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Are the units of energy the same in higher dimensions?
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In 3 spatial dimensions, $$[E] = [ML^2 T^-2]$$
Would it change in higher dimensions? If yes, then what would be the dimensions for 4 spatial dimensions?
homework-and-exercises units mass-energy dimensional-analysis spacetime-dimensions
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AccidentalFourierTransform
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up vote
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down vote
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In 3 spatial dimensions, $$[E] = [ML^2 T^-2]$$
Would it change in higher dimensions? If yes, then what would be the dimensions for 4 spatial dimensions?
homework-and-exercises units mass-energy dimensional-analysis spacetime-dimensions
edited 10 mins ago
AccidentalFourierTransform
24k1365119
asked 12 hours ago
Shubham Gothwal
694
New contributor
Shubham Gothwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
In the 4-dimensional case, would the fourth dimension be pretty much the same thing as the first three spatial dimensions?
– Nat
52 mins ago
add a comment |Â
up vote
13
down vote
favorite
up vote
13
down vote
favorite
In 3 spatial dimensions, $$[E] = [ML^2 T^-2]$$
Would it change in higher dimensions? If yes, then what would be the dimensions for 4 spatial dimensions?
homework-and-exercises units mass-energy dimensional-analysis spacetime-dimensions
edited 10 mins ago
AccidentalFourierTransform
24k1365119
asked 12 hours ago
Shubham Gothwal
694
New contributor
Shubham Gothwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
In 3 spatial dimensions, $$[E] = [ML^2 T^-2]$$
Would it change in higher dimensions? If yes, then what would be the dimensions for 4 spatial dimensions?
homework-and-exercises units mass-energy dimensional-analysis spacetime-dimensions
homework-and-exercises units mass-energy dimensional-analysis spacetime-dimensions
edited 10 mins ago
AccidentalFourierTransform
24k1365119
asked 12 hours ago
Shubham Gothwal
694
New contributor
Shubham Gothwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 10 mins ago
AccidentalFourierTransform
24k1365119
asked 12 hours ago
Shubham Gothwal
694
New contributor
Shubham Gothwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 10 mins ago
AccidentalFourierTransform
24k1365119
edited 10 mins ago
AccidentalFourierTransform
24k1365119
edited 10 mins ago
AccidentalFourierTransform
24k1365119
24k1365119
asked 12 hours ago
Shubham Gothwal
694
New contributor
Shubham Gothwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 12 hours ago
Shubham Gothwal
694
asked 12 hours ago
Shubham Gothwal
694
694
New contributor
Shubham Gothwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Shubham Gothwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Shubham Gothwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
In the 4-dimensional case, would the fourth dimension be pretty much the same thing as the first three spatial dimensions?
– Nat
52 mins ago
add a comment |Â
1
In the 4-dimensional case, would the fourth dimension be pretty much the same thing as the first three spatial dimensions?
– Nat
52 mins ago
1
1
In the 4-dimensional case, would the fourth dimension be pretty much the same thing as the first three spatial dimensions?
– Nat
52 mins ago
In the 4-dimensional case, would the fourth dimension be pretty much the same thing as the first three spatial dimensions?
– Nat
52 mins ago
add a comment |Â
1 Answer
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Suppose for a moment that we're specifically interested in the kinetic energy of a single, non-relativistic particle, so that $E=frac12mvecv^2$. I include the vector notation for the velocity here because it's useful to keep in mind that $vecv^2=vecvcdotvecv$.
If the particle is moving in two dimensions, then $vecv=v_xhatx+v_yhaty$ and $vecvcdotvecv=v_x^2+v_y^2$, which has units of velocity squared. In three dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz$ and $vecvcdotvecv = v_x^2+v_y^2+v_z^2$, which still has the units of velocity squared. In four dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz+v_whatw$, and $vecvcdotvecv=v_x^2+v_y^2+v_z^2+v_w^2$, which, again, still has the units of squared velocity. This is because adding two quantities of the same units together does not change their units. Indeed, this holds for any number of dimensions, and suggests that the units of energy should be unchanged.
