A Swift Introduction to Spacetime Algebra

Transcript

if you're like me your formal education in special relativity has been pretty lacking I remember starting with the postulat of Relativity and then moving on to a few of the direct consequences of these postulates like time dilation length contraction and the relativity of simultaneity I also heard something about Makowski space but I remember being confused about how it works from this description I remember thinking that special relativity is just an annoying problem to deal with when speeds get too too fast however as I started studying on my own I realized that there are many mathematical techniques that make working with relativity much simpler to the point that working with relativity seems simpler than not working with relativity in this video I will be presenting one of these techniques SpaceTime algebra which is the merging of geometric algebra with special relativity before I get into it I want to mention that there are a few prerequisites for this video first you need to know the basics of special relativity such as how the laws of physics and the speed of light are the same in all inertial reference frames in the equations for a Lawrence boost if you haven't heard of Lawrence boost before most introductory textbooks incorrectly call them Lawrence Transformations the term Lawrence transformation actually refers to something that is much more General than Lawrence boost and will be defined near the end of the video while there is much more to relativity than just Lawrence boosts we won't need to know anything else for this video the other main topic that you need to know about is geometric algebra you don't need to know much about geometric algebra what you do need to know is what multiv vectors are what the geometric product is and how geometric algebra describes rotations in both two dimensions and higher Dimensions if you don't know anything about geometric Algebra I made a video introducing exactly these Concepts from geometric algebra which should be sufficient for understanding this video so if you need to you can watch that video and then you should be good to go here I'll provide a link to it in the description so here's a general outline for this video we will first focus on discovering the algebra what I mean by this is figuring out what basic algebraic rules we are going to want to use to describe relativity it's a little different than vanilla geometric algebra so we're going to want to figure out why we have these differences most of this section is going to be covering things like Makowski space and the SpaceTime interval with no geometric algebra until the end so if you've learned those things before you can skip to the last part of this section if you wish after doing this we will look at space-time splits the space-time split is probably the biggest unique contribution that SpaceTime algebra makes in our understanding and use of special relativity finally we will look in detail at how Lawrence Transformations are described in SpaceTime algebra while the final results here might be familiar to to people who are already quite acquainted with minkowski space SpaceTime algebra makes things much simpler now let's start trying to discover space-time algebra however to do this we need to figure out why we want space-time algebra in the first place to answer this let's have a brief discussion about symmetry think about this Square we say that it's a symmetrical shape but what do we really mean by that well when we rotate the square by 90° it looks the same as before we say that this 90° rotation is a symmetry of the square there are several other symmetries of the square such as rotating by 180° and flipping the square in various ways what we see from this is that a symmetry is a transformation that leaves something unchanged now this something can be very general let's look at a more abstract example that is closer to what I want to talk about consider this simple Collision of billiard balls during this Collision the balls followed these paths but now what if we rotated this whole situation of course the same thing happens with the balls still following the same rotated paths now this might not seem that special to you but there is actually a very important point to make here the laws of physics are unchanged by rotations notice that I'm not saying that physical situations are unchanged by rotations in our previous example of a collision in one situation the ball came from the right while in the other situation the ball came from the top so that can't be the case instead I am saying that the laws of physics themselves are unchanged by rotations while this is a much more abstract example it is still a symmetry thus we can say that the laws of physics are rotationally symmetric in this context we often say that the laws of physics are rotationally invariant rather than rotationally symmetric although it means the same thing now going back to special relativity remember that one of the fundamental postulates of Relativity is that the laws of physics are the same in all inertial reference frames this is another example of a symmetry more explicitly the laws of physics are unchanged by a change in reference frame which is just a Lawrence boost we often say that the laws of physics are lawence invariant because Lawrence boosts are an important symmetry