What is condition for always be negative/ positive quartic equation?
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I have a parametric quartic equation. It is potential of a black hole that I want to always be negative. I thought to make it to two quadratic equation. But it is very difficult to solve. What can I do?
My equation is:
$-(alpha) r^4 +(alpha( L-a)^2 -2) r^2 +(L^2) r -2(L-a)^2$
This is black hole effective potential which I want to be always negative. α is the intensity of dark energy, a is spin of the black hole and L is particle angular momentum. For Kerr black hole effective potential is a quadratic equation and we know for always negative potential, discriminant must be negative when the coefficient of $r^2$ is negative. Then we get a relation for $L$. For this black hole, I want such $L$ in other variables except for $r$. It means $L(alpha, a)$. ((In Kerr black hole it was $L(a)$.))
inequality physics
edited Nov 27 at 18:45
Alex Ravsky
36.8k32078
asked Nov 17 at 10:24
R.t
62
|
show 1 more comment
up vote
1
down vote
favorite
I have a parametric quartic equation. It is potential of a black hole that I want to always be negative. I thought to make it to two quadratic equation. But it is very difficult to solve. What can I do?
My equation is:
$-(alpha) r^4 +(alpha( L-a)^2 -2) r^2 +(L^2) r -2(L-a)^2$
This is black hole effective potential which I want to be always negative. α is the intensity of dark energy, a is spin of the black hole and L is particle angular momentum. For Kerr black hole effective potential is a quadratic equation and we know for always negative potential, discriminant must be negative when the coefficient of $r^2$ is negative. Then we get a relation for $L$. For this black hole, I want such $L$ in other variables except for $r$. It means $L(alpha, a)$. ((In Kerr black hole it was $L(a)$.))
inequality physics
edited Nov 27 at 18:45
Alex Ravsky
36.8k32078
asked Nov 17 at 10:24
R.t
62
1
Please add more information to this post. Could you show the expression you want to keep negative explicitly?
– Yuriy S
Nov 17 at 10:27
2
See this en.wikipedia.org/wiki/Quartic_function#Nature_of_the_roots
– Prakhar Nagpal
Nov 17 at 10:28
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 17 at 10:33
You mean you want to keep some quartic polynomial to be negative always? If so there is no equation. Can you write in more detail exactly what is it that you want to do? If in general, I would just say the negative of that quartic can be expressed as a sum of squares at least one of which is positive, and you are done.
– Macavity
Nov 17 at 12:44
1
I've edited my text. Thanks for your attention.
– R.t
Nov 17 at 13:14
|
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have a parametric quartic equation. It is potential of a black hole that I want to always be negative. I thought to make it to two quadratic equation. But it is very difficult to solve. What can I do?
My equation is:
$-(alpha) r^4 +(alpha( L-a)^2 -2) r^2 +(L^2) r -2(L-a)^2$
This is black hole effective potential which I want to be always negative. α is the intensity of dark energy, a is spin of the black hole and L is particle angular momentum. For Kerr black hole effective potential is a quadratic equation and we know for always negative potential, discriminant must be negative when the coefficient of $r^2$ is negative. Then we get a relation for $L$. For this black hole, I want such $L$ in other variables except for $r$. It means $L(alpha, a)$. ((In Kerr black hole it was $L(a)$.))
inequality physics
edited Nov 27 at 18:45
Alex Ravsky
36.8k32078
asked Nov 17 at 10:24
R.t
62
I have a parametric quartic equation. It is potential of a black hole that I want to always be negative. I thought to make it to two quadratic equation. But it is very difficult to solve. What can I do?