We can also see this more generally by the formal definition of work, which is the definition of change in energy:
$$W=int_C vecFcdot dvecs$$
for an object being acted on by a force $vecF$ moving along a path $C$ with arclength parameter $dvecs$. This is the formal, most general definition, and holds no matter how many dimensions of space you have. Once again, you can see that this definition involves a dot product. The dot product takes two vectors in any number of dimensions and outputs a one-dimensional quantity (i.e. a number). The work only cares about one component of the force, namely, the one pointing along the one-dimensional path that the particle is taking. No matter how many spatial dimensions this one-dimensional path is embedded in, the work only cares about things happening along one of those dimensions. As such, the units of work should always be the same, and, since work has the same units as energy (otherwise we wouldn't be able to add them together), the units of energy should also always be the same.
answered 12 hours ago
probably_someone
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If you like the answer, mark it as accepted!
– bRost03
10 hours ago
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
25
down vote
Suppose for a moment that we're specifically interested in the kinetic energy of a single, non-relativistic particle, so that $E=frac12mvecv^2$. I include the vector notation for the velocity here because it's useful to keep in mind that $vecv^2=vecvcdotvecv$.
If the particle is moving in two dimensions, then $vecv=v_xhatx+v_yhaty$ and $vecvcdotvecv=v_x^2+v_y^2$, which has units of velocity squared. In three dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz$ and $vecvcdotvecv = v_x^2+v_y^2+v_z^2$, which still has the units of velocity squared. In four dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz+v_whatw$, and $vecvcdotvecv=v_x^2+v_y^2+v_z^2+v_w^2$, which, again, still has the units of squared velocity. This is because adding two quantities of the same units together does not change their units. Indeed, this holds for any number of dimensions, and suggests that the units of energy should be unchanged.
We can also see this more generally by the formal definition of work, which is the definition of change in energy:
$$W=int_C vecFcdot dvecs$$
for an object being acted on by a force $vecF$ moving along a path $C$ with arclength parameter $dvecs$. This is the formal, most general definition, and holds no matter how many dimensions of space you have. Once again, you can see that this definition involves a dot product. The dot product takes two vectors in any number of dimensions and outputs a one-dimensional quantity (i.e. a number). The work only cares about one component of the force, namely, the one pointing along the one-dimensional path that the particle is taking. No matter how many spatial dimensions this one-dimensional path is embedded in, the work only cares about things happening along one of those dimensions. As such, the units of work should always be the same, and, since work has the same units as energy (otherwise we wouldn't be able to add them together), the units of energy should also always be the same.
answered 12 hours ago
probably_someone
13.7k12350
1
If you like the answer, mark it as accepted!
– bRost03
10 hours ago
add a comment |Â
up vote
25
down vote
Suppose for a moment that we're specifically interested in the kinetic energy of a single, non-relativistic particle, so that $E=frac12mvecv^2$. I include the vector notation for the velocity here because it's useful to keep in mind that $vecv^2=vecvcdotvecv$.
If the particle is moving in two dimensions, then $vecv=v_xhatx+v_yhaty$ and $vecvcdotvecv=v_x^2+v_y^2$, which has units of velocity squared. In three dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz$ and $vecvcdotvecv = v_x^2+v_y^2+v_z^2$, which still has the units of velocity squared. In four dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz+v_whatw$, and $vecvcdotvecv=v_x^2+v_y^2+v_z^2+v_w^2$, which, again, still has the units of squared velocity. This is because adding two quantities of the same units together does not change their units. Indeed, this holds for any number of dimensions, and suggests that the units of energy should be unchanged.
We can also see this more generally by the formal definition of work, which is the definition of change in energy:
$$W=int_C vecFcdot dvecs$$
for an object being acted on by a force $vecF$ moving along a path $C$ with arclength parameter $dvecs$. This is the formal, most general definition, and holds no matter how many dimensions of space you have. Once again, you can see that this definition involves a dot product. The dot product takes two vectors in any number of dimensions and outputs a one-dimensional quantity (i.e. a number). The work only cares about one component of the force, namely, the one pointing along the one-dimensional path that the particle is taking. No matter how many spatial dimensions this one-dimensional path is embedded in, the work only cares about things happening along one of those dimensions. As such, the units of work should always be the same, and, since work has the same units as energy (otherwise we wouldn't be able to add them together), the units of energy should also always be the same.
answered 12 hours ago
probably_someone
13.7k12350
1
If you like the answer, mark it as accepted!