in physics let's look at them in closer detail here are the equations for a Lawrence boost along the X AIS before we move on I want to simplify these equations a bit first off the top equation uses different units than the rest so let's make the units consistent we can do this by multiplying that equation by the speed of light now notice that most occurrences of time are multiplied by the speed of light and if we include the expression for the lawence factor most expressions for the velocity are divided by the speed of light the only exception to this is the equation for x Prime which we can force into this form by multiplying and dividing by C in light of this let's make two new variables that represent time and speed but with this factor of C included notice that the letters used for these variables are a slightly different looking T and V you might think that using similar looking letters is confusing but the reason for using these letters is that conceptually they represent the same physical quantity just with different units time now has units of length and speed is dimensionless by using these new variables the equations for a lawence boost become much simpler at this point we can forget about the original definitions of time and speed I won't be using the old variables anymore for the rest of this video so things shouldn't be confusing at this point you might think that these equations for Lawrence boost are nice and simple but there are still several issues let's think back to rotations which are another symmetry of the laws of physics we can do a rotation with this formula from geometric algebra in light of this equation we can start to see where Lawrence boost are lacking first while rotations simply mess with space Lawrence boost makes space and time we normally think of time as a scalar and space as a vector and having a symmetry that mixes scalers and vectors together is cumbersome second while rotations don't change the length of vectors Lawrence boost do most of the symmetries of the laws of physics keep Vector lengths the same so Lawrence boosts are much harder to use finally we can do rotations in a coordinate independent way which makes rotations easy to work with however these equations that we have for the Lawrence boost are coordinate dependent these are the problems that SpaceTime algebra solves let's let's go through the problems one by one first Lawrence boosts mix space and time this is a problem because most Transformations keep objects the same for example if you rotate a vector you'll get another Vector if you translate a vector you'll get another Vector but if you apply a Lawrence boost to a vector you'll get a new time which is a scalar in addition to a vector we could add them but it still produces a different object than what we started with since this mix of space and time is unavoidable let's try something that might seem strange What If instead of thinking of time as a scaler we think of time as a vector we normally have three basis vectors representing the three spatial directions so what if we added a fourth basis Vector representing Time by thinking of time as a vector Lawrence boost would transform vectors to vectors just like every other symmetry of the laws of physics this is the first step we need to take towards SpaceTime algebra we need to switch from thinking of three-dimensional space to four-dimensional SpaceTime now we need to have a way to distinguish between three-dimensional spatial vectors and these new four-dimensional SpaceTime vectors so we're going to need to introduce some new notation first of all instead of calling the basis vectors by these names we usually use the Greek letter gamma with a subscript this might seem confusing since the Lawrence factor is also called gamma but notice that these basis vectors use a subscript while the Lawrence Factor never uses a subscript you can blame dra for this confusion also while we use the little Vector Arrow to represent space vectors we will not use that Arrow to represent a vector in SpaceTime now this might cause us to confuse scalers and vectors so we need to make a further notational convention other than the Basis vectors in SpaceTime every time we see a Greek letter it will be a scaler note that we will continue to use Latin letters to represent certain scalers that we've already defined so there can still be some confusion between scalers and vectors in certain situations everything should be clear from Context though now one issue with having four basis vectors is that true four-dimensional visualization is impossible however we can often get away with only visualizing one dimension of space and one dimension of time in a plane we usually let the horizontal axis be the spatial axis and the vertical axis be the temporal axis thus these two vectors are gamma 0er and Gamma 1 in this setting we can actually get a geometric picture of what a Lawrence boost does consider this Vector how will it be affected by a Lawrence boost well with a Lawrence boost at half the speed of light this is the result if we play around with the velocity a bit more we can see that the vector traces out this hyperbola as we apply different Lawrence booths the ASM tootes for this hyperbola are these diagonal lines thus under Lawrence boosts this Vector is trapped in the upper region of the plane however this situation is more General