4 added 5 characters in body edited May 21 at 15:16 J.G. 10.3k22 gold badges1717 silver badges3131 bronze badges A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T\mu\upsilon =0$$$$T_{\mu\upsilon}=0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R\mu\upsilon$$$$R_{\mu\upsilon}$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. What is the purpose of the holding function We don't have the metric at the start, what we do have is this horror below, the Riemann tensor elements (all of them based on the metric): $${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$ The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry). The holding functions are no value in themselves, they are literally just letters, they help remind us of the real functions we need to find (and we can manipulate them to reduce their number). In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric, as the functions are not equal to one in places where there is a gravitational mass. A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T\mu\upsilon =0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R\mu\upsilon$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. What is the purpose of the holding function We don't have the metric at the start, what we do have is this horror below, the Riemann tensor elements (all of them based on the metric): $${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$ The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry). The holding functions are no value in themselves, they are literally just letters, they help remind us of the real functions we need to find (and we can manipulate them to reduce their number). In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric, as the functions are not equal to one in places where there is a gravitational mass. A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T_{\mu\upsilon}=0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R_{\mu\upsilon}$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. What is the purpose of the holding function We don't have the metric at the start, what we do have is this horror below, the Riemann tensor elements (all of them based on the metric): $${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$ The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry). The holding functions are no value in themselves, they are literally just letters, they help remind us of the real functions we need to find (and we can manipulate them to reduce their number). In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric, as the functions are not equal to one in places where there is a gravitational mass. 3 added 130 characters in body edited Dec 2 '16 at 2:28 user108787 A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T\mu\upsilon =0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R\mu\upsilon$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. What is the purpose of the holding function We don't have the metric at the start, what we do have is this horror inbelow, the Riemann tensor elements (all of them based on the metric): $${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$ The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry). The holding functions are no value in themselves, they are literally just letters, they help remind us ofof the real functions we need to find and(and we can manipulate them to reduce their number). In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. InIn flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric, as the functions are not equal to one in places where there is a gravitational mass. A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T\mu\upsilon =0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R\mu\upsilon$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. What is the purpose of the holding function We don't have the metric at the start, what we do have is this horror in the Riemann tensor elements (all of them based on the metric): $${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$ The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry). The holding functions are no value in themselves, they just remind us of the functions we need to find and we can manipulate them to reduce their number. In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric. A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T\mu\upsilon =0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R\mu\upsilon$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. What is the purpose of the holding function We don't have the metric at the start, what we do have is this horror below, the Riemann tensor elements (all of them based on the metric): $${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$ The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry). The holding functions are no value in themselves, they are literally just letters, they help remind us of the real functions we need to find (and we can manipulate them to reduce their number). In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric, as the functions are not equal to one in places where there is a gravitational mass. 2 added 1031 characters in body edited Dec 2 '16 at 2:22 user108787 A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T\mu\upsilon =0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R\mu\upsilon$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. What is the purpose of the holding function We don't have the metric at the start, what we do have is this horror in the Riemann tensor elements (all of them based on the metric): $${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$ The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry). The holding functions are no value in themselves, they just remind us of the functions we need to find and we can manipulate them to reduce their number. In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric. A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T\mu\upsilon =0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R\mu\upsilon$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T\mu\upsilon =0$$). Also drop the cosmological constant. Then you need to connect this place holding metric with $$R\mu\upsilon$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole. Source: Wormhole embedding diagram How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage. What is the purpose of the holding function We don't have the metric at the start, what we do have is this horror in the Riemann tensor elements (all of them based on the metric): $${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$ The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry). The holding functions are no value in themselves, they just remind us of the functions we need to find and we can manipulate them to reduce their number. In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric. 1 answered Dec 2 '16 at 1:00 user108787