8.2 Introduction to 4-Vector Notation

Key Takeaways

Welcome back to A40 special relativity. In this short section, we want to introduce the new notation for vectors. And you know, if you look at previous discussions, this is actually not that new. We have seen that we need to treat time and space in a consistent manner. And you have often, you know, applied Lorentz transformation, for example, to a vector of time and the next component space. Now, we just want to do this with X, Y, and Z here and not treat the Y component and Z component as zero. So, we, you know, as a starting point, you can just simply say, "Okay, we have this new four-vector." Um and, you know, the zeroth component is the time or time times the speed of light. And then the first component, second, and third component are the spatial component X, Y, and Z. Now, I wrote a vector xi mu here um with a mu as being the upper index. I can also introduce xi with a lower index, and you'll see a little while in a little while why this is useful. Where the zeroth component is not T, but minus uh CT, but minus CT. As a reminder for three vectors, we, you know, you learned about the dot product, which is just a multiplication of two three vectors, where all vectors with N components, where you multiply, you know, the same component of each vector and add the those results together. So, the the dot product of a vector A and the vector B is the sum over all indices um for Ai times Bi. Now, for our four-vector, we do the very same thing. We just sum over all four components, and we treat the vectors as a product of the vector with a lower index and the upper index. And you find here then we get minus CT C squared T squared plus X squared Y squared and Z squared. More generally, this is for two vectors of the same two of the same vectors. More generally, for two different vectors, we can write this in this in this way here. Or in short, you can divide define a new notation in which we basically sum over all uh indices which are equal. So, here we have an and upper and lower indices together. So, you sum over this case here where there's the same index mu for both vectors, and one is lower and one is upper. All right. Then we can, you know, continue the introduction and just introduce a few tools to work with those vectors. For example, if you wanted to bring um, you know, the component mu uh from the bottom to the top, you can do this this multiplying the vector with a matrix. And the matrix here is also called a metric matrix. Um and simply what you have to do is multiply the first component with a minus one and the rest with one. You see this here on the diagonal. All other components are zero. What this does, you can check this if you want, is bringing um the component the the index of the vector from a lower to a upper one. All right. An interesting example is, you know, the product of a four-vector with itself, and we have already seen this because we we saw this as our invariant interval. Um here, the four-vector is the distance in space and time between two events. So, we looked at delta xi mu times delta xi mu. I think delta xi mu is the difference between event A and B. And so, we have seen this already and calculated the invariant and showed that this is um this um squared of a distance of two events is actually invariant under the Lorentz transformation. But there's other examples for vectors. The first one we'll investigate some more in the next sections to come is the energy momentum four-vector, where we place in the first component the energy and in the in the zeroth component the energy, and then the first, second, and third components the three-vector of the momentum. All right. But there's others, for example, the four-potential, where in the zeroth component you have the potential electric potential, and in the first, second, and third component you have this new field A, which is related to the magnetic and electric field. So, E and M is not part of this course, but we'll come back to this in the in the last week um and and discuss consequences and and ideas a little bit more. All right. But if you then, you know, look at this the invariant four-vector, which is a product of the energy momentum vector, we find that the first component the energy squared minus the energy squared over C squared plus the three-component vector of the momentum squared. And that's constant. We can just here name this mass or minus mass squared times C squared. So, if you rewrite this, you find this energy momentum mass relation E squared is equal to P squared C squared plus M squared C to the fourth power. And if you look at this for particles of zero momentum, in which case this component here is zero, you find the equation E is equal to MC squared.