than you might think look at how all of these vectors are affected by lorence boosts all of these vectors end up traveling along hyperbolas whose ASM tootes are the same diagonal lines thus all vectors in the upper region of the plane stay up there when a lawence boost happens however this situation is even more General than you might think look at how all of these other vectors are affected by Lawrence boosts all of these vectors end up traveling along hyperbolas whose ASM tootes are these same diagonal lines as well thus we see that Lawrence Booth split the plane up into several regions we have the upper region which contains vectors pointing roughly towards the future the lower region which contains vectors pointing roughly towards the past and the left and right regions which contain vectors pointing roughly in a spatial Direction in higher Dimensions these spatial regions are connected to each other so it's better to think of them as the same region we call vectors in the top and bottom regions time like vectors and we call the vectors in the left and right regions space- like vectors but what about vectors on the boundary between these two regions if you were to send a beam of light from the origin it would travel along this boundary so we call vectors along the boundary lightlike vectors anyway as we can see by considering time to be a vector the fact that Lawrence boost makx space and time is no longer an issue but now what about our next problem Lawrence boost change the length of vectors while this problem might seem insurmountable it actually is possible to deal with it as long as we are willing to bend the rules a bit before we talk about how to do this I want to remind you of a fact from geometric algebra remember that the square of a vector is simply its length squared while I have previous ly stated this as a theorem in more advanced contexts we usually consider the geometric product to be more fundamental than the length and then we Define the length of a vector using the square of the vector now the square of a vector is pretty easy to find it's just the sum of the squares of the components because the square of a vector is much easier to work with than its length from now on let's only consider the square of a vector not its length this will make things considerably simpler the two quantities are easily convertible to each other so we haven't really lost anything anyway our problem is that when we have a vector its Square changes every time we change our reference frame however one thing we do know is that everybody will agree on what the square of the vector is in one particular reference frame what if we decided to pick one particular reference frame and say that the square of a vector in any reference frame is the square of the vector in that particular reference frame this would be changing the definition of the square of a vector but it would trivially cause it to be unchanged by a Lawrence boost but this directly raises another question which reference frame do we pick to answer this question we need to look at another way to think about SpaceTime vectors the way we usually think about space-time vectors is as representing an event specifying the time and place that something happens but here's another way to think of space-time vectors let's say I was at rest at the origin as time passes I would move up in SpaceTime the path that I took is this space-time Vector if I instead started at the origin but was moving to the right at half the speed of light I would move along this path in space time the path that I took this time is this space time Vector what we see is that SpaceTime vectors correspond to moving at particular constant velocities because moving at a constant velocity is the definition of a reference frame vectors in SpaceTime correspond to reference frames so going back to finding a reference frame in which to calculate the square of a vector why not use the reference frame that the vector represents to be precise to define the square of this Vector we would do a Lawrence boost to make the vector Point straight up where it represents a reference frame at rest and then calculate the square of the vector in this reference frame everybody agrees on what the square of a vector is in its own reference frame so by calculating this value we can get a definition for the square of a vector that is unchanged by Lawrence boosts we can get an exact expression for this definition of the square of a vector with some algebra in the original reference frame let's call the vector U it is some linear combination of gamma 0 and Gamma 1 when we calculate a Lawrence boost T and X are given by these coefficients now the square of this Vector in the final reference frame is T Prime s + x Prime s the reference frame we want to move to is the one where the vector is pointing straight up which is when X Prime is zero we can now just play with these equations to get a simple expression for the square of the vector the first thing we can do is put this expression for T Prime in the equation for the square remember that by definition gamma s is 1 1 - B squ we want to find an expression for this Square purely in terms of Alpha and beta so we need to find a way to get rid of V we can do this with the other equation here notice that because gamma can never be zero it it must be the case that beta minus V Alpha is