My equation is:
$-(alpha) r^4 +(alpha( L-a)^2 -2) r^2 +(L^2) r -2(L-a)^2$
This is black hole effective potential which I want to be always negative. α is the intensity of dark energy, a is spin of the black hole and L is particle angular momentum. For Kerr black hole effective potential is a quadratic equation and we know for always negative potential, discriminant must be negative when the coefficient of $r^2$ is negative. Then we get a relation for $L$. For this black hole, I want such $L$ in other variables except for $r$. It means $L(alpha, a)$. ((In Kerr black hole it was $L(a)$.))
inequality physics
inequality physics
edited Nov 27 at 18:45
Alex Ravsky
36.8k32078
asked Nov 17 at 10:24
R.t
62
edited Nov 27 at 18:45
Alex Ravsky
36.8k32078
asked Nov 17 at 10:24
R.t
62
edited Nov 27 at 18:45
Alex Ravsky
36.8k32078
edited Nov 27 at 18:45
Alex Ravsky
36.8k32078
edited Nov 27 at 18:45
Alex Ravsky
36.8k32078
36.8k32078
asked Nov 17 at 10:24
R.t
62
asked Nov 17 at 10:24
R.t
62
asked Nov 17 at 10:24
R.t
62
62
1
Please add more information to this post. Could you show the expression you want to keep negative explicitly?
– Yuriy S
Nov 17 at 10:27
2
See this en.wikipedia.org/wiki/Quartic_function#Nature_of_the_roots
– Prakhar Nagpal
Nov 17 at 10:28
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 17 at 10:33
You mean you want to keep some quartic polynomial to be negative always? If so there is no equation. Can you write in more detail exactly what is it that you want to do? If in general, I would just say the negative of that quartic can be expressed as a sum of squares at least one of which is positive, and you are done.
– Macavity
Nov 17 at 12:44
1
I've edited my text. Thanks for your attention.
– R.t
Nov 17 at 13:14
|
show 1 more comment
1
Please add more information to this post. Could you show the expression you want to keep negative explicitly?
– Yuriy S
Nov 17 at 10:27
2
See this en.wikipedia.org/wiki/Quartic_function#Nature_of_the_roots
– Prakhar Nagpal
Nov 17 at 10:28
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 17 at 10:33
You mean you want to keep some quartic polynomial to be negative always? If so there is no equation. Can you write in more detail exactly what is it that you want to do? If in general, I would just say the negative of that quartic can be expressed as a sum of squares at least one of which is positive, and you are done.
– Macavity
Nov 17 at 12:44
1
I've edited my text. Thanks for your attention.
– R.t
Nov 17 at 13:14
1
1
Please add more information to this post. Could you show the expression you want to keep negative explicitly?
– Yuriy S
Nov 17 at 10:27
Please add more information to this post. Could you show the expression you want to keep negative explicitly?
– Yuriy S
Nov 17 at 10:27
2
2
See this en.wikipedia.org/wiki/Quartic_function#Nature_of_the_roots
– Prakhar Nagpal
Nov 17 at 10:28
See this en.wikipedia.org/wiki/Quartic_function#Nature_of_the_roots
– Prakhar Nagpal
Nov 17 at 10:28
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 17 at 10:33
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 17 at 10:33
You mean you want to keep some quartic polynomial to be negative always? If so there is no equation. Can you write in more detail exactly what is it that you want to do? If in general, I would just say the negative of that quartic can be expressed as a sum of squares at least one of which is positive, and you are done.
– Macavity
Nov 17 at 12:44
You mean you want to keep some quartic polynomial to be negative always? If so there is no equation. Can you write in more detail exactly what is it that you want to do? If in general, I would just say the negative of that quartic can be expressed as a sum of squares at least one of which is positive, and you are done.
– Macavity
Nov 17 at 12:44
1
1
I've edited my text. Thanks for your attention.
– R.t
Nov 17 at 13:14
I've edited my text. Thanks for your attention.
– R.t
Nov 17 at 13:14
|
show 1 more comment
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1
Please add more information to this post. Could you show the expression you want to keep negative explicitly?
– Yuriy S
Nov 17 at 10:27
2
See this en.wikipedia.org/wiki/Quartic_function#Nature_of_the_roots
– Prakhar Nagpal
Nov 17 at 10:28
Welcome to MSE. Please read this text about how to ask a good question.
– José Carlos Santos
Nov 17 at 10:33
You mean you want to keep some quartic polynomial to be negative always? If so there is no equation. Can you write in more detail exactly what is it that you want to do? If in general, I would just say the negative of that quartic can be expressed as a sum of squares at least one of which is positive, and you are done.
– Macavity
Nov 17 at 12:44
1
I've edited my text. Thanks for your attention.
– R.t
Nov 17 at 13:14