Extra dimensions string theory inherently requires the...

Extra Dimensions: String theory inherently requires the existence of extra spatial dimensions beyond the familiar three dimensions of space. These extra dimensions are typically compactified or curled up at incredibly tiny scales, making them difficult to detect directly. Discuss the theoretical motivations behind the need for extra dimensions in string theory and how their existence may provide a framework for resolving some of the long-standing issues in particle physics and cosmology.

Transcript

00:01 So we have planck's length.

00:02 We have planck's length is defined as the square root of the gravitational constant times plank's constant divided by the speed of light cubed.

00:12 Now, for gravitational constant, we have 6 .67 times 10 to the negative 11th.

00:21 And then we have meters cubed kilograms to the negative first times seconds to the negative second.

00:30 We have planks constant, which is going to be 6 .63 times 10 to the negative 34th meters squared times kilograms times second to the negative first.

00:47 And then we have the speed of light.

00:49 This is going to be 3 .00 times 10 to the 8th meters per second.

00:57 So meters times second to the negative first.

01:00 And at this point it's asking us to find essentially the make sure that the dimensional analysis checks out so for part a we have a square root of and then essentially we're only going to be using the units so here we'll say a length cubed times a we can say a mass times a time to the negative second and then here for h will have a mass squared, my apology, a length squared times a mass.

01:46 And here, actually, i forgot.

01:49 Here it's going to be mass to the negative first.

01:58 And then for plank's constant is again, a length squared times mass times time to the negative first.

02:09 And then this is going to be divided by a length times a time to the negative first.

02:17 However this is going to be cubed so what we end up with is that t times t to the negative second times t to negative first is going to be equal to t times t to the negative third and that'll be for the numerator for the denominator we have t to the negative first to the third so this is going to be t to negative third so all the ts are going to cancel out so t cancels out and then we have a length we have a mass here and then we have a mass to the negative first.

02:48 So here the mass is going to cancel out.

02:50 So that'll be mass to the zero power or just simply one.

02:54 And then we have a length to the third, a length to the second.

02:57 That'll be a length to the fifth, divided by the length.

03:00 So that'll be the square roots of a length to the length to the third.

03:05 So i'll be linked to the second.

03:08 Because again, length to the third times length to the second, that'll be a length to the fifth...