zero this equation can be solved for V to show that V is equal to Beta over Alpha we can now use this value of v in the equation for the square thankfully this expression can be considerably simplified first the fraction of fractions on the left can be converted into a single fraction the expression on the right can also be converted to a single fraction at this point several terms can be cancelled we now finally have an expression for the square of A Spacetime vector by Design This value is left unchanged by a change in reference frame in other sources this value is often called the space-time interval now one issue here is that this is only for one dimension of space but it's not too much work to show that for three dimensions the formula is similar you might notice that this defines a quadratic form which is actually all you need to do geometric algebra this formula for squaring a vector is the fundamental algebraic difference between SpaceTime algebra and four-dimensional vanilla geometric algebra to get a feel for how it works let's see some examples the square of this Vector is one now as we move the vector as if we were doing a lawence boost it still squares to one even though to us it might look like the vector is getting longer if we were to instead move the vector straight to the right its Square would actually decrease in fact if we go to the right enough the vector squares to zero this is something that is not possible in vanilla geometric algebra the fact that some non-zero vectors squared to zero is a fact that often has to be accounted for for example this Vector is non-invertible so it is no longer the case that all non-zero vectors have an inverse if we continue to move to the right the vector Square becomes negative this is also not possible in vanilla geometric algebra a natural question that arises is which Vector squared to a positive or negative value while looking at the equation for the square of a space-time Vector we can see that the square is zero whenever the absolute value of the space coordinate is the same as the absolute value of the time coordinate so it's any of the vectors along these diagonal lines wait a minute these are just the lightlike vectors now which Vector squared to a positive value for a vector to square to a positive value the absolute value of its time coordinate must be greater than the absolute value of its space coordinate this means that all vectors pointing more vertically squared to a positive value hey again this is the region we've seen before these are the Tim like vectors as you might be expecting the vectors that squared to a negative value are the vectors in the left and right regions which are the space-like vectors so we now have a much simpler definition of time like lightlike and space-like vectors if if a vector squares to a positive value it's Tim like if it squares to zero it's li like and if it squares to a negative value it's space like now there might be one thing nagging at you I earlier said that it's fine to talk about the square of vectors rather than their length because we can easily Define the length of a vector as the square root of the square of that Vector but we've seen now that with this new definition of the square of a vector this value can be negative we don't want an imaginary length and because length should be a scale we wouldn't want to pick something else from geometric algebra that squares to a negative value either there are two possible solutions to this problem the first option is to just ignore lengths entirely most of the things you want to do with the length of a vector you can do with the vector Square in the situations that you really do want a length several options exist however we won't be needing to Define length for this video so I won't dwell on this point any longer anyway we finally have everything we need to describe SpaceTime algebra SpaceTime algebra is simply the geometric algebra of SpaceTime in vanilla geometric algebra we start with a vector space given by some orthonormal basis we then describe the geometric product by saying that the basis vectors anticommute and that each basis Vector squares to one now for SpaceTime algebra we will do something similar we start with our four-dimensional Vector space given by the orthonormal bases gamma 0 Gamma 1 1 gamma 2 and Gamma 3 where gamma 0er represents time and the others represent space once again we say that these basis vectors anti- commute the only difference now is that instead of saying that all of our basis vectors Square to one we will say that gamma 0 squares to one while the other gamma is squar to negative 1 these rules describe how we carry out multiplication in SpaceTime algebra in SpaceTime algebra multiv vectors end up having 16 components one scalar component four Vector components six bi Vector components four Tri Vector components and one pseudoscalar component we can also split the basis vectors into the time like vectors and the space like vectors we can easily extend the definition of Tim like and space like to the other basis multiv vectors by saying that anything that squares to a positive value is Tim like and anything that squares to a negative value is space likee if you want to it's a nice exercise for getting acquainted with SpaceTime algebra to figure out which of these are Tim like and which are space- likee in the end we get