorentz transformation derivation youtube, derive, lorentz transformation, lorentz transformations, lorentz transformation equations, lorentz transformation equation, lorentz transformation explained, lorentz transformation in relativity, lorentz transformation in hindi, lorentz, lorentz transformation equation in hindi, special relativity, Special theory of relativity

Id: 6pj1f8x3APg

Length: 26min 19sec (1579 seconds)

Published: Sun May 13 2018