– bRost03
10 hours ago
add a comment |Â
up vote
25
down vote
up vote
25
down vote
Suppose for a moment that we're specifically interested in the kinetic energy of a single, non-relativistic particle, so that $E=frac12mvecv^2$. I include the vector notation for the velocity here because it's useful to keep in mind that $vecv^2=vecvcdotvecv$.
If the particle is moving in two dimensions, then $vecv=v_xhatx+v_yhaty$ and $vecvcdotvecv=v_x^2+v_y^2$, which has units of velocity squared. In three dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz$ and $vecvcdotvecv = v_x^2+v_y^2+v_z^2$, which still has the units of velocity squared. In four dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz+v_whatw$, and $vecvcdotvecv=v_x^2+v_y^2+v_z^2+v_w^2$, which, again, still has the units of squared velocity. This is because adding two quantities of the same units together does not change their units. Indeed, this holds for any number of dimensions, and suggests that the units of energy should be unchanged.
We can also see this more generally by the formal definition of work, which is the definition of change in energy:
$$W=int_C vecFcdot dvecs$$
for an object being acted on by a force $vecF$ moving along a path $C$ with arclength parameter $dvecs$. This is the formal, most general definition, and holds no matter how many dimensions of space you have. Once again, you can see that this definition involves a dot product. The dot product takes two vectors in any number of dimensions and outputs a one-dimensional quantity (i.e. a number). The work only cares about one component of the force, namely, the one pointing along the one-dimensional path that the particle is taking. No matter how many spatial dimensions this one-dimensional path is embedded in, the work only cares about things happening along one of those dimensions. As such, the units of work should always be the same, and, since work has the same units as energy (otherwise we wouldn't be able to add them together), the units of energy should also always be the same.
answered 12 hours ago
probably_someone
13.7k12350
Suppose for a moment that we're specifically interested in the kinetic energy of a single, non-relativistic particle, so that $E=frac12mvecv^2$. I include the vector notation for the velocity here because it's useful to keep in mind that $vecv^2=vecvcdotvecv$.
If the particle is moving in two dimensions, then $vecv=v_xhatx+v_yhaty$ and $vecvcdotvecv=v_x^2+v_y^2$, which has units of velocity squared. In three dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz$ and $vecvcdotvecv = v_x^2+v_y^2+v_z^2$, which still has the units of velocity squared. In four dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz+v_whatw$, and $vecvcdotvecv=v_x^2+v_y^2+v_z^2+v_w^2$, which, again, still has the units of squared velocity. This is because adding two quantities of the same units together does not change their units. Indeed, this holds for any number of dimensions, and suggests that the units of energy should be unchanged.
We can also see this more generally by the formal definition of work, which is the definition of change in energy:
$$W=int_C vecFcdot dvecs$$
for an object being acted on by a force $vecF$ moving along a path $C$ with arclength parameter $dvecs$. This is the formal, most general definition, and holds no matter how many dimensions of space you have. Once again, you can see that this definition involves a dot product. The dot product takes two vectors in any number of dimensions and outputs a one-dimensional quantity (i.e. a number). The work only cares about one component of the force, namely, the one pointing along the one-dimensional path that the particle is taking. No matter how many spatial dimensions this one-dimensional path is embedded in, the work only cares about things happening along one of those dimensions. As such, the units of work should always be the same, and, since work has the same units as energy (otherwise we wouldn't be able to add them together), the units of energy should also always be the same.
answered 12 hours ago
probably_someone
13.7k12350
answered 12 hours ago
probably_someone
13.7k12350
answered 12 hours ago
probably_someone
13.7k12350
answered 12 hours ago
probably_someone
13.7k12350
13.7k12350
1
If you like the answer, mark it as accepted!
– bRost03
10 hours ago
add a comment |Â
1
If you like the answer, mark it as accepted!
– bRost03
10 hours ago
1
1
If you like the answer, mark it as accepted!
– bRost03
10 hours ago
If you like the answer, mark it as accepted!
– bRost03
10 hours ago
add a comment |Â
Shubham Gothwal is a new contributor. Be nice, and check out our Code of Conduct.
Shubham Gothwal is a new contributor. Be nice, and check out our Code of Conduct.
Shubham Gothwal is a new contributor. Be nice, and check out our Code of Conduct.
Shubham Gothwal is a new contributor. Be nice, and check out our Code of Conduct.
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1
In the 4-dimensional case, would the fourth dimension be pretty much the same thing as the first three spatial dimensions?
– Nat
52 mins ago