three Tim like by vectors and three Space likee by vectors one Tim like Tri vector and three space- likee Tri vectors and the pseudo scaler is space likee like usual we often call the unit pseudoscalar I the span of these multiv vectors is what makes up SpaceTime algebra now that we have finally discovered the algebra we can move on to some of the new things that we can do with it the main thing we will look at in this regard is the SpaceTime split which in my opinion is one of the most useful contributions SpaceTime algebra has made over its Alternatives while I previously introduced SpaceTime as just thinking of space plus an extra dimension of time reality is more complicated than this what is the correspondence between space-time vectors and space vectors to answer this let's think about a vector in one-dimensional space as time passes it doesn't doesn't move anywhere thus if we were to add a second axis for time as time passes the vector would sweep out an area in SpaceTime wait a minute this just looks like a bi Vector in SpaceTime because this bi Vector has a temporal component this is a Tim like bi Vector so we see that our correspondence between space and space time is that a vector in space corresponds to a Tim like B Vector in SpaceTime now I'll admit that this geometric argument is not that persuasive but hopefully by exploring this idea further you will see that this is the most natural correspondence between space and SpaceTime that we can make of course it's nice to think of vectors in SpaceTime as being related to vectors in space as well so how could we convert this Vector to a bi Vector well the product of two perpendicular vectors produces a bi Vector so we could do this by multiplying this vector by gamma 0er what about a vector that only points in a temporal Direction in space we usually think of time as a scalar not a vector so how could we convert this Vector to a scalar well the product of two parallel vectors produces a scalar so we could once again do this by multiplying this vector by gamma zero but now what about a vector that has a mix of spatial and temporal components we should try to split the vector into a spatial and temporal parts and then multiply them both by gamma 0er but wait isn't that what the geomet product already does if we were to multiply this vector by gamma 0er the result would be the sum of the two vectors inner and outer products the inner product is precisely the length of the temporal component of the vector while the outer product is precisely the B Vector made from the spatial part of the vector in gamma 0er which corresponds to the correct Vector in space what we see here is that to split a space-time Vector into a spatial part and a temporal part all we have to do is multiply by gamma zero then the temporal part is given by the inner product and the spatial part is given by the outer product because we are splitting a space-time Vector into space and time we call this process a space-time split but wait it gets better it would seem that the SpaceTime split is Basis dependent because we used one of our basis vectors in the equation however recall that time like vectors correspond to reference frames gamma 0er is simply a plain Tim like vector and this split is doing A Spacetime split in the gamma zero reference frame if we replaced gamma 0er with another reference frame given by a difference normalized Vector gamma 0 Prime everything works the same way in doing a space-time split with a new Vector it will instead split relative to the reference frame given by the new Vector not the old one this means that we now have a coordinate free way to move from SpaceTime to any reference frame without even having to use the Lawrence boost equations but wait it gets even better while our focus on the correspondence between time like B vectors in SpaceTime and vectors in space has been mostly geometric so far it also works algebraically as well think about the four basis vectors in SpaceTime when we do A Spacetime split with these vectors by multiplying by gamma zero we associate the resulting scalar with the scalar from space and the resulting by vectors with the vectors from space but does this correspondence work algebraically the vectors in space follow certain algebraic rules that are important to using them do these B vectors in space time also follow these rules well when we Square Gamma 1 gamma 0er we get a minus sign from swapping the two vectors which then gets canceled by the minus sign from squaring Gamma 1 gamma 0 squar is 1 so the square of Gamma 1 gamma 0 is 1 this is the same as X hat the same thing happens with the other B vectors in correspondence with space vectors as well so these B vectors do satisfy this algebraic rule in keeping with the correspondence now what about the rule saying that the vectors in space anticommute well let's try multiplying Gamma 1 gamma 0 with gamma 2 gamma 0 and see what happens when we swap them to swap the two B vectors we just need to swap the individual vectors several times in the end we see that the two B vectors do anticommute as suggested by this correspondence once again the same thing happens with the other B vectors in correspond with the space vectors this has shown us that this correspondence between B vectors in space time and the vectors in space is not just some geometric trick it is also reflected algebraically as well but why stop here when we multiply the basis vectors of space together we get bi vectors in space let's make these products in space time as well and Associate the bi vectors in space with these products in SpaceTime after a bit of simplification we see that we have a correspondence between the space likee B vectors in space time with the B vectors in space we can also find what corresponds with the pseudoscalar in space by multiplying all of the time like by vectors together and simplifying so we see that the pseudo scalers in space and in space time are the same the multiv vector Spann by what's on the left side of the screen forms a subalgebra of the SpaceTime algebra called the even subalgebra of the SpaceTime algebra while I showed that this correspondence keeps the rules for multiplying vectors intact it turns out that every single possible multiplication is the same on both sides in more formal terms this correspondence induces an isomorphism between the even subalgebra of the space-time algebra and the algebra of physical space we can also see from this correspondence that space-time splits don't just split space-time vectors into spatial and temporal Parts they also split space-time bi vectors into spatial vectors and spatial bi vectors which correspond to the Tim likee and space-like parts of the original bi vector perspectively now that we have explored SpaceTime splits we can finally move on to understanding lawence transformations to start our discussion on Lawrence Transformations let's go back to the three problems that we had with Lawrence boosts we have solves the first two Problems by using SpaceTime algebra however you might have realized that we haven't solved the third problem yet so how can we do that at a first glance you might think that we've already solved it with SpaceTime splits after all doing A Spacetime split splits a vector in the reference frame of the Tim like vector being used so it seems that a Laurence boost has been done implicitly the issue is that SpaceTime splits Force us to leave SpaceTime what we want is a way to do Lawrence boost that keeps us in SpaceTime thus we need to look elsewhere for a way to solve this problem let's look at Lawrence boost in more detail in light of SpaceTime algebra once again here are the equations for a Lawrence boost in one spatial Dimension to express this in the language of SpaceTime algebra let's think of a Lawrence boost as a linear transformation that takes a vector and outputs the result of doing a lawence boost on it we can now just use this equation for Lawrence boosts however it's still coordinate dependent so we need to keep looking when we do a lawence boost we move from one reference frame to another we usually say that the starting reference frame is gamma 0er but what is our final reference frame in terms of our velocity let's say that our velocity was 0.5 * the speed of light this would mean that after traveling one unit up we have traveled half a unit to the right in general this Vector here is V * Gamma 1 the vector representing our final reference frame is just the sum of these two vectors so it is gamma 0 plus v Gamma 1 now we usually like our vectors that represent reference frames to be normalized so let's normalize this Vector Hey look it's our good friend the Lawrence factor that makes this expression considerably simpler let's call this new Vector gamma 0 Prime now when we do the Lawrence boost we see that gamma 0 Prime is brought to where gamma 0 was so we see that the lence Boost brings gamma 0 Prime to gamma 0er thus what we want is a simpler expression for this linear transformation that brings gamma 0 Prime to gamma 0 what linear Transformations bring gamma 0 Prime to gamma 0 while a simple one is multiplying a vector by gamma 0 Prime gamma 0 this brings gamma 0 Prime to gamma 0 because gamma 0 Prime squar is 1 but is this linear transformation the same thing as the Lawrence boost let's see what the effect of this linear transformation is on an arbitrary Vector in two-dimensional SpaceTime we can first expand out the definition of gamma 0 Prime at this point we just need to do some simple but tedious algebraic manipulation we see now that this is precisely the equation for Lawrence boost keeping in mind that this was a generic two-dimensional Vector what we see is that this simple equation represents a two-dimensional Lawrence boost that moves gamma 0 Prime to gamma 0 wait a minute this is exactly the same as the formula for a rotation in two-dimensional space does this mean that Lawrence boost are rotations now your first instinct might be to say no because the effect of a Lawrence boost on the plane looks nothing like a rotation however recall the definition of a rotation a rotation is a linear transformation that preserves lengths and whose determinant is one Lawrence boost are linear Transformations we defined length so that Lawrence boost would preserve length and you can verify that the determinant of a Lawrence boost is one thus lawence boost are rotations the reason that they don't look like normal rotations is because of our different def definition of length however algebraically they still behave almost exactly the same as any other rotation for example with normal rotations you can represent a rotation with an exponential using a unit by Vector I which can be expanded using Oilers formula Oilers formula is valid here because the unit by Vector squares to1 however in two-dimensional SpaceTime the unit by Vector squares to positive 1 instead of negative 1 even though it squares to positive one we can still represent rotations using an exponential because I now squares to positive 1 when expanding this exponential we need to use the hyperbolic cosine and S rather than the trigonometric cosine and S as we saw earlier these rotations cause points to travel along a hyperbola so we call these rotations hyperbolic rotations now one problem with these formulas for lorrence Boost is that because the two-dimensional rotation formulas don't work for high dimensional rotations these formulas for two-dimensional Lawrence boosts don't work for higher dimensional Lawrence boost as well how can we calculate Lawrence boost in higher Dimensions well since they're just rotations we can just use the formulas for n dimensional rotations and that's it we can use these equations to build rotors to perform and compose Lawrence boost like in any other geometric algebra now one important thing to note about Lawrence boosts in higher Dimensions is that not all rotations are Lawrence boosts in general the different ways you can rotate are given by the bi vectors in your algebra because our bi Vector basis has six elements we have six degrees of freedom for rotations the three space like by vectors end up creating normal rotations not hyperbolic rotations they create the normal rotations in Space the three time like by vectors are the ones that end up creating hyperbolic rotations which represent the Lawrence boost in the three spatial directions in a sense we can think of Lawrence boost as being rotations through time so we see that rotations in SpaceTime include both rotations in space and Lawrence boosts it is this set of all rotations in SpaceTime that we call Lawrence Transformations actually this isn't quite right while making this video I was surprised to learn that we usually allow flips in SpaceTime in our definition of Lawrence Transformations as well as rotations which makes Laurence Transformations any orthogonal transformation in SpaceTime given that geometric algebra can handle orthogonal Transformations easily by using the sandwich product SpaceTime algebra is an incredibly useful framework for dealing with Lawrence Transformations now that we've seen the basics of SpaceTime algebra a natural question that arises is what can we do with it I want to finish this video with briefly showcasing several of the applications of SpaceTime algebra that I've seen one application is relativistic mechanics which is the relativistic version of classical mechanics surprisingly relativistic mechanics looks pretty similar to classical mechanics a few definitions change but everything else is pretty much the same another application is an electrodynamics geometric algebra already makes electrodynamics much simpler but space-time algebra makes it even better remember that in vanilla geometric algebra the differential operator and the current are the sum of a scalar and a vector and the electromagnetic field is the sum of a vector and a by Vector remember that when we do A Spacetime split SpaceTime vectors become a scalar plus a vector and SpaceTime by vectors become a vector plus a bi Vector thus in SpaceTime these quantities are simply vectors and bi vectors the differential operator is now simply the gradient in SpaceTime and the Tim likee part of the current now explicitly says that charge density is current moving through time the Lawrence Force law is very simple in SpaceTime algebra as well because the DOT product of a vector and a bi Vector corresponds to projecting the vector onto the bi vector's plane and then rotating in that plane we see that the Laurence Force simply rotates moving charged particles in SpaceTime the space likee component of the electromagnetic field the magnetic field rotates particles in a spatial plane and the timelight component the electric field rotates particles in a temporal plane which we perceive as acceleration another application is in relativistic quantum mechanics it it turns out that the D matrices are simply a matrix representation of the basis vectors in SpaceTime algebra so by using SpaceTime algebra we don't need to use the D matrices then the D equation can be written like this David HZ has studied this form of the D equation in detail and it has led to many interesting ideas including my personal favorite the zitter bong interpretation of quantum mechanics finally some of you may be wondering about general relativity this has been worked out in the context of SpaceTime algebra as well Anthony lasenby Chris Doran and Steven G have worked out a way to express general relativity in SpaceTime algebra using gauge theory in a theory called gauge Theory gravity in this Theory instead of gravity being realized as curvature in SpaceTime gravity is realized as a gauge field derived from translational and rotational gauge symmetries furthermore some work has been done using guge Theory gravity in quantum mechanics and I personally think that this is a promising step towards a theory of quantum gravity for all of these applications I'll leave several links to textbooks and papers related to them in the description whatever it is that you work on I hope that this video will help you in your future endeavors